# Rational number

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

A rational number can be represented as a ratio of two integers with a nonzero denominator.

mko-rational-number · concept · v0.10.0



## 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

[Rational number: different fraction representations of one rational, cross-multiplication and reduction](https://ocme.evemisslab.com/en/derivations/derivation-mko-rational-number-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Integer](https://ocme.evemisslab.com/en/mko/mko-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.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/rational-number.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-rational-number/index.md)