{"id":"rational-number","domain":"number-theory","type":"Concept","title":"Rational Number","slug":"rational-number","url":"/mathematics/number-theory/rational-number/","summary":"Any number expressible as p/q with integers p,q, q≠0. Forms a field (field of fractions of ℤ). Dense but countable.","formal_definition":"ℚ = {p/q : p,q∈ℤ, q≠0}, with p/q=r/s iff ps=qr.","key_properties":["Field (every nonzero element has inverse)","Dense in ℝ","Countable (Cantor)","Terminating or repeating decimals"],"added_to_codex":"2026-05-27","last_verified":"2026-05-27","sources":[{"tier":1,"citation":"Rudin, W. (1976). Principles of Mathematical Analysis. 3rd ed. McGraw-Hill."},{"tier":1,"citation":"Hardy, G.H. and Wright, E.M. (2008). An Introduction to the Theory of Numbers. 6th ed. Oxford University Press."},{"tier":2,"citation":"Plofker, K. (2009). Mathematics in India. Princeton University Press."}],"relationships":[{"type":"extends","target":"integer","label":"Integer"},{"type":"contrasts_with","target":"irrational-number","label":"Irrational Number"},{"type":"algebraic_structure","target":"field","label":"Field"},{"type":"historical_rules_from","target":"brahmagupta","label":"Brahmagupta"}],"entry_status":"live","reading_levels_available":["curious","exploring","deep_dive"],"completeness":1.0,"codex_version":"1.4"}