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Questions tagged [complex-numbers]

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Teaching Clifford Algebra Instead of Imaginary/Complex Numbers

For those unaware, Clifford Algebra (also known as Geometric Algebra) is able to generalize vectors and rotations in n-dimensional space, and simplifies a great many formulas. However, I was curious ...
johnnyb's user avatar
  • 1,159
-4 votes
2 answers

Is it nonsensical to try to 'prove' Euler's 'formula' in real numbers? What is Wikipedia/proofwiki even doing? [closed]

Edit re the close vote: I guess this 1 of those questions whose on-topic-ness depends on the answer. If the answer is no, then well maybe it's off-topic. But if the answer is yes, then I believe it's ...
BCLC's user avatar
  • 574
7 votes
2 answers

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, ...
Steven Gubkin's user avatar
2 votes
4 answers

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 $z$...
Ted Ersek's user avatar
  • 241
16 votes
5 answers

(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 ...
shuhalo's user avatar
  • 433
5 votes
2 answers

Is evaluating a Real Polynomial at a Complex Value a suitable task for Precalculus students?

In Korea, basically every teaching material for 10th grade math(about the level of precalculus) contains this kind of exercises in their first treatment of complex numbers: Evaluate $f(x)=4x^4-8x^3+...
Hyobin Lee's user avatar
9 votes
8 answers

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 ...
MadScientist's user avatar
1 vote
2 answers

Quadratic equations using complex math but with no imaginary roots

Many years ago when learning complex maths we used complex maths as an example in the quadratic equation to find real roots. My nephew is struggling to deal with complex maths as his teacher is ...
AndyW's user avatar
  • 369
12 votes
6 answers

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 ...
Ma Joad's user avatar
  • 1,635
8 votes
2 answers

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 ...
Buster Bie's user avatar
2 votes
0 answers

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 ...
Simd's user avatar
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7 votes
3 answers

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,...
user9054364's user avatar
2 votes
2 answers

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 ...
Mike Shulman's user avatar
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2 votes
0 answers

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)+...
reuns's user avatar
  • 129
5 votes
1 answer

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 ...
J W's user avatar
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1 vote
2 answers

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?
Quinten's user avatar
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9 votes
3 answers

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

I'm an nntaleb 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 '...
BCLC's user avatar
  • 574
17 votes
10 answers

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 ...
YukiJ's user avatar
  • 702
3 votes
0 answers

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 ...
zeraoulia rafik's user avatar
38 votes
14 answers

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 ...
Simply Beautiful Art's user avatar
8 votes
6 answers

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 ...
Addem's user avatar
  • 313
7 votes
2 answers

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 ...
Fraïssé's user avatar
  • 429
11 votes
6 answers

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 ...
celeriko's user avatar
  • 4,900
4 votes
1 answer

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 $...
Ortomala Lokni's user avatar
12 votes
1 answer

The origins of $\operatorname{cis}(\theta)$

There is a abbreviation used in high school mathematics that is almost never seen outside of it: $\operatorname{cis}(\theta) = \cos(\theta) + i \sin(\theta)$, where cis stands for cosine + i sine. As ...
Simon's user avatar
  • 223
22 votes
8 answers

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 ...
Jack M's user avatar
  • 1,297