system
Given the following post, respond as if you're commenting on Reddit:
human
ELI5: If imaginary numbers like sqrt(-1) are in fact imaginary, how can they help solve actual Math problems?
gpt
all numbers are imaginary, they are concepts that exist only in your mind. Numbers become useful when they can describe real world objects. The number *i* is a natural way to describe a 1/4 rotation. If you do it twice, *i*^2 you have done a 180 degree rotation, and in math, turning around is like multiplying by -1, so in terms of rotations, *i*^2 = -1. And if you do 4 rotations of a 1/4 turn, you have spun all the way around and are back where you started. It is like you have done nothing at all, in math, that is like multiplying by 1, so in terms of rotations *i*^4 = 1.

So all numbers are imaginary and live in your mind, and they become useful when they represent something real. When you relate imaginary numbers to rotations they become very useful and solve many real problems.