Euler's Formula

We've seen that multiplying by \cos\theta + i\sin\theta rotates the plane by \theta — and that the angles add when you multiply. Adding when you multiply is the signature behaviour of an exponential. That hint is no accident. It points at the single most celebrated identity in mathematics, Euler's formula:

e^{i\theta} = \cos\theta + i\sin\theta.

Plugging an imaginary number into an exponential turns it into a point on the unit circle. Let's see exactly why, straight from the series.

Deriving it from the series

The Maclaurin series of the exponential is

e^{x} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \frac{x^5}{5!} + \cdots.

Step 1 — substitute x = i\theta. Nothing fancy; just put i\theta everywhere x was:

e^{i\theta} = 1 + i\theta + \frac{(i\theta)^2}{2!} + \frac{(i\theta)^3}{3!} + \frac{(i\theta)^4}{4!} + \frac{(i\theta)^5}{5!} + \cdots.

Step 2 — work out the powers of i. They cycle with period four — this is the engine of the whole derivation:

i^0 = 1, \quad i^1 = i, \quad i^2 = -1, \quad i^3 = -i, \quad i^4 = 1, \quad \dots

Step 3 — apply the cycle term by term. So (i\theta)^2 = -\theta^2, (i\theta)^3 = -i\theta^3, (i\theta)^4 = \theta^4, and so on:

e^{i\theta} = 1 + i\theta - \frac{\theta^2}{2!} - i\frac{\theta^3}{3!} + \frac{\theta^4}{4!} + i\frac{\theta^5}{5!} - \cdots.

Step 4 — split into real and imaginary parts. Gather the terms without an i separately from the terms carrying one:

e^{i\theta} = \underbrace{\left(1 - \frac{\theta^2}{2!} + \frac{\theta^4}{4!} - \cdots\right)}_{\text{real}} + i\underbrace{\left(\theta - \frac{\theta^3}{3!} + \frac{\theta^5}{5!} - \cdots\right)}_{\text{imaginary}}.

Step 5 — recognise the two series. These are precisely the Maclaurin series for cosine and sine:

\cos\theta = 1 - \frac{\theta^2}{2!} + \frac{\theta^4}{4!} - \cdots, \qquad \sin\theta = \theta - \frac{\theta^3}{3!} + \frac{\theta^5}{5!} - \cdots.

Step 6 — read off Euler's formula. The real part is \cos\theta and the imaginary part is \sin\theta:

e^{i\theta} = \cos\theta + i\sin\theta.

The unit-modulus rotators of the previous page were exponentials all along.

The most beautiful equation: e^{iπ} + 1 = 0

Set \theta = \pi. Then \cos\pi = -1 and \sin\pi = 0, so Euler's formula collapses to

e^{i\pi} = -1, \qquad\text{i.e.}\qquad e^{i\pi} + 1 = 0.

Euler's formula also gives complex numbers their slickest form. Since r(\cos\theta + i\sin\theta) = re^{i\theta}, every complex number is just

z = re^{i\theta},

modulus times a pure rotation. And the rotate-and-scale rule becomes one line of exponent arithmetic — multiplying simply adds the exponents:

r_1 e^{i\theta_1} \cdot r_2 e^{i\theta_2} = r_1 r_2\, e^{i(\theta_1 + \theta_2)}.

Moduli multiply, angles add — exactly the result we ground out from the angle-sum identities, now falling straight out of e^{A}e^{B} = e^{A+B}.

For every real angle \theta:

Euler's identity is famous because it ties together the five most important constants in all of mathematics in a single, spare statement:

e^{i\pi} + 1 = 0.

It contains e (the base of natural growth), i (the imaginary unit), \pi (the geometry of the circle), 1 (the multiplicative identity), and 0 (the additive identity) — each from a different corner of the subject, bound by the three basic operations of addition, multiplication, and exponentiation. Geometrically it says something almost mundane: start at 1 and rotate half a turn (\pi radians) around the unit circle, and you land on -1. The wonder is that analysis (the exponential) and geometry (the circle) were the same thing all along.

Walk the unit circle

Turn the \theta slider and watch the point e^{i\theta} = (\cos\theta,\ \sin\theta) ride around the unit circle. Its horizontal coordinate is \cos\theta, its vertical coordinate is \sin\theta. Stop at \theta = 90° to land on i, and at \theta = 180° to land on -1 — Euler's identity, seen.