I'm reading through Serge Lang's Basic Mathematics and I've fallen into trouble with a particular proof exercise:
Let $a = \frac m n$ be a rational number expressed as a quotient of integers m, n with $m\neq0$ and $n\neq0$. Show that there is a rational number $b$ such that $ab = ba = 1$.
I was able to solve the problem in my own way by assuming $b = \frac r s$ and proving the exercise via the cross-multiplication rule, however Lang instead proves the exercise by giving b the value of $\frac n m$ and simply multiplying both $a$ and $b$. How is this valid? Wouldn't the wording of the problem assume that rational number $b$ has its own unique numerator and denominator values?