# Real Number

**Type:** Concept  
**Domain:** number-theory  
**Codex URL:** /mathematics/number-theory/real-number/  
**Entry status:** Live — v1.0 (2026-07-09)

## Summary
Complete ordered field — all points on the continuous number line. Dedekind cuts (1872) and Cauchy sequences (Cantor 1872) as constructions. Least upper bound property. Uncountable (Cantor diagonal). Foundation of calculus and analysis.

## Sources
- [Tier 1] Dedekind, R. (1872). Stetigkeit und irrationale Zahlen. Trans. Beman 1901.
- [Tier 1] Rudin, W. (1976). Principles of Mathematical Analysis. 3rd ed.
- [Tier 2] Spivak, M. (2008). Calculus. 4th ed. Publish or Perish.
- [Tier 3] Stillwell, J. (2010). The Real Numbers. Springer.

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*Mathematics Codex entry v1.0 — added 2026-07-09 — thecodex.expert/mathematics/*