{"id":"complex-number","domain":"analysis","type":"Concept","title":"Complex Number","slug":"complex-number","url":"/mathematics/analysis/complex-number/","summary":"a+bi where i²=-1. Forms an algebraically closed field. Essential to electrical engineering, quantum mechanics, complex analysis.","formal_definition":"C = {a+bi : a,b in R}, i^2=-1","added_to_codex":"2026-05-27","last_verified":"2026-05-27","sources":[{"tier":1,"citation":"Ahlfors, L.V. (1979). Complex Analysis. 3rd ed. McGraw-Hill."},{"tier":1,"citation":"Needham, T. (1997). Visual Complex Analysis. Oxford University Press."},{"tier":2,"citation":"Nahin, P.J. (1998). An Imaginary Tale: The Story of √-1. Princeton University Press."},{"tier":3,"citation":"Renou, M-O. et al. (2021). Quantum theory based on real numbers can be experimentally falsified. Nature, 600, 625-629."}],"relationships":[{"type":"forms","target":"field","label":"Field — algebraically closed"},{"type":"related_to","target":"eulers-identity","label":"Euler's Identity"},{"type":"roots_of","target":"polynomial","label":"Polynomial"},{"type":"used_in","target":"fourier-transform","label":"Fourier Transform"}],"entry_status":"live","reading_levels_available":["curious","exploring","deep_dive"],"completeness":1.0,"codex_version":"1.8"}