The rationals as the field of fractions of the integers

Example

The field of fractions of the ring of integers $\mathbb{Z}$ 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.