# Integer

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

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## Summary

ℤ = {…, −2, −1, 0, 1, 2, …} — whole numbers with negatives and zero. Extends ℕ, making subtraction always possible. Forms a commutative ring but not a field.

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## Algebraic structure

ℤ is:
- **Integral domain** — no zero divisors
- **Euclidean domain** — division algorithm applies
- **Principal ideal domain (PID)** — every ideal is nℤ
- **Unique factorisation domain (UFD)** — Fundamental Theorem of Arithmetic

Not a field — most integers have no multiplicative inverse in ℤ.

## Construction from ℕ

ℤ built as equivalence classes of pairs (a,b) ∈ ℕ×ℕ representing "a−b", with (a,b)~(c,d) iff a+d=b+c. Special case of the Grothendieck group construction.

## Historical acceptance

Diophantus (3rd c. CE) rejected negative solutions. Descartes called them "false roots." Brahmagupta's *Brāhmasphuṭasiddhānta* (628 CE) gives first systematic rules treating negatives on equal footing with positives. Full European acceptance: 19th century (Hankel 1867, Peano 1889).

## Generalisations

- **p-adic integers ℤₚ** — inverse limit of ℤ/pⁿℤ; essential to Wiles's FLT proof
- **Gaussian integers ℤ[i]** — Euclidean domain; sums of two squares
- **Rings of algebraic integers** — may lack unique factorisation (motivated Dedekind's ideals)

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## Sources

### Tier 1
- Hardy, G.H. and Wright, E.M. (2008). *An Introduction to the Theory of Numbers*. 6th ed. Oxford University Press.
- Dummit, D.S. and Foote, R.M. (2004). *Abstract Algebra*. 3rd ed. John Wiley & Sons.

### Tier 2
- Plofker, K. (2009). *Mathematics in India*. Princeton University Press.

### Tier 3
- Kline, M. (1972). *Mathematical Thought from Ancient to Modern Times*. Oxford University Press.

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