Questions tagged [complex-numbers]

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Single variable complex analysis textbook which uses differential forms

Is there any single variable complex analysis textbook which uses $\textrm{d}\bar{z}$? Every single variable text I have found defines what a complex line integral with respect to $\textrm{d}z$ means, ...
3k views

Are there disadvantages to teaching complex numbers as purely geometrical objects?

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 ...
652 views

Are the following topics usually in an introductory Complex Analysis class: Julia sets, Fatou sets, Mandelbrot set, etc?

I'm a Nero fan so I'm glad I learned about the Mandelbrot set, but I notice that said topics are not in Brown-Churchill or 'A First Course in Complex Analysis' while they are in Coursera's '...
715 views

When does thinking $(-8)^{1/3} = -2$ result in problems for an undergraduates?

In high school we learn that the cube root of $-8$ is $-2$. Much later some of us learn about the single valued natural logarithm of a complex number, and that $w^z = e^{z\cdot Lz(w)}$ when $w$ and $z$...
3k views

Are there any proofs of Euler's Formula that do not rely on calculus?

The most common way I have seen Euler's formula $$re^{i\theta} = r(\cos\theta+i\sin\theta)$$ introduced in a classroom environment is to substitute $i\theta$ into the series expansion of the ...
4k views

(How) Do American undergraduate math programs teach complex numbers?

What kind of exposure to complex numbers can you expect in mathematics majors at American colleges? I teach at a very large public university. It occurred to me that it is possible to graduate in ...
163 views

19k views

Why do we teach complex numbers?

In algebra II, USA, we teach our students complex numbers. However, after algebra II, they never use complex numbers until pretty much complex analysis. The whole point of teaching them complex ...
223 views

Polar form before Cartesian form when introducing complex numbers

When I teach complex numbers to undergraduate engineering students, I invariably start, as appears to be customary, with $a + bi$ (or $a + bj$ for electrical engineers) and then follow up with the ...
131 views

Complex numbers [closed]

I would like to learn the subject 'complex numbers'. My goal is to study this on my own. Are there any good tips, books, sites to study this?
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Complex numbers in high school

Are complex numbers taught in high school in other countries? I am from Germany and complex numbers are next to never touched in high school with the exception of extra-curricular activities, for ...
220 views

How do i deal with students who make these mistakes? [closed]

I came across some interesting mistakes in many area of mathematics with my students and do not let me also to tell you for university students level, I would like to know How do i deal with ...
223 views

Are there more modern or computation oriented applications of complex analysis in science and engineering?

No doubt that complex analysis is a tremendously useful with plenty of applications in engineering and physics. Common raw applications of complex analysis includes: evaluation of ordinary and ...
637 views

Convincing a high schooler that $i$ is a number

I would like to convince a high school student that $i$ is a number, broadly put. I'm not going to define what I mean by "number" unless he asks, but I just want to convince him that it's somehow ...
Pedagogical quandary with the definition of $i$
I'm not sure how the concept of $i$ is taught in other places, but in our district the curriculum defines $i = \sqrt{-1}$, which is how it has been traditionally taught (for a while now) and also how ...
In a lecture, the teacher told us that the notion of imaginary number can be introduced using the geometric mean. We have: $i=\sqrt{-1}=\sqrt{(-1)\cdot1}$ which is the geometric mean of $-1$ and \$...