Topic · Analysis

Euler's Identity

Five numbers, one line.

e^(iπ) + 1 = 0 — five constants from five different corners of mathematics, locked into a single equality. The Explorer next door lets you watch e^(iθ) sweep around the unit circle in real time, so you can see, with your own eyes, the moment at θ = π when the identity actually happens.

→ Open the Explorer ← Back to the atlas

eⁱᵖⁱ + 1 = 0

First encounter

Five strangers. One line. Forever.

The big idea

Five of mathematics' most important numbers — e, i, π, 0, 1 — never run into each other in everyday life. Yet one equation, e^(iπ) + 1 = 0, ties all five together with only addition, multiplication, and exponentiation. Nothing else. No constants left over, no decoration. That is the whole identity.

A concrete example

Imagine walking around a circle of radius 1, centred at the origin of a flat plane. Start at the point 1 on the right. Multiplying by e^(iθ) means: walk θ radians around the circle. So walk π radians — half the way round — and you arrive at the point −1. Add 1 and you are back at 0.

start · 1

× e^(iπ) · half-turn

land · −1

+ 1 · 0

Why it matters

Feynman called the formula e^(iθ) = cos θ + i sin θ, of which this identity is the θ = π case, "the most remarkable formula in mathematics", "our jewel." A 1990 Mathematical Intelligencer poll of working mathematicians voted this identity the most beautiful theorem ever proved. It is a bridge: algebra writes the symbols, geometry walks the circle, analysis supplies the infinite series that makes it all click.

Below: slide θ around the circle yourself, and watch the Taylor series build it term by term.

Section 01 · Five characters

Who is in this equation

Each of the five numbers arrives from a different country of mathematics. 0 is the additive identity — adding it changes nothing. 1 is the multiplicative identity — multiplying by it changes nothing. e ≈ 2.71828 is the natural base, the number whose own derivative is itself, born in calculus from compound interest and continuous growth. i is the imaginary unit, defined by i² = −1, born in algebra when Cardano needed a square root of a negative to solve cubic equations. π ≈ 3.14159 is the half-turn of the circle, the ratio of circumference to diameter, born in geometry. They normally never share a room. The identity is the room.

The five constants

0	Additive identity	Arithmetic
1	Multiplicative identity	Arithmetic
e	Base of natural growth ≈ 2.71828	Calculus
i	Imaginary unit, i² = −1	Algebra
π	Half-turn of the circle ≈ 3.14159	Geometry

Section 02 · Complex multiplication

Why i is a quarter-turn

Forget that i is "imaginary" for a moment and look at what multiplying by i does. Start at 1 on the real axis. Multiply by i and you land at i — a quarter-turn up. Multiply by i again and you land at i² = −1 — another quarter-turn, exactly opposite where you started. Two quarter-turns make a half-turn, which is exactly what i² = −1 means geometrically. Complex multiplication is rotation (by the argument) times scaling (by the magnitude). The numbers on the unit circle, where the magnitude is 1, are pure rotation. They are the engine of the identity.

i · i · i · i

1→×i→i→×i→−1→×i→−i→×i→1

Four quarter-turns close the loop. Each ×i is a 90° rotation; i² = −1 is two of them.

Interactive · slider

Walk θ around the circle

Drag the slider from 0 to 2π. The yellow dot is e^(iθ); the cyan and violet markers are its real and imaginary projections — the cos θ and sin θ components. When θ ≈ π you land on −1, and the identity glows.

Drag θ · e^(iθ) on the unit circle

θ0.785 (0.25π)

cos θ · Re

0.7071

sin θ · Im

0.7071

e^(iθ) = 0.707 + 0.707i

Section 03 · Euler's formula

e^(iθ) = cos θ + i sin θ

Euler published this formula in his 1748 Introductio in analysin infinitorum, and it is the larger truth of which the identity is a single value. The left-hand side, e^(iθ), is an exponential with an imaginary exponent — a kind of growth nobody had ever pictured before. The right-hand side, cos θ + i sin θ, is the point on the unit circle at angle θ. Euler's claim is that those two completely different-looking things are equal. Set θ = π and the right-hand side collapses to −1 + 0i, so e^(iπ) = −1, and adding 1 gives the identity. The identity is Euler's formula at θ = π.

Proof in four steps

1.e^(iθ) = cos θ + i sin θ

2.θ ← π

3.e^(iπ) = −1 + i·0 = −1

4.e^(iπ) + 1 = 0 ∎

Section 04 · Why it's true

Taylor series, substituted

The cleanest proof goes through power series. Each of the three functions has a Taylor expansion that converges on the entire real line: e^x = Σ x^k / k!, cos x = Σ (−1)^k x^(2k) / (2k)!, sin x = Σ (−1)^k x^(2k+1) / (2k+1)!. Now formally substitute x = iθ into the series for e^x. Because i^k cycles through 1, i, −1, −i, the real terms of the resulting series are exactly the Taylor series for cos θ, and the imaginary terms (without the i) are exactly the Taylor series for sin θ. So e^(iθ) = cos θ + i sin θ — not by definition, but as a theorem about converging series. Set θ = π and the algebra finishes the proof.

Substitute x = iθ into the Taylor series of eˣ and watch the real and imaginary parts split.

e^x   = Σₖ xᵏ / k!
e^(iθ) = Σₖ (iθ)ᵏ / k!
       = Σₖ iᵏ θᵏ / k!

iᵏ cycles:  k mod 4 = 0 → +1
            k mod 4 = 1 → +i
            k mod 4 = 2 → −1
            k mod 4 = 3 → −i

real part      = Σ (−1)ʲ θ²ʲ / (2j)!     = cos θ
imaginary part = Σ (−1)ʲ θ²ʲ⁺¹ / (2j+1)! = sin θ

Interactive · slider

Build the proof, term by term

Each curve is a partial Taylor sum — the first N+1 terms only. Slide N up and the partial sums close in on the true exp, cos and sin curves. Watch the small panel underneath: the complex partial sum of (iπ)^k / k! drifts toward −1 as N grows.

Slide N · partial Taylor sums

N3 terms

exp

cos

sin

Complex walk · e^(iπ) partial sum

Σ_k=0..2 (iπ)ᵏ / k! = -3.9348 + 3.1416i

→ −1 + 0i as N → ∞

Section 05 · Connection everywhere

Where the identity lives in the wild

Once you have e^(iθ), it stops being a curiosity and becomes an indispensable tool. Electrical engineers write the impedance of a circuit as a complex number Z = R + iX, so that voltages and currents become rotating phasors V·e^(iωt) and Kirchhoff's laws turn into linear algebra. Quantum mechanics writes the time-evolution of a state as ψ(t) = e^(−iHt/ℏ) ψ(0) — the Schrödinger equation, with e^(iθ) doing every bit of the rotation in Hilbert space. Fourier analysis decomposes any reasonable signal into a sum (or integral) of e^(iωt) modes, and every JPEG, MP3, MRI scanner and Wi-Fi handshake on the planet is built on top of that decomposition. The identity is the boundary stone where all four roads — arithmetic, algebra, geometry, analysis — meet.

Where the identity shows up

Electrical engineering

Impedance Z = R + iX; phasors V·e^(iωt) turn AC circuit analysis into linear algebra.

Quantum mechanics

Time evolution ψ(t) = e^(−iHt/ℏ) ψ(0); every state rotates in Hilbert space via e^(iθ).

Fourier analysis

Every signal decomposes into e^(iωt) modes — the basis of MP3, JPEG, MRI, Wi-Fi.

Number theory

Roots of unity e^(2πik/n) parameterise every solution to z^n = 1 and seed cyclotomic fields.

Section 06 · Where it sits

A short progression

Step back and see the chain. Real-line arithmetic gives you +, ×, and the constants 0 and 1. Calculus turns growth into a number and names it e. Algebra solves cubics, runs out of real square roots, invents i, and discovers the complex plane. Geometry parameterises the unit circle by angle and meets π. Euler's formula then notices, with two pages of power series, that growth in the imaginary direction is rotation in the plane — that e and i secretly do the same thing. Set the angle to a half-turn and the five constants finally meet, with nothing in between. The identity is not a coincidence; it is the destination.

The arc that leads here

1.Real-line arithmetic gives you +, ×, 0, 1.
2.Calculus turns continuous growth into a number and names it e.
3.Algebra runs out of real square roots and invents i; the complex plane appears.
4.Geometry parameterises the unit circle by angle and meets π.
5.Euler's formula shows growth in the imaginary direction = rotation in the plane.
6.Set θ = π. All five constants meet.

Take it further

Open the Explorer.

The Explorer lets you sweep θ live, watch e^(iθ) trace the circle, and decompose any complex number into its rotating, scaling pieces. Everything you just read is one click away.

→ Open the Explorer