# Demystifying complex numbers

At the end of this month I start teaching complex analysis to 2nd year undergraduates, mostly from engineering but some from science and maths. The main applications for them in future studies are contour integrals and Laplace transform, but this should be a "real" complex analysis course which I could later refer to in honours courses. I am now confident (after this discussion, especially Gauss’s complaints given in Keith’s comment) that the name "complex" is discouraging to average students.

Why do we need to study numbers which do not belong to the real world?

We all know that the thesis is wrong and I have in mind some examples where the use of complex variable functions simplify solving considerably (I give two below). The drawback is that all them assume some knowledge from students already.

So I would be happy to learn elementary examples which may convince students that complex numbers and functions of a complex variable are useful. As this question runs in the community wiki mode, I would be glad to see one example per answer.

Here are the two promised examples. I was reminded of the second one by several answers and comments about trigonometric functions (and also by the notification that "the bounty on your question Trigonometry related to Rogers–Ramanujan identities expires within three days"; it seems to be harder than I expected).

Example 1. What is the Fourier expansion of the (unbounded) periodic function $$f(x)=\ln\Bigl\lvert\sin\frac x2\Bigr\rvert\ ?$$

Solution. The function $$f(x)$$ is periodic with period $$2\pi$$ and has poles at the points $$2\pi k$$, $$k\in\mathbb Z$$.

Consider the function on the interval $$x\in[\varepsilon,2\pi-\varepsilon]$$. The series $$\sum_{n=1}^\infty\frac{z^n}n, \qquad z=e^{ix},$$ converges for all values $$x$$ from the interval. Since $$\Bigl\lvert\sin\frac x2\Bigr\rvert=\sqrt{\frac{1-\cos x}2}$$ and $$\operatorname{Re}\ln w=\ln\lvert w\rvert$$, where we choose $$w=\frac12(1-z)$$, we deduce that $$\operatorname{Re}\Bigl(\ln\frac{1-z}2\Bigr)=\ln\sqrt{\frac{1-\cos x}2} =\ln\Bigl\lvert\sin\frac x2\Bigr\rvert.$$ Thus, $$\ln\Bigl\lvert\sin\frac x2\Bigr\rvert =-\ln2-\operatorname{Re}\sum_{n=1}^\infty\frac{z^n}n =-\ln2-\sum_{n=1}^\infty\frac{\cos nx}n.$$ As $$\varepsilon>0$$ can be taken arbitrarily small, the result remains valid for all $$x\ne2\pi k$$.

Example 2. Let $$p$$ be an odd prime number. $$\newcommand\Legendre{\genfrac(){}{}}$$For an integer $$a$$ relatively prime to $$p$$, the Legendre symbol $$\Legendre ap$$ is $$+1$$ or $$-1$$ depending on whether the congruence $$x^2\equiv a\pmod{p}$$ is solvable or not. Using the elementary result (a consequence of Fermat's little theorem) that $$\Legendre ap \equiv a^{(p-1)/2}\pmod p, \tag{*}\label{star}$$ show that $$\Legendre 2p=(-1)^{(p^2-1)/8}.$$

Solution. In the ring $$\mathbb Z+\mathbb Zi=\Bbb Z[i]$$, the binomial formula implies $$(1+i)^p\equiv1+i^p\pmod p.$$ On the other hand, $$(1+i)^p =\bigl(\sqrt2e^{\pi i/4}\bigr)^p =2^{p/2}\biggl(\cos\frac{\pi p}4+i\sin\frac{\pi p}4\biggr)$$ and $$1+i^p =1+(e^{\pi i/2})^p =1+\cos\frac{\pi p}2+i\sin\frac{\pi p}2 =1+i\sin\frac{\pi p}2.$$ Comparing the real parts implies that $$2^{p/2}\cos\frac{\pi p}4\equiv1\pmod p,$$ hence from $$\sqrt2\cos(\pi p/4)\in\{\pm1\}$$ we conclude that $$2^{(p-1)/2}\equiv\sqrt2\cos\frac{\pi p}4\pmod p.$$ Then using the elementary result \eqref{star}: $$\Legendre2p \equiv2^{(p-1)/2} \equiv\sqrt2\cos\frac{\pi p}4 =\begin{cases} 1 & \text{if } p\equiv\pm1\pmod8, \cr -1 & \text{if } p\equiv\pm3\pmod8, \end{cases}$$ which is exactly the required formula.

• Maybe an option is to have them understand that real numbers also do not belong to the real world, that all sort of numbers are simply abstractions. Commented Jul 1, 2010 at 14:50
• Probably your electrical engineering students understand better than you do that complex numbers (in polar form) are used to represent amplitude and frequency in their area of study. Commented Jul 1, 2010 at 15:36
• Not an answer, but some suggestions: try reading the beginning of Needham's Visual Complex Analysis (usf.usfca.edu/vca/) and the end of Levi's The Mathematical Mechanic (amazon.com/Mathematical-Mechanic-Physical-Reasoning-Problems/dp/…). Commented Jul 1, 2010 at 17:05
• Your example has a hidden assumption that a student actually admits the importance of calculating F.S. of $\ln\left|\sin{x\over 2}\right|$, which I find dubious. The examples with an oscillator's ODE is more convincing, IMO. Commented Jul 2, 2010 at 3:02
• @Mariano, Gerald and Qiaochu: Thanks for the ideas! Visual Complex Analysis sounds indeed great, and I'll follow Levi's book as soon as I reach the uni library. @Paul: I give the example (which I personally like) and explain that I do not consider it elementary enough for the students. It's a matter of taste! I've never used Fourier series in my own research but it doesn't imply that I doubt of their importance. We all (including students) have different criteria for measuring such things. Commented Jul 2, 2010 at 5:06

The nicest elementary illustration I know of the relevance of complex numbers to calculus is its link to radius of convergence, which student learn how to compute by various tests, but more mechanically than conceptually. The series for $$1/(1-x)$$, $$\log(1+x)$$, and $$\sqrt{1+x}$$ have radius of convergence 1 and we can see why: there's a problem at one of the endpoints of the interval of convergence (the function blows up or it's not differentiable). However, the function $$1/(1+x^2)$$ is nice and smooth on the whole real line with no apparent problems, but its radius of convergence at the origin is 1. From the viewpoint of real analysis this is strange: why does the series stop converging? Well, if you look at distance 1 in the complex plane...

More generally, you can tell them that for any rational function $$p(x)/q(x)$$, in reduced form, the radius of convergence of this function at a number $$a$$ (on the real line) is precisely the distance from $$a$$ to the nearest zero of the denominator, even if that nearest zero is not real. In other words, to really understand the radius of convergence in a general sense you have to work over the complex numbers. (Yes, there are subtle distinctions between smoothness and analyticity which are relevant here, but you don't have to discuss that to get across the idea.)

Similarly, the function $$x/(e^x-1)$$ on the real line is smooth, but its power series at $$x = 0$$ has finite radius of convergence $$2\pi$$ (not sure if you can make this numerically apparent). Again, on the real line the reason for this is not visible, but in the complex plane there is a good explanation. If someone is not happy about that function looking initially problematic at $$x = 0$$, where its value is $$1$$, use $$x/(e^x+1)$$ instead and the radius of convergence of its power series at $$x = 0$$ has radius of convergence $$\pi$$ rather than $$2\pi$$.

• Thanks, Keith! That's a nice point which I always mention for real analysis students as well. The structure of singularities of a linear differential equation (under some mild conditions) fully determines the convergence of the series solving the DE. The generating series for Bernoulli numbers does not produce sufficiently good approximations to $2\pi$, but it's just beautiful by itself. Commented Jul 2, 2010 at 5:14
• Ī̲ wouldn’t say it demystifies anything. Commented Apr 14, 2021 at 4:24
• @IncnisMrsi my answer was posted almost 11 years ago, so I had to remind myself what the OP's original request was. It was not about the title, but about the following, taken from the OP's post: "to learn elementary examples which may convince students in usefulness of complex numbers and functions in complex variable." Судя по этой просьбе, я думаю, что все в порядке Commented Apr 14, 2021 at 5:18
• My guess is that the main example you cite is the sort of thing that is actually appealing to the already-convinced, i.e. mathematicians who are reading this question, and yet doesn't necessarily appeal to someone who feels strong and comfortable with some real analysis but doesn't want to do complex analysis. i.e. it is not at all "strange" from the point of view of real analysis as to why the series doesn't converge: The tests that a student has learned for convergence of basic series will show it easily.
– SBK
Commented Jun 11 at 14:41
• @SBK I of course am already convinced, but I still think an appropriately thoughtful student might wonder why the real analysis convergence tests are saying the radius of convergence is a value $R$ when nothing appears to be going wrong at $x = \pm R$ (a contrast to the series for $1/(1-x)$ and $\log(1+x)$ at $x = 0$, where $R = 1$). Commented Jun 11 at 23:57

You can solve the differential equation $$y''+y=0$$ using complex numbers. Just write $$(\partial^2 + 1) y = (\partial +i)(\partial -i) y$$ and you are now dealing with two order one differential equations that are easily solved $$(\partial +i) z =0,\qquad (\partial -i)y =z.$$ The multivariate case is a bit harder and uses quaternions or Clifford algebras. This was done by Dirac for the Schrodinger equation ($$-\Delta \psi = i\partial_t \psi$$), and that led him to the prediction of the existence of antiparticles (and to the Nobel prize).

• Thanks, coudy! This is really nice (Pietro's answer only hints a possible use of 2nd order DEs). Commented Jul 1, 2010 at 12:24
• I was taught a similar trick as a second-year honours undergraduate (same target that the OP suggests), and I don't find it helpful at all. The problem is that you'll probably need to spend a full lecture trying to formalize it and explain why it works, otherwise it just looks like a magical unmotivated trick that abuses notation. Commented Jan 27, 2014 at 11:30
• @Frederico: why do you say it abuses notation? That would be true in higher dimensional case without introducing Clifford algebras. But here it's perfectly valid. You just need to know what complex numbers are. Commented Jan 27, 2014 at 15:09
• How is Dirac theory based on (parabolic) Schrödinger equation? We know that a solution to Dirac equation is a solution to Klein–Gordon equation which is hyperbolic. Commented Apr 14, 2021 at 4:29
• @Marek I see only now this old comment but I'll answer anyway just in case. The 'abuse of notation' is that it at that point students do not know that $\partial$ belongs to an operator algebra and that it makes sense to factor out $\partial^2 y +y = (\partial^2 +1)y$, let alone decompose it as $(\partial + i)(\partial -i)$. The fact that the derivative notation $\partial^2 y$ reads like a square followed by a product by $y$ at that point is just notation, so this seems to belong to the same category as simplifying out $dx$ to change variables in integrals. Commented Apr 14, 2021 at 6:43

Students usually find the connection of trigonometric identities like $\sin(a+b)=\sin a\cos b+\cos a\sin b$ to multiplication of complex numbers striking.

• Not sure about the students, but I do. :-) Commented Jul 1, 2010 at 12:21
• Well, I am not claiming this about all students, but whenever I mentioned this (or same things in terms of rotation matrices) in class, I always get excited feedback at least from some of them. Commented Jul 1, 2010 at 12:24
• This is an excellent suggestion. I can never remember these identities off the top of my head. Whenever I need one of them, the simplest way (faster than googling) is to read them off from $(a+ib)(c+id)=(ac-bd) + i(ad+bc)$.
– alex
Commented Jul 1, 2010 at 20:35
• When I first started teaching calculus in the US, I was surprised that many students didn't remember addition formulas for trig functions. As the years went by, it's gotten worse: now the whole idea of using an identity like that to solve a problem is alien to them, e.g. even if they may look it up doing the homework, they "get stuck" on the problem and "don't get it". What is there to blame: calculators? standard tests that neglect it? teachers who never understood it themselves? Anyway, it's a very bad omen. Commented Jul 2, 2010 at 1:43
• @Victor: It can be worse... When I taught Calc I at U of Toronto to engineering students, I was approached by some students who claimed they had heard words "sine" and "cosine" but were not quite sure what they meant. Commented Jul 2, 2010 at 8:51

From "Birds and Frogs" by Freeman Dyson [Notices of Amer. Math. Soc. 56 (2009) 212--223]:

• As discussed at physics.stackexchange.com/questions/428033/… putting in an imaginary diffusion coefficient into the heat equation changes the equation to become invariant against time reversal $t \rightarrow -t$ like a wave equation. This is what Dyson means with "it suddenly made sense". Commented Oct 7, 2018 at 19:57

If the students have had a first course in differential equations, tell them to solve the system

$$x'(t) = -y(t)$$ $$y'(t) = x(t).$$

This is the equation of motion for a particle whose velocity vector is always perpendicular to its displacement. Explain why this is the same thing as

$$(x(t) + iy(t))' = i(x(t) + iy(t))$$

hence that, with the right initial conditions, the solution is

$$x(t) + iy(t) = e^{it}.$$

On the other hand, a particle whose velocity vector is always perpendicular to its displacement travels in a circle. Hence, again with the right initial conditions, $x(t) = \cos t, y(t) = \sin t$. (At this point you might reiterate that complex numbers are real $2 \times 2$ matrices, assuming they have seen this method for solving systems of differential equations.)

• Thanks, Qiaochu! They have some background in DEs, and this is a very good way to get DEs, trig identities and matrices at the same time. Commented Jul 2, 2010 at 5:43
• I just used basically this in a first course in differential equations to prove Euler's formula for them. One of us thought it was really cool, anyway... Commented Mar 22, 2013 at 14:10

Here are two simple uses of complex numbers that I use to try to convince students that complex numbers are "cool" and worth learning.

1. (Number Theory) Use complex numbers to derive Brahmagupta's identity expressing $(a^2+b^2)(c^2+d^2)$ as the sum of two squares, for integers $a,b,c,d$.

2. (Euclidean geometry) Use complex numbers to explain Ptolemy's Theorem. For a cyclic quadrilateral with vertices $A,B,C,D$ we have $$\overline{AC}\cdot \overline{BD}=\overline{AB}\cdot \overline{CD} +\overline{BC}\cdot \overline{AD}$$

• And even more amazingly, one can completely solve the diophantine equation $x^2+y^2=z^n$ for any $n$ as follows: $$x+yi=(a+bi)^n, \ z=a^2+b^2.$$ I learned this from a popular math book while in elementary school, many years before studying calculus. Commented Jul 2, 2010 at 1:21
• @Byron: Thanks! 2 examples in one answer: I can vote only once. :-( @Victor: I am indeed amazed! This elementary knowledge is new to me. I probably spent too much on complex DEs... Commented Jul 2, 2010 at 5:21
• How does the vector calculus identity explain complex numbers? It is about an inner product space, true in every dimension (and not necessary for a positive-definite scalar product). There is dot product in every $\mathbb{R}^n, n\ge 1$, but only for $n = 1, 2, 4$ multiplication exists (and only for $n = 1, 2$ is it commutative). Downvote. Commented Apr 14, 2021 at 4:49

One cannot over-emphasize that passing to complex numbers often permits a great simplification by linearizing what would otherwise be more complex nonlinear phenomena. One example familiar to any calculus student is the fact that integration of rational functions is much simpler over $$\mathbb C$$ (vs. $$\mathbb R$$) since partial fraction decompositions involve at most linear (vs quadratic) polynomials in the denominator. Similarly one reduces higher-order constant coefficient differential and difference equations to linear (first-order) equations by factoring the linear operators over $$\mathbb C$$. More generally one might argue that such simplification by linearization was at the heart of the development of abstract algebra. Namely, Dedekind, by abstracting out the essential linear structures (ideals and modules) in number theory, greatly simplified the prior nonlinear theory based on quadratic forms. This enabled him to exploit to the hilt the power of linear algebra. Examples abound of the revolutionary power that this brought to number theory and algebra - e.g. for one little-known gem see my recent post explaining how Dedekind's notion of conductor ideal beautifully encapsulates the essence of elementary irrationality proofs of n'th roots.

One of my favourite elementary applications of complex analysis is the evaluation of infinite sums of the form $$\sum_{n\geq 0} \frac{p(n)}{q(n)}$$ where $p,q$ are polynomials and $\deg q > 1 + \deg p$, by using residues.

• Thanks, José! This was not my list (even I use this very often in serious problems). I only wonder whether it is possible to start a course with such example. Commented Jul 1, 2010 at 10:01

If you really want to "demystify" complex numbers, I'd suggest teaching what complex multiplication looks like with the following picture, as opposed to a matrix representation:

If you want to visualize the product "z w", start with '0' and 'w' in the complex plane, then make a new complex plane where '0' sits above '0' and '1' sits above 'w'. If you look for 'z' up above, you see that 'z' sits above something you name 'z w'. You could teach this picture for just the real numbers or integers first -- the idea of using the rest of the points of the plane to do the same thing is a natural extension.

You can use this picture to visually "demystify" a lot of things:

• Why is a negative times a negative a positive? --- I know some people who lost hope in understanding math as soon as they were told this fact
• i^2 = -1
• (zw)t = z(wt) --- I think this is a better explanation than a matrix representation as to why the product is associative
• |zw| = |z| |w|
• (z + w)v = zv + wv
• The Pythagorean Theorem: draw (1-it)(1+it) = 1 + t^2 etc.

One thing that's not so easy to see this way is the commutativity (for good reasons).

After everyone has a grasp on how complex multiplication looks, you can get into the differential equation: $\frac{dz}{dt} = i z , z(0) = 1$ which Qiaochu noted travels counterclockwise in a unit circle at unit speed. You can use it to give a good definition for sine and cosine -- in particular, you get to define $\pi$ as the smallest positive solution to $e^{i \pi} = -1$. It's then physically obvious (as long as you understand the multiplication) that $e^{i(x+y)} = e^{ix} e^{iy}$, and your students get to actually understand all those hard/impossible to remember facts about trig functions (like angle addition and derivatives) that they were forced to memorize earlier in their lives. It may also be fun to discuss how the picture for $(1 + \frac{z}{n})^n$ turns into a picture of that differential equation in the "compound interest" limit as $n \to \infty$; doing so provides a bridge to power series, and gives an opportunity to understand the basic properties of the real exponential function more intuitively as well.

But this stuff is less demystifying complex numbers and more... demystifying other stuff using complex numbers.

Here's a link to some Feynman lectures on Quantum Electrodynamics (somehow prepared for a general audience) if you really need some flat out real-world complex numbers

They're useful just for doing ordinary geometry when programming.

A common pattern I have seen in a great many computer programs is to start with a bunch of numbers that are really ratios of distances. Theses numbers get converted to angles with inverse trig functions. Then some simple functions are applied to the angles and the trig functions are used on the results.

Trig and inverse trig functions are expensive to compute on a computer. In high performance code you want to eliminate them if possible. Quite often, for the above case, you can eliminate the trig functions. For example $\cos(2\cos^{-1} x) = 2x^2-1$ (for $x$ in a suitable range) but the version on the right runs much faster.

The catch is remembering all those trig formulae. It'd be nice to make the compiler do all the work. A solution is to use complex numbers. Instead of storing $\theta$ we store $(\cos\theta,\sin\theta)$. We can add angles by using complex multiplication, multiply angles by integers and rational numbers using powers and roots and so on. As long as you don't actually need the numerical value of the angle in radians you need never use trig functions. Obviously there comes a point where the work of doing operations on complex numbers may outweigh the saving of avoiding trig. But often in real code the complex number route is faster.

(Of course it's analogous to using quaternions for rotations in 3D. I guess it's somewhat in the spirit of rational trigonometry except I think it's easier to work with complex numbers.)

• Thanks! Your answer and Victor's comments reminded me about a related deduction in number theory (I'll try to add one more example to the original question). Commented Jul 2, 2010 at 8:46

Here is a kind of mime show.

Silently write the formulas for $\cos(2x)$ and $\sin(2x)$ lined up on the board, something like this: $$\cos(2x) = \cos^2(x) \hphantom{+ 2 \cos(x) \sin(x)} - \sin^2(x)$$ $$\sin(2x) = \hphantom{\cos^2(x)} + 2 \cos(x) \sin(x) \hphantom{- \sin^2(x)}$$

Do the same for the formulas for $\cos(3x)$ and $\sin(3x)$, and however far you want to go: $$\cos(3x) = \cos^3(x) \hphantom{+ 3 \cos^2(x) \sin(x)} - 3 \cos(x) \sin^2(x) \hphantom{- \sin^3(x)}$$ $$\sin(3x) = \hphantom{\cos^3(x)} + 3 \cos^2(x) \sin(x) \hphantom{- 3 \cos(x) \sin^2(x)} - \sin^3(x)$$

Maybe then let out a loud noise like "hmmmmmmmmm... I recognize those numbers..."

Then, on a parallel board, write out Pascal's triangle, and parallel to that write the application of Pascal's triangle to the binomial expansions $(x+y)^n$. Make some more puzzling sounds regarding those pesky plus and minus signs.

Then maybe it's time to actually say something: "Eureka! We can tie this all together by use of an imaginary number $i = \sqrt{-1}$". Then write out the binomial expansion of $$(\cos(x) + i\,\sin(x))^n$$ break it into its real and imaginary parts, and demonstrate equality with $$\cos(nx) + i\, \sin(nx).$$

Several motivating physical applications are listed on wikipedia

Why do we need to study numbers which do not belong to the real world?

You may want to stoke the students' imagination by disseminating the deeper truth - that the world is neither real, complex nor p-adic (these are just completions of Q). Here is a nice quote by Yuri Manin picked from here

On the fundamental level our world is neither real nor p-adic; it is adelic. For some reasons, reflecting the physical nature of our kind of living matter (e.g. the fact that we are built of massive particles), we tend to project the adelic picture onto its real side. We can equally well spiritually project it upon its non-Archimediean side and calculate most important things arithmetically. The relations between "real" and "arithmetical" pictures of the world is that of complementarity, like the relation between conjugate observables in quantum mechanics. (Y. Manin, in Conformal Invariance and String Theory, (Academic Press, 1989) 293-303 )

• Thanks for the tip! I'll better not cite Yuri Ivanovich to my electrical engineers; this will hardly encourage them to do complex analysis. :-) Commented Jul 1, 2010 at 13:24
• There is some hype about alleged physical relevance of p-adic numbers (in fact, Ī̲ know in person several researchers who built their career upon it), but—as for $\mathbb{Q}_p$—these are either overreaching generalizations or barren conjectures, whereas complex numbers are justified by quantum mechanics alone. Commented Apr 14, 2021 at 5:04
• If they have a suitable background in linear algebra, I would not omit the interpretation of complex numbers in terms of conformal matrices of order 2 (with nonnegative determinant), translating all operations on complex numbers (sum, product, conjugate, modulus, inverse) in the context of matrices: with special emphasis on their multiplicative action on the plane (in particular, "real" gives "homotety" and "modulus 1" gives "rotation").

• The complex exponential, defined initially as limit of $(1+z/n)^n$, should be a good application of the above geometrical ideas. In particular, for $z=it$, one can give a nice interpretation of the (too often covered with mystery) equation $e^{i\pi}=-1$ in terms of the length of the curve $e^{it}$ (defined as classical total variation).

• A brief discussion on (scalar) linear ordinary differential equations of order 2, with constant coefficients, also provides a good motivation (and with some historical truth).

• Related to the preceding point, and especially because they are from engineering, it should be worth recalling all the useful complex formalism used in Electricity.

• Not on the side of "real world" interpretation, but rather on the side of "useful abstraction" a brief account of the history of the third degree algebraic equation, with the embarrassing "casus impossibilis" (three real solutions, and the solution formula gives none, if taken in terms of "real" radicals!) should be very instructive. Here is also the source of such terms as "imaginary".

• Thanks a lot, Pietro, for so many fruitful suggestions! Except for the point on $(1+z/n)^n$ (unfortunately, this is not the usual way to define the exponential), I can use all them. Commented Jul 1, 2010 at 11:52
• In fact the equivalence with the definition of exp(z) by the exponential series may be a nice exercise about dominated convergence for series. BTW, I realized only now that you asked for one suggestion for answer...sorry :-) Commented Jul 1, 2010 at 12:49
• No worries, Pietro, this is a standard requirement for wiki community questions. Yes, it's a nice exercise, but most probably not for the level I get. (Last year I taught ODEs and, in particular, the linear systems where I needed to compute the exponential of matrices. The limit definition was mentioned as equivalent form of the series, and it was needed for proving $e^{\operatorname{tr}(A)}=\operatorname{det}(e^A)$. But that wasn't really accepted... :-( ) Commented Jul 1, 2010 at 13:00
• @Wadim: The $(1+z/n)^n$ definition of the exponential is exactly what you get by applying Euler's method to the defining diff Eq of the exponential function, if you travel along the straight line from 0 to z in the domain, and use n equal partitions. Commented Aug 27, 2012 at 13:24

This is a specific example where complex numbers aid a task in elementary real analysis; I haven't thought about the extent to which it generalizes.

In my first year, I was given the task of formally proving that the Taylor series for arctan is $$\arctan(x) = x - \frac{x^3}{3} + \frac{x^5}{5} - \ldots,$$ where "formally" meant not simply integrating the series for $1/(1+x^2)$ termwise, since we hadn't yet seen any theorems that said you could do that. We had, however, seen Taylor's theorem.

Hence the problem was to determine the values of all derivatives of $f(x)=\arctan(x)$, or of $f'(x)=1/(1+x^2)$, at $x=0$. However, it's not so easy to find a closed-form expression for the $n$-th derivative of $1/(1+x^2)$, unless you write it as $$f'(x) = \frac{1}{2i} \left( \frac{1}{x-i}-\frac{1}{x+i} \right),$$ which then immediately yields $$f^{(n)}(x) = \frac{(-1)^{n-1} (n-1)!}{2i} \cdot \left( \frac{1}{(x-i)^n}-\frac{1}{(x+i)^n} \right)$$ for $n>0$, which then gives the answers $f^{(2n)}(0)=0$ and $f^{(2n+1)}(0)=(-1)^n \cdot (2n)!$. Combining this with Taylor's theorem gives the desired series.

I still think this is pretty neat. There really isn't any obvious way to cut the complex numbers and still have as painless a calculation as the one above.

"Why do we need to study numbers which do not belong to the real world?"

you might simply state that quantum mechanics tells us that complex numbers arise naturally in the correct description of probability theory as it occurs in our (quantum) universe.

I think a good explanation of this is in Chapter 3 of the third volume of the Feynman lectures of physics, although I don't have a copy handy to check. (In particular, similar to probability theory with real numbers, the complex amplitude of one of two independent events A or B occuring is just the sum of the amplitude of A and the amplitude of B. Furthermore, the complex amplitude of A followed by B is just the product of the amplitudes. After all intermediate calculations one just takes the magnitude of the complex number squared to get the usual (real number) probability.)

• Perhaps you are referring to Feynman's book QED? Commented Jul 2, 2010 at 4:41
• Yes, I think it was there. (The "strange theory of light and matter" book.)
– Jon
Commented Jul 2, 2010 at 16:19
• This is a conceptual mess. Basic probability theory absolutely doesn’t need complex numbers but—applied to the real world—it is a simplification just like any other theory, QM included. Surely we can (and do) reduce complex amplitudes to dumb probability measures using |·|², but it has nothing to do with “correct description of probability theory”. It is indeed about real-world randomness which extends far beyond Kolmogorov-style probability. Can anybody rewrite, please? Commented Apr 14, 2021 at 5:22

I never took a precalculus class because every identity I've ever needed involving sines and cosines I could derive by evaluating a complex exponential in two different ways. Perhaps you could tell them that if they ever forget a trig identity, they can rederive it using this method?

• I especially like the complex derivation of $\cos^n x$ and $\sin^n x$ in terms of trig functions of multiple angle, which is very useful if you need to integrate them. Commented Jul 2, 2010 at 1:32
• @Dylan, thanks! You expand Yuri's answer. @Victor: aren't these standard problems for 1st year in algebra? ;-) Commented Jul 2, 2010 at 5:38

Tristan Needham's book Visual Complex Analysis is full of these sorts of gems. One of my favorites is the proof using complex numbers that if you put squares on the sides of a quadralateral, the lines connecting opposite centers will be perpendicular and of the same length. After proving this with complex numbers, he outlines a proof without them that is much longer.

• The same as for mathoverflow.net/a/30185/56921 — vector calculus is not complex calculus! Complex numbers certainly are a good example of an inner product space, and to teach vectors you can use the example of ℂ, but complex multiplication and (especially) division go beyond the vector-based intuition. Commented Apr 14, 2021 at 5:38

How about how the holomorphicity of a function $f(z)=x+yi$ relates to, e.g., the curl of the vector $(x,y)\in\mathbb{R}^2$? This relates nicely to why we can solve problems in two dimensional electromagnetism (or 3d with the right symmetries) very nicely using "conformal methods." It would be very easy to start a course with something like this to motivate complex analytic methods.

• Jeremy, can you expand your answer or provide a reference where your example is worked in details? Commented Jul 1, 2010 at 10:30
• This is in a number of advanced math methods for physics, and graduate electromagnetism texts, but one offhand I know it's in is "Electrodynamics of Continuous Media" by Landau and Lifshitz, chapter 1 section 3, on methods of solving electrostatic problems. But there are other books out there with more involved and more sophisticated treatments, but offhand I don't know any titles. If you search amazon's "search inside this book" for "the method of conformal mapping" you can find part of the discussion. But the basics of it are elementary enough that that book should be sufficient. Commented Jul 1, 2010 at 11:20
• Thanks, Jeremy! I'll definitely do the search, the magic word "the method of conformal mapping" is really important here. Commented Jul 1, 2010 at 11:45
• I think most older Russian textbooks on complex analysis (e.g. Lavrentiev and Shabat or Markushevich) had examples from 2D hydrodynamics (Euler-D'Alambert equations $\iff$ Cauchy-Riemann equations). Also, of course, the Zhukovsky function and airwing profile. They serve more as applications of theory than motivations, since nontrivial mathematical work is required to get there. Commented Jul 2, 2010 at 2:04
• Yes, those examples are great! I was thinking about those examples, too, when I said "conformal methods," but they are a little less basic than the E&M example. There are many more examples, too, though, such as classical gravity, or just about anything that can be described with a potential. They would make excellent topics to visit after developing some of the formalism more carefully, since they can lead to a lot of intuition about why things are constructed like they are, and why they're useful! Commented Jul 2, 2010 at 3:02

From the perspective of complex analysis, the theory of Fourier series has a very natural explanation. I take it that the students had seen Fourier series first, of course. I had mentioned this elsewhere too. I hope the students also know about Taylor theorem and Taylor series. Then one could talk also of the Laurent series in concrete terms, and argue that the Fourier series is studied most naturally in this setting.

First, instead of cos and sin, define the Fourier series using complex exponential. Then, let $f(z)$ be a complex analytic function in the complex plane, with period $1$.

Then write the substitution $q = e^{2\pi i z}$. This way the analytic function $f$ actually becomes a meromorphic function of $q$ around zero, and $z = i \infty$ corresponds to $q = 0$. The Fourier expansion of $f(z)$ is then nothing but the Laurent expansion of $f(q)$ at $q = 0$.

Thus we have made use of a very natural function in complex analysis, the exponential function, to see the periodic function in another domain. And in that domain, the Fourier expansion is nothing but the Laurent expansion, which is a most natural thing to consider in complex analysis.

I am am electrical engineer; I have an idea what they all study; so I can safely override any objections that this won't be accessible to electrical engineers. Moreover, the above will reduce their surprise later in their studies when they study signal processing and wavelet analysis.

This answer doesn't show how the complex numbers are useful, but I think it might demystify them for students. Most are probably already familiar with its content, but it might be useful to state it again. Since the question was asked two months ago and Professor Zudilin started teaching a month ago, it's likely this answer is also too late.

If they have already taken a class in abstract algebra, one can remind them of the basic theory of field extensions with emphasis on the example of $\mathbb C \cong \mathbb R[x]/(x^2+1).$

It seems that most introductions give complex numbers as a way of writing non-real roots of polynomials and go on to show that if multiplication and addition are defined a certain way, then we can work with them, that this is consistent with handling them like vectors in the plane, and that they are extremely useful in solving problems in various settings. This certainly clarifies how to use them and demonstrates how useful they are, but it still doesn't demystify them. A complex number still seems like a magical, ad hoc construction that we accept because it works. If I remember correctly, and has probably already been discussed, this is why they were called imaginary numbers.

If introduced after one has some experience with abstract algebra as a field extension, one can see clearly that the complex numbers are not a contrivance that might eventually lead to trouble. Beginning students might be thinking this and consequently, resist them, or require them to have faith in them or their teachers, which might already be the case. Rather, one can see that they are the result of a natural operation. That is, taking the quotient of a polynomial ring over a field and an ideal generated by an irreducible polynomial, whose roots we are searching for.

Multiplication, addition, and its 2-dimensional vector space structure over the reals are then consequences of the quotient construction $\mathbb R[x]/(x^2+1).$ The root $\theta,$ which we can then relabel to $i,$ is also automatically consistent with familiar operations with polynomials, which are not ad hoc or magical. The students should also be able to see that the field extension $\mathbb C = \mathbb R(i)$ is only one example, although a special and important one, of many possible quotients of polynomial rings and maximal ideals, which should dispel ideas of absolute uniqueness and put it in an accessible context. Finally, if they think that complex numbers are imaginary, that should be corrected when they understand that they are one example of things naturally constructed from other things they are already familiar with and accept.

Reference: Dummit & Foote: Abstract Algebra, 13.1

• Thanks for this point, Anthony! Although "my" students struggle abstract algebra as well, I would incorporate these ideas in my next year lectures (I should teach complex analysis again). Commented Oct 10, 2010 at 22:05

This is not exactly an answer to the question, but it is the simplest thing I know to help students appreciate complex numbers. (I got the idea somewhere else, but I forgot exactly where.)

It's something even much younger students can appreciate. Recall that on the real number line, multiplying a number by -1 "flips" it, that is, it rotates the point 180 degrees about the origin. Introduce the imaginary number line (perpendicular to the real number line) then introduce multiplication by i as a rotation by 90 degrees. I think most students would appreciate operations on complex numbers if they visualize them as movements of points on the complex plane.

Another classic I haven't seen mentioned yet is the proof of the Machin formula $$\frac{\pi}{4} = 4\arctan \frac{1}{5}-\arctan \frac{1}{239}.$$ I honestly don't know a proof of this that avoids complex numbers, but surely it can be nowhere near as simple as the elementary computation needed to prove the identity $$2+2i = \frac{(5+i)^4}{239+i}.$$ Taking $\arg$ on both sides yields Machin's formula.

• It follows from the arctan summation formula: $\operatorname{arctan}(x) + \operatorname{arctan}(y) = \operatorname{arctan}\frac{x+y}{1-xy}$, which in turn can be derived from the sine angle addition formula. Commented Aug 29, 2016 at 1:38
• But surely $e^{i(x+y)}=e^{ix}e^{iy}$ is easier to remember than the addition formulae for sine and cosine. Commented Sep 28, 2017 at 15:51
• A simpler formula like $\pi/4=2\arctan(1/3)+\arctan(1/7)$ can be easily shown without words: Draw the integer coordinate points $A:=(9,0)$, $B:=(9,3)$, $C:=(8,6)$, $D:=(7,7)$ and argue on the angles $\hat {AOB}$, $\hat{BOC}$, $\hat{COD}$, $\hat{AOD}$ by similarity of rectangle triangles. An analogous picture shows Machin formula (although one need to start from $(625,0)$ to get all required points with integer coordinates). Of course this is just the elementary geometry version of the complex number computation. Commented Apr 23, 2019 at 19:49

Motivating complex analysis

The physics aspect of motivation should be the strongest for engineering students. No complex numbers, no quantum mechanics, no solid state physics, no lasers, no electrical or electronic engineering (starting with impedance), no radio, TV, acoustics, no good simple way of understanding of the mechanical analogues of RLC circuits, resonance, etc., etc.

Then the "mystery" of it all. Complex numbers as the consequence of roots, square, cubic, etc., unfolding until one gets the complex plane, radii of convergence, poles of stability, all everyday engineering. Then the romance of it all, the "self secret knowledge", discovered over hundreds of years, a new language which even helps our thinking in general. Then the wider view of say Smale/Hirsch on higher dimensional differential equations, chaos etc. They should see the point pretty quickly. This is a narrow door, almost accidentally discovered, through which we see and understand entire new realms, which have become our best current, albeit imperfect,descriptions of how to understand and manipulate a kind of "inner essence of what is" for practical human ends, i.e. engineering. (True, a little over the top, but then pedagogical and motivational).

For them to say that they just want to learn a few computational tricks is a little like a student saying, "don't teach me about fire, just about lighting matches". It's up to them I suppose, but they will always be limited.

There might be some computer software engineer who needs a little more, but then I suppose there is also modern combinatorics. :-)

• Thanks! I'll borrow some of your wording for my lecture notes. (If you don't object.) Commented Jul 2, 2010 at 8:54
• Absolutely, Sure. Commented Jul 2, 2010 at 15:43

Maybe artificial, but a nice example (I think) demonstrating analytic continuation (NOT just the usual $\mathrm{Re}(e^{i \theta})$ method!) I don't know any reasonable way of doing this by real methods.

As a fun exercise, calculate $$I(\omega) = \int_0^\infty e^{-x} \cos (\omega x) \frac{dx}{\sqrt{x}}, \qquad \omega \in \mathbb{R}$$ from the real part of $F(1+i \omega)$, where $$F(k) = \int_0^\infty e^{-kx} \frac{dx}{\sqrt{x}}, \qquad \mathrm{Re}(k)>0$$ (which is easily obtained for $k>0$ by a real substitution) and using analytic continuation to justify the same formula with $k=1+i \omega$.

You need care with square roots, branch cuts, etc.; but this can be avoided by considering $F(k)^2$, $I(\omega)^2$.

Of course all the standard integrals provide endless fun examples! (But the books don't have many requiring genuine analytic continuation like this!)

• I rather suspect analytic continuation is a conceptual step above what the class in question coud cope with... Commented Jul 8, 2010 at 1:22
• Thanks, Zen! Although this seems to be hard for the class, I keep your complexified example for my personal collection. Commented Jul 8, 2010 at 3:47
• That is a very nice example. Commented Jul 8, 2010 at 3:59
• Thanks, and hi again to my friend Yemon! I agree that this is too hard for most classes, but I couldn't resist showing it! It needs a clear understanding of analytic continuation to do it, so it's only appropriate for courses which actually cover that rigorously (or at semi-rigorously). Commented Jul 10, 2010 at 12:33

Consider the function f(x)=1/(1+x^2) on the real line. Using the geometric progression formula, you can expand f(x)=1-x^2+... . This series converges for |x|<1 but diverges for all other x. Why this is so? The function looks nice and smooth everywhere on the real line.

This example is taken from the Introduction of the textbook by B. V. Shabat.

Try this: compare the problems of finding the points equidistant in the plane from (-1, 0) and (1, 0), which is easy, with finding the points at twice the distance from (-1, 0) that they are from (1, 0). The idea that "real" concepts are the only ones of use in the "real world" is of course a fallacy. I suppose it is more than a century since electrical engineers admitted that complex numbers are useful.

• Charles, you easily guessed that "my" engineers are electrical! The problem is (and I clearly see it after discussions with colleagues from EE) that they needed it one century ago but hardly need (in a reasonable generality) nowadays. Only concepts and rules of manipulation are required. :-( +1 As for your example, I don't see serious benefits for changing $\mathbb R^2$ by $\mathbb C$ (what happens if I am later asked about the same or similar problem in $\mathbb R^3$?). Commented Jul 1, 2010 at 10:28
• I do see an undeniable benefit. If you are later asked about it in $\mathbb{R}^3$ then you use vectors and dot product. The historical way would have been to use quaternions; indeed, this is how the notion of dot product crystallized in the work of Gibbs, and more relevantly for your EE students, Oliver Heaviside. Commented Jul 2, 2010 at 1:26

I always like to use complex dynamics to illustrate that complex numbers are "real" (i.e., they are not just a useful abstract concept, but in fact something that very much exist, and closing our eyes to them would leave us not only devoid of useful tools, but also of a deeper understanding of phenomena involving real numbers.) Of course I am a complex dynamicist so I am particularly partial to this approach!

Start with the study of the logistic map $x\mapsto \lambda x(1-x)$ as a dynamical system (easy to motivate e.g. as a simple model of population dynamics). Do some experiments that illustrate some of the behaviour in this family (using e.g. web diagrams and the Feigenbaum diagram), such as:

• The period-doubling bifurcation
• The appearance of periodic points of various periods
• The occurrence of "period windows" everywhere in the Feigenbaum diagram.

Then let x and lambda be complex, and investigate the structure both in the dynamical and parameter plane, observing

• The occurence of beautiful and very "natural"-looking objects in the form of Julia sets and the (double) Mandelbrot set;
• The explanation of period-doubling as the collision of a real fixed point with a complex point of period 2, and the transition points occuring as points of tangency between interior components of the Mandelbrot set;
• Period windows corresponding to little copies of the Mandelbrot set.

Finally, mention that density of period windows in the Feigenbaum diagram - a purely real result, established only in the mid-1990s - could never have been achieved without complex methods.

There are two downsides to this approach: * It requires a certain investment of time; even if done on a superficial level (as I sometimes do in popular maths lectures for an interested general audience) it requires the better part of a lecture * It is likely to appeal more to those that are mathematically minded than engineers who could be more impressed by useful tools for calculations such as those mentioned elsewhere on this thread.

However, I personally think there are few demonstrations of the "reality" of the complex numbers that are more striking. In fact, I have sometimes toyed with the idea of writing an introductory text on complex numbers which uses this as a primary motivation.

• Thanks a lot, Lasse, for the nice example. Can you give a link to a place where the dynamics is discussed in the complex case as well as the density of period windows in the Feigenbaum diagrams is discussed? Commented Jul 28, 2010 at 9:42
• Hm, I am not sure what the best place is - I guess you don't want me to reference a textbook on holomorphic dynamics! Devaney's "A first course in chaotic dynamical systems" has some things on Julia sets and the Mandelbrot set; I'll have a closer look if I get a chance. His "An introduction to chaotic dynamical systems", which is on a slightly more advanced level, also does, according to the table of contents - I don't have it handy. There are many other references, but I'll have to think a little bit whether I can come up with one that makes it easy for you to find what you need. Commented Jul 28, 2010 at 10:54
• Density of period windows in the Feigenbaum diagram is the celebrated result on "density of hyperbolicity" (or "density of Axiom A") in the quadratic family. It was established independently by Lyubich and by Graczyk and Swiatek in the 90s. Lyubich had an article on the quadratic family in the AMS Notices a while back, I think on his stronger "Regular or Stochastic" theorem. Commented Jul 28, 2010 at 10:56
• If you look for even more recent results in the same vein - all of which use complex methods - density of hyperbolicity was established for real polynomials by Kozlovski, Shen and van Strien a few years ago (appeared in the Annals of Mathematics). Even more recently, Sebastian van Strien and I extended this to some families of real transcendental entire functions such as $x\mapsto a\sin(x)+b\cos(x)$. In this proof, we, in particular, consider complex points that tend to $\infty$ under iteration by such a map - even though the map is bounded on the real line. (Available on the arXiv.) Commented Jul 28, 2010 at 11:00
• Thanks for the tips, Lasse! I promise to follow them, although slowly, without bothering myself about the students, more educating myself. Thank you very much. Commented Jul 28, 2010 at 22:54

From the point of view of enginieers, the most obvious application of complex numbers is computing alternating currents.

Consider first direct current. If you have a network of resistors, and want to compute the current in this network, or the potential of a node, then Kirchhoff's rules reduce this problem to a system of linear equations. Kirchhoff's rules are obvious, essentially saying that ellectric current cannot just disappear.

If you have alternating current, you have capacities and inductions in addition to the resistors, but if you consider them as imaginary resistance depending on the frequency, the computations are exactly the same as in the direct case, just over another field. The alternative would be computing the phase shift separately from the current, which is much more effort and only works for very simple networks, e.g. oscillators.

Once you learned Fourier analysis, this approach immediately tells you how a filter works, and whether a given network acts as a filter.

"Why do we need to study numbers which do not belong to the real world?"

I don't think you can answer this in a single class. The best answer I can come up with is to show how complicated calculus problems can be solved easily using complex analysis.

As an example, I bet most of your students hated solving the problem $$\int e^{-x}\cos(x) dx$$. Solve it for them the way they learned it in calculus, by repeated integration by parts and then by $$\int e^{-x}\cos(x) dx\ \ =\ \ \Re \int e^{-x(1-i)}dx$$. They should notice how much easier it was to use complex analysis. If you do this enough they might come to appreciate numbers that do not belong to the real world.

• That idea to compute the integral does not use complex analysis :) Commented Jan 20, 2011 at 2:07

As an example to demonstrate the usefulness of complex analysis in mechanics (which may seem counterintuitive to engineering students, since mechanics is introduced on the reals), one may consider the simple problem of the one dimensional harmonic oscillator, whose Hamiltonian equations of motion are diagonalized in the complex representation, equivalently one needs to integrate a single (holomorphic) first order ODE instead of a single second order or two first order ODEs.

• Thanks, David! It goes in line with some other answers but it's interesting to see a different point of view on the linearised DE for the harmonic oscillator. Commented Jul 2, 2010 at 8:49