Questions tagged [complex-numbers]

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7
votes
2answers
157 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 ...
1
vote
0answers
95 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 ...
5
votes
3answers
221 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,...
1
vote
2answers
109 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 ...
2
votes
0answers
105 views

Complex logarithm and $\mathbb{C}/2i\pi \mathbb{Z}$

Is it possible and is it a good idea to introduce the additive group and metric space $\mathbb{C}/2i\pi \mathbb{Z}$ very soon, at the same time as the complex logarithm $\log(r e^{i \theta}) = \ln(r)+...
4
votes
1answer
125 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 ...
1
vote
2answers
124 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?
7
votes
3answers
560 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 '...
16
votes
10answers
4k views

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 ...
3
votes
0answers
206 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 ...
33
votes
13answers
14k 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 ...
8
votes
6answers
529 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 ...
7
votes
2answers
160 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 ...
11
votes
6answers
499 views

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 ...
3
votes
1answer
178 views

How to introduce the notion of imaginary number using the geometric mean?

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 $...
11
votes
1answer
193 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 ...
19
votes
8answers
2k 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 ...