Why do we require that $\displaystyle a$ be a non-negative number in the general expression of an exponential function,

$\displaystyle y = C \cdot a^x \, .$

I understand that a negative value for $\displaystyle a$ would give a very strange function that would fluctuate between negative and positive $\displaystyle y$-values which in a certain context would make no sense. But I would like to know, purely mathematically speaking, why $\displaystyle a$ has to non-negative.