{"id":"eulers-identity","domain":"named-results","also_in_domain":"analysis","type":"Theorem","title":"Euler's Identity","slug":"eulers-identity","url":"/mathematics/named-results/eulers-identity/","statement":"e^(i*pi) + 1 = 0","summary":"Special case of Euler's formula e^(ix)=cos(x)+i*sin(x) at x=pi. Connects e, i, pi, 1, 0. Often voted most beautiful equation in mathematics.","added_to_codex":"2026-05-27","last_verified":"2026-05-27","sources":[{"tier":1,"citation":"Euler, L. (1748). Introductio in analysin infinitorum."},{"tier":1,"citation":"Ahlfors, L.V. (1979). Complex Analysis. 3rd ed. McGraw-Hill."},{"tier":2,"citation":"Maor, E. (1994). e: The Story of a Number. Princeton University Press."},{"tier":3,"citation":"Crease, R.P. (2004). The greatest equations ever. Physics World, October 2004."}],"relationships":[{"type":"special_case_of","target":null,"label":"Euler's formula"},{"type":"uses","target":"irrational-number","label":"Irrational Number — e, π transcendental"}],"entry_status":"live","reading_levels_available":["curious","exploring","deep_dive"],"completeness":1.0,"codex_version":"1.5"}