---
tag: 05K3J8J8T2
kind: example
name: The rationals as the field of fractions of the integers
msc: [13B30]
instance_of: [0F2PTGX1A2]
uses: [0VF11ZTTMD, 0JHM6JK7FD]
sources:
  - url: https://en.wikipedia.org/wiki/Field_of_fractions
    locator: Examples
---

## Example

The [[field of fractions|0VF11ZTTMD]] of the [[ring of integers|0JHM6JK7FD]] $\mathbb{Z}$ is the field $\mathbb{Q}$ of rational numbers.
Each element of $\operatorname{Frac}(\mathbb{Z})$ is a class $a/b$ with $a, b \in \mathbb{Z}$ and $b \neq 0$.
Two pairs give the same class, $a/b = c/d$, if and only if $a d = b c$.
The map $\mathbb{Z} \to \mathbb{Q}$, $n \mapsto n/1$, is the usual inclusion of the integers in the rationals.

## Remarks

Every rational number has a unique form $a/b$ with $b > 0$ and $\gcd(a, b) = 1$.
Every subfield of $\mathbb{C}$ contains $\mathbb{Z}$, so it contains $\mathbb{Q}$, by the universal property of the field of fractions.

## Explanations

### L1 human

The rational numbers are exactly the fractions of integers.
The field of fractions makes this precise for any integral domain.
The fractions $1/2$, $2/4$ and $-3/(-6)$ all name the same rational number.