Rational numbers - Knowlpedia

Reflexive: a ∼ a a\sim a a ∼ a .
Symmetric: if a ∼ b a\sim b a ∼ b , then b ∼ a b\sim a b ∼ a .
Transitive: if a ∼ b a\sim b a ∼ b and b ∼ c b\sim c b ∼ c , then a ∼ c a\sim c a ∼ c .

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 $pq'=p'q$ .

Remarks

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 $n/1$ .