[complex_and_trig] Editorial Comments

[HumphreyYang]

Comments by @pgrosser1:

Content

• It would be useful at the very start to define what a complex number is, ie. $i = \sqrt{-1}$ or equivalently the solution to the quadratic equation $x^2 + 1 = 0$. Then we could say something along the lines of: Clearly, this is not part of our repertoire of real ordinary numbers, because we expect that squaring a number will always make it positive and therefore there is no $x$ that should satisfy $x^2 + 1 = 0$. Hence, we must introduce a new mathematical formulation called complex numbers…
• Give a graphical representation at the start when introducing the various forms of a complex number so that the values $x$, $y$, $r$, and $\theta$ can be visualized and related.
• It could be useful to show how trigonometric functions in terms of complex numbers come out of the definition $e^{i \theta} = cos \theta + i sin \theta$ (would take approximately 2 lines to show, so it is brief and allows for some familiarity with Euler’s formula before examples are introduced).
• A note in 6.3.5 that $i$ can be treated as a constant (which it is anyway, but just for clarity’s sake) when integrating and differentiating might assist. Again, I don’t think the code at the end of 6.3.5 presents a very intuitive verification, and a graphical representation (as outlined above) would be more helpful.