---
tag: 0FFH3HSRBA
kind: concept
name: Construction of the rationals as fractions
msc: [13B30]
sources:
  - url: https://en.wikipedia.org/wiki/Field_of_fractions
    locator: Definition
  - url: https://en.wikipedia.org/wiki/Rational_number
    locator: Formal construction
---

## Summary

The rational numbers can be constructed as classes of pairs of [[integers|0C4JS9RF5M]].
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|0NCJ5WXN6Z]] 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.