Since , I don't think you'll have a lot of luck proving that it's not a complex number. In particular, for , taking is a valid choice for the complex number .
Try reading Construction of the real numbers - Wikipedia, the free encyclopedia
Edit: Proving that a number is real requires a definition for a real number. Of course, you could always rely on your natural intuition for what a real number is, but that seems to defeat the purpose a bit in what you're trying to do . There are a few ways to define (or construct) the real number system, and the method by Dedekind cuts doesn't require a substantial amount of mathematical background to understand (ie, it doesn't rely on a precise definition of, say, Cauchy sequences).
Anyway, what I'm trying to get at is that defining a number as real is a bit more involved than defining rational numbers (which seems to be the analogous proof in this case).