Rational numbers

The rational numbers are the set $\mathbb{Q}$ of fractions $p/q$ with $p,q\in\mathbb{Z}$ and $q\ne 0$, where $p/q$ and $p'/q'$ represent the same rational number exactly when $pq'=p'q$.

One way to formalize this identification is to view $\mathbb{Q}$ as a quotient set of ordered pairs of integers under an equivalence relation . The rationals embed into the real numbers .

Examples:

• $\frac{2}{3}$ and $-\frac{7}{4}$ are rational numbers.
• Every integer $n$ corresponds to the rational number $n/1$.