# Rational Number

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

## Summary
Any number expressible as p/q with p,q∈ℤ, q≠0. ℚ forms a field — the field of fractions of ℤ.

## Key properties
- Field: every nonzero rational has a multiplicative inverse
- Dense: between any two rationals lies another
- Countable: same cardinality as ℕ (Cantor)
- Decimal expansions terminate or repeat

## Historical development
Egyptian unit fractions (Rhind Papyrus, c. 1650 BCE). Babylonian sexagesimal fractions. Indian general fraction rules (Brahmagupta 628 CE, Bhāskara II).

## Ostrowski's theorem (1916)
The only completions of ℚ are ℝ (usual absolute value) and ℚₚ for each prime p (p-adic absolute value) — nothing else.

## Sources
### Tier 1
- Rudin, W. (1976). *Principles of Mathematical Analysis*. 3rd ed. McGraw-Hill.
- Hardy, G.H. and Wright, E.M. (2008). *An Introduction to the Theory of Numbers*. 6th ed.
### Tier 2
- Plofker, K. (2009). *Mathematics in India*. Princeton University Press.

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