Euler's Identity: Why e^(iπ) + 1 = 0 Is the Most Beautiful Equation

How one single formula unites five fundamental constants: e, i, π, 1, and 0.

The Five Fundamental Constants of Math

Leonhard Euler discovered the identity uniting the five most critical constants in all of mathematics: • e: Base of natural logarithms (2.71828...) • i: Imaginary unit (√-1) • π: Ratio of circle circumference to diameter (3.14159...) • 1: Multiplicative identity • 0: Additive identity

Euler's General Exponential Formula

Euler proved that imaginary exponents describe circular rotation in the complex plane using Taylor series expansions of eˣ, cos(x), and sin(x).

Setting x = π

Substitute x = π: e^(iπ) = cos(π) + i · sin(π) = -1 + i(0) = -1. Adding 1 to both sides yields e^(iπ) + 1 = 0.

Applications in Physics and Engineering

Euler's formula forms the foundation of AC electrical circuit impedance, quantum mechanics wavefunctions Ψ(x,t) = e^(i(kx-ωt)), and Fast Fourier Transforms (FFT) in audio processing.