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# Questions tagged [complex-numbers]

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### Should Euler's formula $e^{ix}=\cos x+i\sin x$ be seen as a definition rather than something to prove?

There are a lot of "proofs" of the identity $e^{ix}=\cos x+i\sin x$ in textbooks, using either differential equations or power series. However, I find those proofs often misleading, because it appears ...
196 views

### Examples of application problems of coordinate geometry in the complex plane?

I am currently writing some basic introductory texts to complex numbers for third-year high school students (Denmark). My main goal is to introduce complex numbers as a practical tool that both ...
99 views

### A compelling example of what complex numbers are for, before teaching them [duplicate]

When talking to kids before they are taught complex numbers, I would really like to give some examples of why it will be exciting to learn them. I am comfortable explaining the intellectual ...
225 views

### Complex numbers and encourage justification

In remedial algebra, we learn that the graph of $y=(\sqrt x)^2$ is only in the first quadrant. We know this is the correct graph for the equation. This is because we know $y=x$ and $x \ge 0$. However,...
110 views

### Lower-division complex analysis textbook

I'm looking for recommendations for a good textbook to use for a hypothetical lower-division course in complex analysis, at a level of sophistication comparable to a second or third semester course in ...
107 views

### The origins of $\mathrm{cis}(\theta)$
There is a abbreviation used in high school mathematics that is almost never seen outside of it: $\mathrm{cis}(\theta) = \cos(\theta) + i \sin(\theta)$, where cis stands for cosine + i sine. As soon ...
Complex numbers are, or at least were to me, generally introduced like this: There's no number whose square is negative. That's a shame! Well, whatever - we'll make one up! Set $i^2=-1$ and declare ...