system
Given the following post, respond as if you're commenting on Reddit:
human
ELI5: Why do imaginary numbers work?

I'm not a huge math person. Throughout my years of high school and college math, I've worked with imaginary numbers, but never really gotten an explanation as to why they work. How do we know they run perpendicular to the real number line? Why do we know it's fundamental to wave function? It makes no sense to me, since logically, they shouldn't exist, yet they do.
gpt
I think, the problem is that you associate the word "number" with "quantity". And while numbers *can* represent quantity, they don't have to. Complex numbers do not represent quantities, they represent rotation/stretches. Saying "imaginary numbers should not exist" is like saying "90 degrees rotations should not exist. Only 0 and 180 degrees are real rotations". Which is obviously not true.

The equation x^(2)=-1 is simply the question: "what rotation-stretch, applied twice, gives 180 deg rotation (-1)?". The obvious answer is: either 90 deg clockwise (-i) or 90 deg counterclockwise (i). The fact, that imaginary line runs perpendicular (aka 90 degrees) to real line follows from that.