What does m/n = m'/n' iff mn'=m'n mean when talking about Rational numbers in Infinite sets?

Question

I couldn't find anything anywhere about what this condition stated meant?

Thanks in advance!

Answer 1

Definition: A pre-rational number is a tuple $(m,n)\in\mathbb{Z}\times(\mathbb{Z}\setminus\{0\})$.

Definition: Two pre-rational numbers $(m,n)$ and $(m^\prime,n^\prime)$ are said to be equivalent if $mn^\prime = m^\prime n$. In this case, we write $(m,n) \sim (m^\prime,n^\prime)$.

Exercise: $\sim$ is an equivalence relation.

Definition: We define the rational numbers as the equivalence classes $\mathbb{Q} \equiv (\mathbb{Z} \times (\mathbb{Z}\setminus\{0\}))\,\,/\sim$.

Now, instead of writing $[(m,n)]$ to denote a member of $\mathbb{Q}$, it is common practice to write instead $a/b$, where $(a,b)\in [(m,n)]$.