Given $a \in \mathbb Q$ which is non-zero, show that $1/a \in \mathbb Q$.

Question

Any idea?

I've started with $a= p/q$, meaning $q$ has to be rational.

Then $1/a= q/p$ But surely this could be irrational if, say $p=9$, $q=4$?

Answer 1

By definition, a real number $x$ is irrational if and only if there are integers $m$ and $n\not=0$ such that $x=m/n$. If $x\not=0$, then $m\not= 0$, so, as you pointed out $1/x=n/m$. This number is by definition rational!