It doesn't exist as there is no number that can be squared to equal a negative number, however, we can imagine such a number even though it cannot exist on the number line. We can use any reference we choose to designate such an imaginary number, ergo, I use i.
There are practical uses for i in scientific calculations as one can have the i's cancel out and come up with an answer that is a real number.
A CAT scan of my poor brain would reveal a hole where most people have their math ability.