“Imaginary” in mathematics shouldn't be confused with imaginary in its non-math sense. It simply means a number outside those numbers called “real”, which are integers, fractions, irrational numbers, etc. The square root of a number can be found as long as the number is greater than or equal to zero. Mathematicians like things to be complete, so they invented something that would represent the square root of a negative number. They came up with the letter i which represents the square root of -1. The square root of any negative number can now be represented by a multiple of i. So the square root of -4 is 2i. Complex numbers are those with a real part and an imaginary part, like i+1. There's a whole branch of mathematics devoted to complex numbers, and some problems can be solved using complex numbers. Whereas real numbers can be thought of as points on a continuous line (one dimension), complex numbers can be represented by points in a continuous plane (2 dimensions). This makes them geometrically interesting and leads on to trigonometry and vectors, among other things. So imaginary becomes very real and useful!