[complex_and_trig] Editorial Comments
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Assessment
- Difficulty
- 3/5
- Estimated time
- 1-2 days
- Newbie friendliness
- 48/100
- Issue type
- Documentation
- Clarity
- Mostly clear
- Activity status
- Stale
- Tech stack
- jupyter-notebook
- Domain
- documentation
Research direction
Open the complex_and_trig lecture and review the introduction and section 6.3.5 first. Address the checklist by improving the complex-number definition, adding a visualization of x, y, r, and θ, briefly motivating Euler’s formula, and clarifying the treatment of i in calculus; done means each requested editorial point is covered.
Written by the indexing model from the issue text.
Description
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.
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