Mystical mathematics

Okay, I am up-close and personal with negative numbers, exponents and rational numbers (square roots to the uninitiated), but I draw the line at imaginary numbers. That's right, imaginary numbers, like negative square roots. Wait a minute, if you multiply two negative numbers you get a positive number! Square roots cannot be negative! Oh, yes they can. You just express it as a complex number (real and imaginary) using an "i" (for imaginary, I suppose). Since I live in a real world, it seems hardly necessary to be mucking around with imaginary numbers. However, the teacher seems to think there is a use for all this somewhere out there, like around Pluto, which is now an imaginary planet, too. And perhaps the whole world is imaginary, so the imaginary numbers are really the real numbers? You think?