What is Euler's Identity?

eiπ + 1 = 0
Five fundamental constants. One equation. Nothing else needed.
The five constants
e
Euler's number≈ 2.71828…
Base of natural logarithms. Governs growth and decay.
i
Imaginary unit= √(−1)
Satisfies i² = −1. Foundation of complex numbers.
π
Pi≈ 3.14159…
Ratio of a circle's circumference to its diameter.
1
One
The multiplicative identity. Any number × 1 = itself.
0
Zero
The additive identity. Any number + 0 = itself.

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.

Step by step
Euler's formulaeⁱˣ = cos(x) + i·sin(x)
Set x = πeⁱπ = cos(π) + i·sin(π)
Evaluateeⁱπ = −1 + 0i
Simplifyeⁱπ = −1
Add 1eⁱπ + 1 = 0 ✓
The unit circle view
+1−1π

eiθ traces the unit circle. Rotating by π lands at −1. Add 1, get 0.

Why mathematicians love it

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."

History

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.

Learn about e →Learn about π →
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Which five mathematical constants appear in Euler's identity?
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