In an interestingly unrelated example, consider the general depressed cubic equation a*x^3 + b*x + c = 0. The algorithm that solves this equation is capable of producing "imaginary" numbers at intermediate steps. While these imaginary numbers are imaginary in a significantly different sense than a unicorn is, they provide an interesting analogy.
Of course, any mathematician will tell you that imaginary numbers are no more imaginary than negative numbers, or square roots or what have you. What confuses most of us is that in general, we can not find an intuitive physical counterpart to offer an explanation as to what imaginary numbers are trying to describe. And thus, we have the reason for the unfortunate nomenclature.
What makes this whole deal interesting is the fact that, although imaginary numbers are honestly no more abstractly ridiculous than any other numbers, we usually have to regard an imaginary solution to a real problem as meaningless.
If we define things as unreal, they may still be real in their consequences; encountering a "meaningless" value in the course of solving an equation does not guarantee that the technique you're using is invalid. The procedure for solving general, cubic equations with real coefficients does exactly this, and it works perfectly every time.