system
Given the following post, respond as if you're commenting on Reddit:
human
ELI5: Uses of complex numbers.

I recently got interested in the topic of complex numbers, I watched a few videos on YouTube about the subject and I think I got the general idea of what they are. But I still don't understand what uses they have in real life.
gpt
The original use of complex numbers was to solve the dilemma of the fundamental theorem of algebra. Way back in the distant past, mathematicians argued that every polynomial had a number of roots (places where the function crossed the x-axis) depending on the order (highest power) of the polynomial. Unfortunately for those mathematicians, there were a host of counter-examples to this very elegant principle. So instead of just taking their lumps, they decided to invent complex numbers so they could be right - with the addition of complex numbers, the fundamental theorem of algebra ends up being true.

However, a more interesting application of complex numbers is with respect to the concept of *rotation*. You can use a complex number to represent a vector in two dimensional space (i.e. a set of x,y coordinates). If you multiply two of these vectors together, you end up with a new vector whose angle is the sum of the angles of the original vectors and whose magnitude is the product of the magnitudes of the original vectors.

Rotation also leads into *periodicity* - things that repeat. So we can represent periodic phenomenon with complex numbers. Since waves - including sound, light, etc. - are periodic phenomenon, complex numbers are a way to model them.