# Real number

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

The real numbers contain the rational numbers and complete the rational system by adding its missing limit points.

mko-real-number · concept · v0.10.0



## Mathematical statement

The real-number system R can be constructed as a Cauchy completion of the rationals. Every rational number embeds into R, while different constructions still require proofs of equivalence.





## Representative formula (exact TeX)

x\in R





## Assumptions


- The setting is a Cauchy completion of the rationals with an explicit rational embedding.





## Symbol scope


- x — number under classification or operation
- R — representative symbol for the real-number system





## Intuition, definition, and misconceptions

Every integer and rational number can be regarded as a real number. A terminating decimal can be converted to an exact fraction and is not a separate number system.

The real numbers may be constructed as equivalence classes of rational Cauchy sequences. “Adding limit points” is constructive intuition rather than a replacement for the precise definition and proof of completeness.

This MKO first establishes exact representation, comparison, and interval language for rational points on the real line. Irrational-number classification, order completeness, and continuity remain later obligations.

Common mistakes include treating a terminating decimal as irrational, treating a decimal approximation as an exact value, or claiming completeness merely by saying the line has no gaps.





## Adopted derivation guide

[Real number: proving in the Cauchy completion that 'repeating nines' and the rational 1 are the same real](https://ocme.evemisslab.com/en/derivations/derivation-mko-real-number-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Rational number](https://ocme.evemisslab.com/en/mko/mko-rational-number/) — This entry uses the rational embedding and completion as its entry into the real numbers.





## Verification boundary

Exact recomputation for finitely many rational points verifies only the specified representation, ordering, and interval claims; it does not prove real completeness or all properties of irrational 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 Mathlib Real as a Cauchy completion together with the rational embedding, linear order, completeness statements, and 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/real-number.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-real-number/index.md)