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 ...
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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 ...
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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 ...
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  • 13.4k
19 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 $x$ ...
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18 votes

Pedagogical quandary with the definition of $i$

A bit of personal opinion: Any time I find myself debating over a bit of pedantry while preparing course material, I ask myself, "Is this something that is really important to a student taking this ...
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  • 181
18 votes

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

I think there are serious pedagogical problems with such an approach. Here is a good general rule for explaining any kind of math: Skipping over the motivation doesn't make something easier to ...
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  • 7,930
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 ...
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16 votes

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

It's funny you should ask this now, because I just taught a Math Ed graduate class on this topic the other day, so a lot of these thoughts are fresh in my mind. The pedagogical sequence we ...
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  • 16.1k
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 ...
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  • 1,426
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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15 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 ...
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13 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 ...
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13 votes

Pedagogical quandary with the definition of $i$

There are issues with how to define $i$ no matter which you choose, as the answers to the linked questions attest. To begin with, it is not actually true that $\sqrt 4 = \pm 2$. The square root ...
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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 ...
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11 votes
Accepted

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

The notation exists since a long time. It was used already by Irving Stringham in 'Uniplanar Algebra (1893).' This is claimed to be the earliest use on http://jeff560.tripod.com/trigonometry.html ...
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  • 7,572
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 ...
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10 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 ...
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10 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 ...
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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 ...
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  • 5,887
8 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 ...
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  • 256
8 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 ...
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  • 91
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 ...
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8 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 ...
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  • 6,171
7 votes

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

To answer the literal question of the title: yes, there are some disadvantages in portrayal of the complex numbers as "purely geometric" objects", although perhaps these disadvantages are more ...
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  • 13.4k

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