{"id":"irrational-number","domain":"number-theory","type":"Concept","title":"Irrational Number","slug":"irrational-number","url":"/mathematics/number-theory/irrational-number/","summary":"A real number not expressible as a fraction of integers. Non-terminating, non-repeating decimal expansion. Includes algebraic irrationals (√2) and transcendentals (π, e).","key_examples":["√2 (proved irrational, Pythagorean school c. 500 BCE)","π (irrational: Lambert 1761; transcendental: Lindemann 1882)","e (irrational: Euler 1737; transcendental: Hermite 1873)","φ golden ratio ≈1.618 (algebraic)"],"added_to_codex":"2026-05-27","last_verified":"2026-05-27","sources":[{"tier":1,"citation":"Niven, I. (1956). Irrational Numbers. Carus Mathematical Monographs, MAA."},{"tier":1,"citation":"Baker, A. (1975). Transcendental Number Theory. Cambridge University Press."},{"tier":2,"citation":"Hardy, G.H. and Wright, E.M. (2008). An Introduction to the Theory of Numbers. 6th ed."}],"relationships":[{"type":"contrasts_with","target":"rational-number","label":"Rational Number"},{"type":"proved_via","target":"pythagorean-theorem","label":"Pythagorean Theorem"},{"type":"evidenced_in","target":"sulbasutras","label":"Śulbasūtras — √2 to 5 decimals"}],"entry_status":"live","reading_levels_available":["curious","exploring","deep_dive"],"completeness":1.0,"codex_version":"1.4"}