Rational number

Mathematical statement

A number x is rational if there exist integers p and q with q≠0 such that x=p/q. Different integer pairs can represent the same rational number.

Representative formula (exact TeX)

\frac{p}{q}

Assumptions

• p and q are integers.
• The denominator q is not 0.

Symbol scope

• p — integer numerator
• q — nonzero integer denominator
• / — quotient representation

Intuition, definition, and misconceptions

Every integer n is rational because it can be written as n/1. A string with zero denominator does not represent a rational number.

Different fractions may represent the same value. When both denominators are nonzero, p/q and r/s can be compared by cross multiplication.

Reducing a fraction and moving a negative sign to the numerator so the denominator is positive are canonical display conventions. They change the representation but not the rational value.

Common mistakes include allowing a zero denominator, treating differently written fractions as different values, or regarding reduction as approximation.

Adopted derivation guide

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Integer — A rational representation uses an integer numerator and a nonzero integer denominator.

Verification boundary

Finite fraction recomputation verifies only concrete representations; it does not prove the construction, order, density, or field axioms of rational numbers.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.

Next obligation

Bind Rat.divInt, the nonzero denominator condition, representation equivalence, and normalized numerator-denominator statements with exact versioned evidence.

Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.