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32 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 ...
Daniel R. Collins's user avatar
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 ...
Xander Henderson's user avatar
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21 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 $x$ ...
Steven Gubkin's user avatar
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 ...
paul garrett's user avatar
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18 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 ...
John Coleman's user avatar
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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 ...
paul garrett's user avatar
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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, ...
mweiss's user avatar
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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 ...
Steven Gubkin's user avatar
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 ...
Mark McClure's user avatar
14 votes

"Pure Imaginary" or "Purely Imaginary"?

checking the most popular textbooks: Papa Rudin has 3 occurrences of 'pure imaginary' vs. 0 of 'purely' Ahlfors has it 0 vs. 14 Stein-Shakarchi: 0 vs. 9 Conway: 0 vs. 4 Gamelin: 2 vs. 1 Needham:...
ac15's user avatar
  • 497
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 ...
Daniel R. Collins's user avatar
12 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 ...
PGnome's user avatar
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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 ...
James S. Cook's user avatar
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 ...
Steven Gubkin's user avatar
11 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 ...
Daniel R. Collins's user avatar
9 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 ...
guest's user avatar
  • 109
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 ...
Gerald Edgar's user avatar
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9 votes

"Pure Imaginary" or "Purely Imaginary"?

I guess you could get away with either, but strictly speaking, grammatically, the correct term is "purely imaginary." "pure imaginary number" means that the number is being ...
Justin Skycak's user avatar
9 votes

"Pure Imaginary" or "Purely Imaginary"?

I generally take a more descriptivist (rather than prescriptivist) view of language. If a particular phrase is commonly used by competent native speakers of a language, then the phrase is correct, ...
Xander Henderson's user avatar
  • 8,225
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 ...
Benjamin Dickman's user avatar
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'...
Justin Skycak's user avatar
7 votes

Quadratic equations using complex math but with no imaginary roots

Per request, I am expanding my comment into an answer. Given a quadratic equation, $ax^2+bx+c=0$, the quadratic formula $$ x = \frac{-b\pm \sqrt{b^2-4ac}}{2a}$$ will only involve complex numbers ...
Adam's user avatar
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7 votes

Teaching Clifford Algebra Instead of Imaginary/Complex Numbers

There is a reason why mathematics is taught from more concrete to more abstract. Students need to figure out what the role of proofs, axioms, etc. is before they can start abstraction first. Some very ...
Tommi's user avatar
  • 7,202
7 votes

Importance of complex numbers knowledge in real roots

we need knowledge of complex numbers to work with real numbers We need to be honest that at the level of your students, there are no examples of this: in all sufficiently elementary uses of complex ...
Kostya_I's user avatar
  • 1,411
7 votes

Importance of complex numbers knowledge in real roots

If series and radius of convergence are within the reach of your syllabus, then a standard example of "needing to go through the complex numbers" is the radius of convergence of the series ...
Pablo H's user avatar
  • 201

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