I've recently started to think about this, and I'm sure a couple of you out there have, too.
In Algebra, we learned that $|x|\geq0$, no matter what number you plug in for $x$. For example: $$|-5|=5\geq0$$
We also learned that $x^2\geq0$. For example: $$(-5)^2=25\geq0$$ The exception for the $x^2$ rule is imaginary numbers (which we learn later on in Algebra II). Imaginary numbers are unique, in that their square is a negative number. For example: $$4i^2=-4$$ These imaginary numbers can be used when finding the "missing" roots of a polynomial equation.
My question to you is this: Is there any number whose absolute value is negative, and how could it be used?