I'm going through the Harvey Mudd Real Analysis lectures. In the second lecture the professor gives a little problem to check knowledge:
if $m/n$ is a positive quotient, then $mn > 0$
How would you show that this is well defined?
I understand that if $m/n = p/q$ and $mn > 0$ therefore $pq > 0$ but am unsure where to proceed from there.
This is used in construction of $\mathbb Q$ from $\mathbb Z$ as definition of positive.
This condition tells us when two fractions represent the same rational number (belong to the same equivalent class): $$\frac mn = \frac pq \qquad\Leftrightarrow\qquad mq=pn.$$
So under the assumptions that $mn=pq$, $n\ne0$, $q\ne0$ and $m,n,p,q\in\mathbb Z$ we want to show that $mn>0$ implies $pq>0$. We are allowed to use known properties of $\mathbb Z$.
Let us have a look at $mn\ge0$ instead. If we multiply both sides of the inequality by $(mq)^2$, the inequality does not change, since $(mq)^2\ge0$. And we will then use that $(mq)^2=mqpn$, which follows from $mq=pn$. \begin{align*} mn&\ge0\\ mn\cdot(mq)^2&\ge0\\ mn\cdot mqpn &\ge 0\\ m^2n^2pq&\ge0\\ pq&\ge 0 \end{align*}
This solves similar problem with strict inequality. I.e., in this way we can show $mn\ge0 \Leftrightarrow pq\ge0$.
So it remains to check when a fraction is equal to zero and what happens in that case.