Real number

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

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

Prerequisites

• 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.