Questions tagged [complex-numbers]
For questions about teaching complex (numbers like $z = a + b \cdot i$ where $\left\{ a,\, b \right\} \in \mathbb{R} \wedge i^{2} = -1$) numbers.
28
questions
3
votes
11
answers
3k
views
Importance of complex numbers knowledge in real roots
Many students question the importance of complex numbers in real life. We can find many important applications of imaginary numbers in Engineering field and physics. This question is not related to ...
- 955
1
vote
0
answers
84
views
MathArt contest about Aesthetic Conformal Image Mapping [closed]
At the moment a MathArt contest is running about Aesthetic Conformal Image Mapping were individuals and classes can participate:
https://www.freelancer.com/contest/MathArt-Contest-Aesthetic-Conformal-...
1
vote
4
answers
401
views
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 ...
- 1,217
-4
votes
2
answers
327
views
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 ...
- 574
7
votes
2
answers
373
views
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, ...
- 24.4k
2
votes
4
answers
778
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 $z$...
- 241
17
votes
5
answers
5k
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 ...
- 443
5
votes
2
answers
190
views
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+...
- 151
9
votes
8
answers
4k
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 ...
- 921
1
vote
2
answers
356
views
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 ...
- 369
12
votes
6
answers
759
views
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 ...
- 1,663
8
votes
2
answers
461
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 ...
- 251
3
votes
0
answers
133
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 ...
- 311
7
votes
3
answers
335
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,...
- 79
2
votes
2
answers
205
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 ...
- 6,520
2
votes
0
answers
121
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)+...
- 129
6
votes
1
answer
258
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 ...
- 4,625
1
vote
2
answers
137
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?
- 19
9
votes
3
answers
747
views
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 '...
- 574
18
votes
10
answers
7k
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 ...
- 712
3
votes
0
answers
222
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 ...
- 287
40
votes
14
answers
20k
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 ...
- 1,572
8
votes
6
answers
750
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 ...
- 313
7
votes
2
answers
284
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 ...
- 737
11
votes
6
answers
593
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 ...
- 5,030
4
votes
1
answer
405
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 $...
- 355
12
votes
1
answer
479
views
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 ...
- 223
25
votes
8
answers
4k
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 ...
- 1,337