Euler's identity - Wikipedia

Abstraction: e raised to i-pi plus one equals zero linking five fundamental constants

Key points:

• States e^(iπ) + 1 = 0, linking five fundamental constants: 0 (additive identity), 1 (multiplicative identity), π, e, and i
• Special case of Euler's formula e^(ix) = cos(x) + i·sin(x) evaluated at x = π
• Geometric interpretation: multiplying by e^(iπ) rotates any complex number by π radians, equivalent to reflecting across the origin
• First published in Euler's 1748 "Introductio in analysin infinitorum"; whether Euler explicitly stated this specific form is uncertain
• Voted "most beautiful theorem in mathematics" by Mathematical Intelligencer readers (1990); tied with Maxwell's equations in Physics World poll (2004)
• Generalizes: nth roots of unity sum to 0 (Euler's identity is n=2 case); analogous identities hold for quaternions and octonions

Connections: Complex Analysis · Mathematical Constants

Source: http://en.wikipedia.org/wiki/Euler's_identity