Utility in math is an interesting issue :). Ultimately, math is a bunch of things we *can* do, ideas we can think of. Do we need to rotate? Add angles? Move in circles? Not really.

But sometimes we can construct a scenario where that idea is convenient. Complex numbers, which make rotations, angles, and circles easy, come to the rescue. Certain problems that looked impossible (Can negatives have square roots? Can an exponent follow a circular path?) become easy if we let ourselves rotate.

The Fourier Transform is probably the best example of complex numbers being useful. It's very difficult to express the formula otherwise.

Thanks all! Yes, one of my goals for the site is to explore ways to make collaborative areas to find/surface the best analogies, diagrams, examples for the topics we're trying to learn.