My cousin asked me if I could provide him with a practical example with complex numbers. I found it hard to do, so does anyone have a easy practical example with the use of complex numbers?
I tried to show him that complex numbers is needed to solve $x^2 = -1$, but he was not impressed.
Draw the complex plane. Draw the point at $(1,0)$. Start multiplying by $i$:
$$\{1,i,-1,-i,1,i,-1,-i,\ldots\}.$$
You will notice that this action will start rotating around the plane in a circular fashion.
Circles are fundamentally important. Normally, we might look at circles by setting up parametric equations of trigonometric terms, but what if we had a better way?
Being able to draw circles by simply multiplying terms ends up being super-important in mathematics and engineering. It allows us to look at rotations as just multiplications of two objects. Multiplication is very familiar!
There are other ways to do this, of course, but by looking at solutions in the complex plane we gain two key things: the ability to look at paths around the plane as rotations, scalings, and translations; and the ability to do so in a world where every polynomial has as many solutions as its degree.
This very powerful notion generalizes to more advanced mathematical concepts like contour integration, which is a tool that allows us to compute challenging real-valued integrals; conformal mapping, which allows us to define complicated geometric transformations as basic multiplications of numbers while preserving important properties; and transformations of functions into different domains, where their properties can be better understood.
Practicality is in the eye of the beholder. Further, like many others, I hesitate to dignify the short-sighted misapprehension that mathematics needs to justify itself through practical applications.
That said, since pictures can be appreciated without complete understanding of their subtleties, your cousin might enjoy pondering the question:
Find two families of curves that fill out the plane (possibly with isolated exceptional points) in such a way that a curve from one family meets curves from the other family at right angles.
Horizontal and vertical lines obviously work. Circles centered at the origin and lines through the origin work. Are there other examples...?
The secret agenda is that there are scads of examples, some exceedingly complex, and of great beauty: Every complex-differentiable (a.k.a. holomorphic) function $f:\mathbf{C} \to \mathbf{C}$ is angle-preserving (a.k.a., conformal), and so gives rise to such a pair of families of curves by taking images (or preimages) of horizontal and vertical lines. A web search for "conformal planar meshes" should turn up numerous visually-compelling examples.
One possible "punch line" to this idea is the beautiful analysis, by Hendrik Lenstra and Bart de Smit, of the Droste effect in Print Gallery by M. C. Escher, particularly the zooming animations.
(Students of complex analysis should note that Print Gallery is a visual model of the Riemann surface of $\log$, with the removed region near the center hiding the ramification point. Unfortunately, Escher made the print "backward", with a clockwise path leading "up" one sheet.)
Conformality of holomorphic mappings also underlies the self-similarity of the Mandelbrot set.
If this is not practical enough, you can point out that these conformal meshes model $2$-dimensional electrostatics (field lines and constant-potential curves) and incompressible fluid flow. :)
Here is how I'd explain complex numbers to a 13 year old.
The number $i$ is just a gadget that keeps track of how many counterclockwise quarter turns you make in the plane.
Stand in your living room (or whatever room you're in) facing east. Let's invent a funny language. Instead of "east," from now on I'm going to call that $1$. It's just a name. So if I'm facing $1$, I'm facing east. Just a language game.
Now I make a quarter turn to the left so that I'm facing north. Instead of calling it "north" my new name for it is $i$. Funny name, but as Shakespeare said, what's in a name. The underlying direction is the same.
Make another quarter turn to the left. Now you're facing west, but let's call that $-1$. I know, funny names.
Another quarter turn and we're facing south. I call that $-i$.
What happens if we make another quarter turn? We're facing $1$, which is where we started.
Now here's some more funny language. Instead of saying that we make a quarter turn to the left, let's say that we "multipy by $i$".
So if we're facing $1$ and we multiply by $i$, we're facing $i$. If we multiply by $i$ again we're facing $-1$. What's a good notation for multiplying by $i$ and then multiplying by $i$ again? We can use the usual notation of squaring. So we just proved that
$$i^2 = -1$$
How about that!!
Then we see that $i^3 = -i$ and $i^4 = 1$.
Quiz: What is $i^{17}$? Five minutes ago that would have been an incomprehensibly difficult problem. But now we see that just as $i^4 = 1$, it must also be the case that $i^{12} = i^{16} = 1$. And then $i^{17}$ must be $i$. This is simple! Multiplying by $i$ just keeps track of how many quarter turns you've made. And you can always work this out "mod $4$" if you know what that means. In fact the set $\{1, i,-1, -i\}$ is an instance of the group of integers mod $4$. So we can teach 13 year olds a little group theory while we're at it.
We could go on. Briefly, multiplying any complex number by a real number just scales it -- stretches or shrinks it. If $1$ means the point $1$ unit away in the eastern direction, then $5$ is the point $5$ units away in the eastern direction; and $5i$ is the point $5$ units away in the northern direction.
Finally, if we have a complex number like $3 + 5i$ that's just a point $3$ units east and $5$ units north. Or if you know coordinate geometry, the point $(3,5)$ in the Cartesian plane.
That's all there is to it. And with a little bit more work we can develop all of high school trigonometry along the same lines. I wish they'd teach this to 13 year olds, it's not very complicated. Math education is in dire need of exactly this type of reform. Complex numbers were very mysterious when we first discovered them, but today we understand their essentially geometric nature and we could teach them in a much simpler way if we wanted to.
Practical example:
Rotate the point $(x,x^2)$ $90$ degrees counterclockwise.
Multiplying a complex number $c$ by $1$ equates to a $360$ degree turn when graphing a complex number by its real part on the $x$ axis and its imaginary part on the $y$ axis because $1$ is the multiplicative identity and thus multiplying by it does not change $c$. Note:
$$i^4=1$$
So multiplying by $i$ four times equate $4$ turns who sum to $360$ degrees. Thus multiplying by $i$ gives a $90$ degree turn. Furthermore (from Euler's formula) as $i=e^{+\frac{\pi}{2}i}$ it will rotate counterclockwise.
Thus consider:
$$c=x+x^2i$$
$Re$ denotes the real part and $Im$ denotes the coefficient of the imaginary part.
$$(Re(c),Im(c))=(x,x^2)$$
Multiplying by $i$ we have:
$$i(x+ix^2)=xi+(-1)x^2=-x^2+xi$$
And thus the result of the rotation is the point $(-x^2,x)$
And we are done.
Note complex numbers can be used to rotate a point in the two dimensional plane $\mathbb{R^2}$ any degrees clockwise or counterclockwise.
Multiplication by:
$$\cos \theta+i \sin \theta$$
Corresponds to a rotation by $\theta$ degrees or radians clockwise depending on how you calculate sine and cosine. As from euler's formula we are just adding $\theta$ to our argument. For clockwise rotations plug in the negative of the angle you want to rotate by in the formula to get the complex number to multiply by.
You mentioned in your comments that he is 13 years old. I'm only a couple years older than that, and don't have any knowledge of practical uses.
However, I can tell you what imaginary numbers are used for (more generically): to describe numbers that aren't real.
I think it is best described with a quadratic equation with no solutions.
We have the equation $ y = x^2 + 1 $. We want to find the zeros (where such parabola intersects the x-axis).
We end up with the following equation: $ 0 = x^2 + 1 $ Solving for $x$, we end up with this: $ x = \pm\sqrt{-1} $. Using imaginary numbers, we can rewrite this as $ x = \pm i $.
If we were to graph the equation, we would see that the parabola never intersects the x-axis. Our solution, which is compromised of imaginary numbers, tells us that there is no real solution.
Complex analysis is one of the most beautiful creations of the human mind. It really does not need practical justification.
To provide a practical justification that the 13 year old mind might be receptive too, you could point out that the cubic formula uses complex numbers even when it is producing real solutions! So the complex numbers are essential to solving real equations with real solutions.
See http://en.wikipedia.org/wiki/Cubic_function#General_formula_for_roots
A*A=-Iis solved with complex numbers in matrix sense, and matrices could be objects' transformation and rotation. For example,which translation and rotation of object should be performed twice to be equivalent to mirroring?(not sure if this formulation is absolutely correct) – medvedNick Dec 10 '13 at 15:46