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+---
+id: a-note-on-the-final-digit-of-pi
+title: A Note on the Final Digit of $\pi$
+date: 2024-05-23
+author: k3jph
+layout: post
+permalink: /2024/05/23/a-note-on-the-final-digit-of-pi
+featured_image: /assets/img/2024/a-note-on-the-final-digit-of-pi.webp
+categories:
+- Blog 
+tags:
+- mathematics
+- number theory
+- proofs
+---
+
+I recently saw someone wearing a t-shirt that said, "My password is the
+last 8 digits of $\pi$." The joke, of course, lies in the fact that
+$\pi$ (pi) is an irrational number, meaning its decimal expansion goes
+on forever without repeating. Consequently, there are no "last" digits
+to know. However, a fascinating twist exists when considering $\pi$ in
+binary form.
+
+> **Theorem.** _The final digit of $\pi$ in binary is 1._
+>
+> _Proof._ To understand why this is true, let's consider the nature of
+> binary representations. In binary (base-2), every number is
+> represented as a sequence of 0s and 1s.
+>
+> Now, consider $\pi$ in its binary form. If we suppose the last digit is
+> 0, we encounter a logical inconsistency. If there were a final 0, it
+> could be removed, as trailing zeros in binary (or any number base) do
+> not affect the value of the number. This removal would imply that the
+> number is not infinite, contradicting the fact that $\pi$ is
+> irrational and has an endless, non-repeating sequence of digits.
+>
+> Therefore, the only consistent option is that the final digit in the
+> binary representation of $\pi$ must be 1. This ensures that $\pi$ retains
+> its infinite nature.
+
+This elegant and humorous observation underlines the intriguing
+properties of irrational numbers and their representations in different
+numeral systems.
+
+The joke about the t-shirt cleverly plays on our understanding of
+infinity and irrational numbers. While it's true that we cannot specify
+the "last" digits of $\pi$ in decimal form, the binary perspective offers a
+unique insight.
+
+For those who love diving deeper into mathematics, exploring the
+properties of $\pi$ across different bases can be a delightful exercise. It
+enriches our appreciation of mathematical beauty and provides a fun way
+to engage with infinity and number theory concepts.
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