Why was $i$ introduced to satisfy this $\sqrt{-1}$? Can someone explain to me why $$\sqrt{-1} = i$$ I love math and I'm looking at doing it to higher levels. I know that we can NEVER have a square root of a negative number as per my reading hence if I was given $$\sqrt{-4}$$ the answer would be $$2i$$
and if I was given
$$\sqrt{4}$$ 
the answer would be $\pm2$, can I get sources explaining why $i$ was introduced?
Thanks. 

on the history, my impression is that people were fairly happy having quadratic equations without (realnumber) roots. Then for cubic equations, Cardano, about 1545, introduced a rudimentary version of complex numbers, because his method lead directly to using cube roots of one of these objects.  https://en.wikipedia.org/wiki/Complex_number
good book: http://www.maa.org/publications/periodicals/convergence/an-imaginary-tale-the-story-of-1
from the book description: 

The solution of cubic equations in the sixteenth century forced
  mathematicians to deal with the square root of -1, and in the first
  major chapter of the book, the author handles this episode judiciously
  and leisurely.

