# Irrational Number

**Type:** Concept  
**Domain:** Number Theory  
**Codex URL:** /mathematics/number-theory/irrational-number/  
**Entry status:** Live — v1.0 (2026-05-27)

## Summary
A real number not expressible as p/q. Non-terminating, non-repeating decimal. Includes algebraic irrationals (√2) and transcendentals (π, e).

## Proof √2 is irrational
By contradiction: if √2=p/q (lowest terms), then p²=2q² makes p even, then q even too — contradicting gcd(p,q)=1. (Euclid, Elements Book X)

## Algebraic vs transcendental
- Algebraic: root of integer polynomial (√2, φ)
- Transcendental: not algebraic (π, e)

π irrational: Lambert 1761. π transcendental: Lindemann 1882.
e irrational: Euler 1737. e transcendental: Hermite 1873.

## Open problems
Unknown whether π+e or πe is irrational. Unknown whether Euler-Mascheroni constant γ is irrational.

## Sources
### Tier 1
- Niven, I. (1956). *Irrational Numbers*. Carus Mathematical Monographs, MAA.
- Baker, A. (1975). *Transcendental Number Theory*. Cambridge University Press.
### Tier 2
- Hardy, G.H. and Wright, E.M. (2008). *An Introduction to the Theory of Numbers*. 6th ed.

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*Mathematics Codex entry v1.0 — added 2026-05-27 — thecodex.expert/mathematics/*