Explain complex numbers 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.

Prelude
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.

Short Answer
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.

Example
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.
