DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-07-24

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The rationals as equivalence classes of pairs of integers

Definition

On the set of pairs $(a, b)$ with $a, b \in Z$ and $b \neq 0$ , define

$(a, b) \sim (c, d) ⟺ a d = c b in Z .$

This is an equivalence relation (The rational-defining relation is an equivalence relation ↗). The rationals are the quotient $Q$ , and $[(a, b)]$ is written $a / b$ .

Remarks

The pair $(a, b)$ encodes the formal quotient $a / b$ ; the relation $a d = c b$ is " $a / b = c / d$ " cleared of denominators, using only the ring arithmetic of $Z$ (The integers form a commutative ring).

Depends on

Dependency tree · 2 levels

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