√(x) = -1

[nova spruce]

okay so I've heard that this has absolutely no solution, which was pretty weird to me cuz im pretty sure there can be an imaginary solution to this

That makes sense to me tbh

i know it's a stupid question

but idk just food for thought ig

[fair oriole]

-1 is a solution to x^2 = 1, but sqrt(x) specifically picks out the positive real root if it exists. We have to restrict it that way so that it's a function.

In short, -1 is a square root of 1, but the square root of 1 is 1.

This is exactly why we add a plus or minus when we take the square root on both sides of an equation.

Every complex number actually has exactly n distinct complex nth roots.

Apart from 0, obviously.

[tawdry needle]

a notion sometimes referred to as "the principal square root"