i have always thought $\displaystyle i$ was defined as $\displaystyle \sqrt{-1}$, where $\displaystyle \sqrt{}$ is the principal square root. Now I heard that you can't take the square root of negative numbers. That makes me confused.

How do you solve this equation for example:

$\displaystyle x^2 = -4$

Do you have to specify that the solution can be complex as well? Is it okay then to take the square root of negative numbers? (Or is it called the complex square root, and how do you know when it is the complex square root that's intended?)

And when is a number negative? is it when the real part of it is negative? Or is negative numbers only possible for real numbers?