I read that if $a = \frac mn$ is a positive rational number, it can be expressed in "lowest form" by cancelling common factors of $m$ and $n$, so that $a = \frac rs$ where r and s are relatively prime.
I'm wondering if we define the "lowest form" representation for a positive rational number to be $a = \frac pq$ where p and q are relatively prime, would this definition work? for a given rational number are these p and q uniquely determined?
On
Yes, they are. This follows from the fact that, if $m,n\in\mathbb N$, then $\frac m{\gcd(m,n)}$ and $\frac n{\gcd(m,n)}$ are relatively prime. Since$$\frac mn=\frac{\frac m{\gcd(m,n)}}{\frac n{\gcd(m,n)}},$$you can always express $\frac mn$ as the quotient of two relatively prime numbers. And there's just a way of doing it: if $a$ and $b$ are relatively prime, if $c$ and $d$ are relatively prime too, and if $\frac ab=\frac cd$, then $a=b$ and $c=d$. In fact, $\frac ab=\frac cd\iff ad=bc$. Now, since $a\mid ad=bc$ and since $a$ and $b$ are relatively prime, $a\mid c$, that is $c=ka$ for some $k\in\mathbb N$. By the same argument, $d=k'b$ for some $k'\in\mathbb N$. But then$$ad=bc\iff abk'=abk\iff k=k'.$$But $c$ and $d$ are relatively prime and, at the same time, $c=ka$ and $d=kb$; terefore, $k=1$, which means that $a=c$ and that $b=d$.
Suppose, $$\frac{a}{b}=\frac{c}{d}$$ with coprime positive integers $a,b$ and coprime positive integers $c,d$. Then, we have $$ad=bc$$
Since $a$ and $b$ are coprime, we can conclude $a|c$ because of $a|bc$ and $b|d$ because of $b|ad$
Since $c$ and $d$ are coprime, we can conclude $c|a$ because of $c|ad$ and $d|b$ because of $d|bc$
So we get $a=c$ and $b=d$