{"id":"integer","domain":"number-theory","type":"Concept","title":"Integer","slug":"integer","url":"/mathematics/number-theory/integer/","summary":"The whole numbers together with their negatives and zero: …,-2,-1,0,1,2,…. Extends the natural numbers, making subtraction always possible. Forms a commutative ring (Euclidean domain, PID, UFD) but not a field.","formal_definition":"ℤ = {…,-2,-1,0,1,2,…}, constructed from ℕ×ℕ as equivalence classes under (a,b)~(c,d) iff a+d=b+c.","key_properties":["Integral domain (no zero divisors)","Euclidean domain (division algorithm)","Principal ideal domain (every ideal is nℤ)","Unique factorisation domain"],"added_to_codex":"2026-05-27","last_verified":"2026-05-27","sources":[{"tier":1,"citation":"Hardy, G.H. and Wright, E.M. (2008). An Introduction to the Theory of Numbers. 6th ed. Oxford University Press."},{"tier":1,"citation":"Dummit, D.S. and Foote, R.M. (2004). Abstract Algebra. 3rd ed. John Wiley & Sons."},{"tier":2,"citation":"Plofker, K. (2009). Mathematics in India. Princeton University Press."},{"tier":3,"citation":"Kline, M. (1972). Mathematical Thought from Ancient to Modern Times. Oxford University Press."}],"relationships":[{"type":"extends","target":"natural-number","label":"Natural Number"},{"type":"includes","target":"zero","label":"Zero"},{"type":"contains","target":"prime-number","label":"Prime Number"},{"type":"algebraic_structure","target":"ring","label":"Ring (prototypical example)"},{"type":"historical_rules_from","target":"brahmagupta","label":"Brahmagupta (628 CE)"}],"entry_status":"live","reading_levels_available":["curious","exploring","deep_dive"],"completeness":1.0,"codex_version":"1.3"}