54
Some of your students will become engineers, and engineers use complex numbers all the time, e.g., to represent impedance. This kind of thing is by far the most common application. Complex numbers are also used in quantum mechanics.
after algebra II, they never use complex numbers until pretty much complex analysis.
I assume you mean "they never use ...
27
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 precisely n roots (including multiplicities). When made, it serves as a capstone and culmination of all the work that the student has done in elementary algebra. Of ...
18
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 subsequently. In particular, there is no mention of Euler's identity expressing sine and cosine in terms of complex exponentials, and, therefore, no mention of the ...
17
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 course, or is it just interesting to me because I'm a mathematician?"
I personally would put the question about how to define $i$ in the "only important to me" ...
16
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 constant coefficients in terms of the characteristic equation. Without complex numbers the theory becomes somewhat ad-hoc, with different solutions depending on ...
16
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 understand.
As math educators, our instinct is always to simplify the math that we are presenting as much as possible. We always want students to understand the ...
16
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 $x$ is by the limit $(1+\frac{x}{n})^n$ as $n$ grows without bound. This is often motivated with a compound interest story.
Substituting $i\theta$ in for $x$ into ...
15
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 conventionally follow with respect to complex numbers -- first introducing them in a "just pretend" way, and only later (for some students) providing a geometric ...
15
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, multiply, and divide complex numbers; to solve quadratic equations with no real roots; and to find all $n$ roots of an $n$th degree polynomial (usually, ...
14
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 others is a choice of convenience, or taste, rather than being canonical. Showing that all of these definitions are equivalent is the real work. Here are some ...
13
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 series, we discuss Euler's identity:
$$e^{i\theta} = \cos(\theta)+i\sin(\theta).$$
Still in Calculus II as an application of the preceding example, we derive ...
13
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 function is defined (by mutual consent of the maths community) to only produce non-negative outputs, so $\sqrt 4 = 2$.
There are two reasons for this. Firstly, a ...
13
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 not include such topics: things about iteration of mappings in the complex domain is a much younger topic (as old as it is by this year) than most of the rest ...
11
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 giving Cajori vol. 2, page 133 as reference.
In this book, Uniplanar Algebra, the notation is used first, as far as I can see, in chapter III (The algebra of ...
11
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 textbooks that I'm familiar with).
We might take a guess and say that they're in the Coursera MOOC because it provides a fluffy, visual, easy-to-promote ...
10
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 cubic $x^3=cx+d$ has, in general, a solution
$$x=\sqrt[3]{d/2+\sqrt{(d/2)^2-(c/3)^3}}+\sqrt[3]{(d/2)-\sqrt{(d/2)^2-(c/3)^3}}.$$
(This formula can be verified by ...
9
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 took people so long to think about using it as a number. Think about it…0 behaves VERY DIFFERENTLY from any of the other familiar numbers. Admitting it "numberhood"...
9
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 definition for the exponential because it is special to the context of $\mathbb{C}$. In other algebras the interplay between the exponential function and ...
8
I just want to convince him that it's somehow meaningful and useful.
This is probably going to require coming up with an example that is applied to something in the real world, not just an example that is applied to some abstract topic in mathematics (roots of polynomials, or some of the more advanced examples referred to in Paul Garrett's answer). You only ...
8
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 characteristic equation. [IOW very standard part of standard ODE course; OFTEN part of even a second semester calculus course during the diffyQ survey...was ...
8
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 best we can do without calculus.
Some textbooks (not assuming calculus) use a notation $\mathrm{cis}\;\theta$ meaning $\cos\theta + i\sin\theta$ and do all ...
7
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 philosophical than mathematical. E.g., it is unclear whether or not we "have license" to declare that "the plane" is "numbers"... although we might have reasons to ...
7
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 schools have 3 years, whereas U.S. high schools have 4 years) and the end of the book discussed complex numbers. This was a standard public school in a second-tier ...
7
The title of the question asks for a proof, but the actual question seems to be more about providing some motivation. For students at this level, I would aim more for motivation that they will understand rather than a proof that they will just see as a mysterious magic trick.
Here's how I motivate it. I have the students work through $i^0$, $i^1$, $i^2$, ......
6
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 when they go to college.
Some examples of things I haven't had to do since high school:
Name the 5 (I think 5, maybe it changed?) biological kingdoms
Diagram a ...
6
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 point in his education, your student probably defines a "number" to be "a real number". (But he might define it as "a positive integer", "a non-negative ...
6
Here is a possibility, taken from
Tristan Needham,
Visual Complex Analysis (Oxford Univ. Press).
The advantages of this theorem are:
it is certainly not obvious,
"it would require a great deal
of ingenuity" to prove this without complex numbers,
it is
elementary planar geometry, and
it is ...
6
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 $\sin$ and $\cos$ in ways that might not quite pass muster today in terms of rigor, but were still pretty cool. According to the linked article, it used 3. in Steven's ...
6
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 exactly when the discriminant $b^2-4ac<0$. In this case the two roots come in a conjugate pair and there are no real solutions. However, when $b^2-4ac \geq 0$, ...
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