What are imaginary numbers? At school, I really struggled to understand the concept of imaginary numbers.  My teacher told us that an imaginary number is a number that has something to do with the square root of $-1$.  When I tried to calculate the square root of $-1$ on my calculator, it gave me an error.  To this day I still do not understand imaginary numbers.  It makes no sense to me at all.  Is there someone here who totally gets it and can explain it?
Why is the concept even useful?

Imaginary numbers were invented to make calculations easier. Everyone knows the quadratic formula; when Cardano was working on the formula for cubics (known as Cardano's formula), he found out that it was extremely hard to write down a formula unless you out down some symbol as a placeholder for $\sqrt{-1}$, which you manipulate like a number and which always cancelled out in the end. So he left it in. He was embarrassed by it, and called it imaginary, but the formula worked. Mathematicians later found out that imaginary numbers made a lot of formulas easier, like finding a formula for $\sin(3x)$, and so they found consistent rules for them. Ever since then, they've kept making formulas easier.
