Construction of the rationals as fractions

Summary

The rational numbers can be constructed as classes of pairs of integers. The idea is that the pair $(a, b)$ with $b \neq 0$ stands for the fraction $a / b$. On the set of pairs $(a, b)$ with $a, b \in \mathbb{Z}$ and $b \neq 0$, define $(a, b) \sim (c, d)$ when $a d = b c$. This relation is an equivalence relation. The set $\mathbb{Q}$ of rational numbers is the set of the equivalence classes, and we write $a / b$ for the class of $(a, b)$. The operations are $\frac{a}{b} + \frac{c}{d} = \frac{a d + b c}{b d}$ and $\frac{a}{b} \cdot \frac{c}{d} = \frac{a c}{b d}$. They do not depend on the choice of the pairs.

Remarks

The same construction works for each integral domain $R$. It gives the field of fractions of $R$, the smallest field that contains $R$. The map $a \mapsto a / 1$ embeds $\mathbb{Z}$ in $\mathbb{Q}$.

Explanations

L1 human

A fraction is a pair of a top number and a nonzero bottom number. The pairs $(1, 2)$ and $(2, 4)$ are different pairs, but $1 \cdot 4 = 2 \cdot 2$, so they give the same rational number. The construction makes this precise: a rational number is the collection of all the pairs that name it.