29
votes
Why do we teach complex numbers?
We owe students a presentation of the Fundamental Theorem of Algebra -- that every nonconstant polynomial has a root; or, equivalently, the marvelous fact that every polynomial of nth degree has ...
- 24.3k
28
votes
(How) Do American undergraduate math programs teach complex numbers?
The following is anecdotal, but based on experience as a student and instructor in American high schools in three states, as well as my undergraduate and graduate experiences.
High School: In the ...
- 7,485
20
votes
Accepted
Are there any proofs of Euler's Formula that do not rely on calculus?
Can you define $e^x$ for real $x$ without calculus? You need to take a limit, I think.
If you allow "informal limit reasoning", then there is such an argument.
One way to define $e^x$ for real $x$ ...
- 24.3k
19
votes
Complex numbers in high school
Although there is a tradition in the U.S. of nominal mention of complex numbers in the high school curriculum, in my observation it is invariably superficial, and complex numbers are not mentioned ...
- 14.2k
17
votes
Accepted
Are the following topics usually in an introductory Complex Analysis class: Julia sets, Fatou sets, Mandelbrot set, etc?
No, these topics are not usually included in courses on complex analysis, for several reasons, which I will explain below. At the same time, it is easier to understand why relatively old textbooks did ...
- 14.2k
16
votes
Complex numbers in high school
In the United States, complex numbers are a standard part of the typical high school Algebra 2 course. Students (normally in grades 10 or 11, corresponding approximately to ages 15-17) learn to add, ...
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16
votes
Why do we teach complex numbers?
I am surprised that nobody has mentioned differential equations. If you know about complex numbers and Euler's formula then there is a beautiful unified theory of linear differential equations with ...
- 1,506
16
votes
Should Euler's formula $e^{ix}=\cos x+i\sin x$ be seen as a definition rather than something to prove?
I think it is worth acknowledging that there is a huge list of equivalent definitions of the complex exponential. Any presentation should be explicit about the fact that choosing one definition over ...
- 24.3k
14
votes
Why do we teach complex numbers?
I think the fundamental tenet of this question is simply false. Here are some of the many encounters that undergraduate college students have in my classes:
In Calculus II, as an application of power ...
- 360
13
votes
Are the following topics usually in an introductory Complex Analysis class: Julia sets, Fatou sets, Mandelbrot set, etc?
A brief stab at an answer before someone more knowledgeable comes along: My brief experience says that: no, fractal sets are not usually a topic in complex analysis (they're not in any of the 3 ...
- 24.3k
11
votes
Convincing a high schooler that $i$ is a number
One of the historically compelling arguments for complex numbers was that they can be used to find real valued solutions to polynomials.
There's a nice discussion on this site. For example, the ...
- 11.6k
11
votes
Should Euler's formula $e^{ix}=\cos x+i\sin x$ be seen as a definition rather than something to prove?
In my view Euler's Formula $e^{ix} = \cos(x)+ i \sin(x)$ and the resulting extension ( this is (5.) in Steven Gubkin's excellent complex answer ) of $e^{x+iy} = e^x( \cos y + i \sin y)$ is not a good ...
- 10.7k
11
votes
(How) Do American undergraduate math programs teach complex numbers?
At my institution we do not discuss complex numbers at all in calculus, but we assume that students are somehow familiar with them in differential equations and linear algebra (to analyze real-linear ...
- 24.3k
10
votes
Why do we teach complex numbers?
Most people are not going to use most of what they use in high school. High school (in the US, at least) is about building a broad foundation for students to be able to jump into any specialization ...
- 276
10
votes
Teaching Clifford Algebra Instead of Imaginary/Complex Numbers
One of the pinnacle goals in high school and basic college algebra is to be able to express the fundamental theorem of algebra, which is everywhere expressed in terms of complex numbers.
Complex ...
- 24.3k
9
votes
Convincing a high schooler that $i$ is a number
If I were faced with this situation, I'd have a discussion with the student about what a number is. We believe that 1,2,3,4,… are numbers. It is a stretch to say that 0 is a number…which is why it ...
- 6,045
9
votes
Are there any proofs of Euler's Formula that do not rely on calculus?
There is a proof of de Moivre's formula
$$
(\cos\theta + i \sin\theta) ^n = \cos(n\theta) + i \sin(n\theta), \qquad n \in \mathbb Z
$$
by induction (for $n > 0$) and symmetry. Maybe that is the ...
- 7,369
8
votes
Complex numbers in high school
Complex numbers are also taught in China. I observed a mathematics classroom in Nanjing, Jiangsu Province, China in 2008-09 for students in their penultimate year of secondary school (their high ...
- 18.4k
8
votes
Importance of complex numbers knowledge in real roots
Another topic that uses complex numbers in service of real numbers, and is immediately accessible to a high schooler who has learned or is learning calculus, is integration using Euler's formula.
Here'...
- 6,111
7
votes
Why do we teach complex numbers?
Just another question from a math guy showing his ignorance of math as a service course.
2nd order diffyQ with constant coeffiecients (most important diffyQ for applications) has complex roots in the ...
- 89
6
votes
Convincing a high schooler that $i$ is a number
You probably cannot get away with having a different definition of "what a number is" than your student has. Your student almost certainly has a very clear understanding of what a number is. By this ...
- 3,148
6
votes
Are the following topics usually in an introductory Complex Analysis class: Julia sets, Fatou sets, Mandelbrot set, etc?
It seems to me there is a fundamental disconnect between what one might call "traditional complex analysis" and the study of fractals.
Traditional complex analysis, as used in physics and engineering,...
- 889
6
votes
Accepted
Examples of application problems of coordinate geometry in the complex plane?
Here is a possibility, taken from
Tristan Needham,
Visual Complex Analysis (Oxford Univ. Press).
The advantages of this theorem are:
it ...
- 29.5k
6
votes
Should Euler's formula $e^{ix}=\cos x+i\sin x$ be seen as a definition rather than something to prove?
I'll put out there that it might not be helpful to have this as a definition because it wasn't discovered in that way. That is to say, Euler discovered this formula by clever manipulation of $\sin$ ...
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