What is the precise definition of $i$?

Solution 1:

I've made my comment above into an answer.

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

Solution 2:

let $e^{z\pi}=-1$, then $z=(2k-1)i, k\in \mathbb N$

Solution 3:

There is no property of the arithmetic and calculus of the complex numbers possessed by $\mathbf{i}$ that is not also possessed by $-\mathbf{i}$; one is just as good as the other.

For example, if you started with $-\mathbf{i}$ as the complex unit and went through the usual development of arithmetic, you would find that the principal square root is $\sqrt{-1} = -\mathbf{i}$.

In fact, a key feature of the complex numbers is complex conjugation, an operation that swaps $\mathbf{i}$ and $-\mathbf{i}$.