Euler's identity follows from Euler's formula: eix = cos(x) + i·sin(x). Setting x = π gives eiπ = cos(π) + i·sin(π) = −1, so eiπ + 1 = 0.
eiθ traces the unit circle. Rotating by π lands at −1. Add 1, get 0.
It connects arithmetic (0 and 1), algebra (i), geometry (π), and analysis (e) — four different branches of mathematics — in a single equation of stunning simplicity. Richard Feynman called it "the most remarkable formula in mathematics."
Leonhard Euler (1707–1783) published the formula eix = cos(x) + i·sin(x) in his Introductio in analysin infinitorum (1748). The identity is the special case at x = π. Euler introduced or popularised the notation e, i, f(x), Σ, and π.
The Taylor series for eˣ groups into cos(π) for the real terms and i·sin(π) for the imaginary terms. Since cos(π) = −1 and sin(π) = 0, we get e^(iπ) = −1, so e^(iπ) + 1 = 0.
The formula e^(i*theta) traces a unit circle on the complex plane as theta increases. e^(i*pi) is a rotation of exactly pi radians (180 degrees) from 1, landing at -1. Adding 1 brings you back to 0. This is why e^(i*pi) + 1 = 0: it is a half-turn of the complex plane expressed as an equation.
e^(iθ) is a rotation operator. At θ=π you have rotated exactly half a circle. The point 1 on the real axis travels to -1. Adding 1 to both sides gives e^(iπ) + 1 = 0.
Euler's identity e^(i*pi) + 1 = 0 unites the five most important constants in mathematics: e (the base of natural logarithms), i (the imaginary unit), pi (the circle constant), 1 (the multiplicative identity), and 0 (the additive identity). It follows directly from Euler's formula e^(i*theta) = cos(theta) + i*sin(theta) by setting theta = pi. Since cos(pi) = -1 and sin(pi) = 0, we get e^(i*pi) = -1. First published by Euler around 1748. Voted the most beautiful equation in mathematics in multiple polls.