What is the precise definition of $i$? This may seem like an extraordinarily trivial question and yet it has completely confounded me.  The technical definition of $i$ is
$$i^2=-1$$
But there are two numbers which fulfill this requirement:
$$\sqrt{-1},-\sqrt{-1}$$
Wouldn't a more precise definition of $i$ simply be $\sqrt{-1}$?
Thank you and forgive the elementary nature of the question.

I've made my comment above into an answer.

This is an excellent question to ask. This section of the Wikipedia page should help.
