My son is taking complex numbers in high school, and he is asking 'What are these good for?' I know they are used in electromagnetic calculations, but how?
That's one of the usual complaints of students taking any sort of mathematics beyond simple arithmetic. Obviously, complex numbers are not required for calculating change or for balancing the checkbook. The simple point is that mathematics is a toolbox, and the layman will normally use very few of the tools in there in his daily life. However, scientists and engineers (and other nerds!) need other tools to do their jobs, and mathematicians need those tools to make new and better tools (although some 'new' tools may *now* look like nobody will ever use them).
Complex numbers are an important tool in several areas of science and engineering. They are very useful whenever one studies any system which has any sinusoidal oscillations. Once you start using a complex exponential [exp(ix)] instead of a cosine [cos(x)] the equations become much simpler to handle. This is because derivatives of complex exponentials are just more complex exponentials, whereas in the case of trigonometric functions you get two different animals, sines and cosines.
For example, this technique is used in electromagnetic theory where calculations involve electromagnetic waves. Complex numbers are also used in AC electrical networks (circuits).
In addition, complex numbers are _mandatory_ in quantum mechanics. As Feynman states and demonstrates in the 'Feynman Lectures on Physics', there is no way one can formulate quantum mechanics without resorting to complex numbers.
There is also a whole field of mathematics called 'complex analysis' which studies functions and calculus on the complex plane rather than real numbers. It provides many mathematical methods where real analysis falls short.
Transformations such as the Fourier and Laplace transforms which are quite important in frequency-domain analysis of functions and many engineering applications also make heavy use of complex numbers (and complex analysis).
After a point, most people end up seeing real numbers as just a 'special case' of complex numbers where the imaginary part happens to be zero. So, I suppose you can tell your son that if he intends to study science or engineering, he should be happy to be introduced to complex numbers as soon as possible.
Yasar Safkan, Ph.D., Software Engineer, Noktalar A.S., Istanbul, Turkey