Why does the definition of a rational number have to include the fact that the denominator is non-zero?

Question

A real number $n$ is rational iff $\exists p,q \in \mathbb{Z}, n=\frac{p}{q} \:$ and $q \neq 0$. Why do we need that last condition? Sure, I understand it makes it more clear that a fraction is undefined if the denominator is 0, but this is an existential statement, so if a number is rational, I would think this implies that there exists a $q$ that is nonzero. My reasoning is that if a real number is rational, we can find a $p$ and $q$ that satisfies $n=\frac{p}{q}$, and for this to be possible, $q$ must be nonzero. I’ve lost points on a homework for not mentioning that condition in a proof, so I’m curious why it’s so necessary.