Mathematics › 🎓 University Year 1 › The Euler Rotator: e^{iθ} and the Unit Circle

The Euler Rotator: e^{iθ} and the Unit Circle

Raise e to an imaginary power and you do not get a bigger number — you get a point on the unit circle, and multiplying by it becomes pure rotation.

University Year 1Calculus II / Linear Algebra

The Euler Rotator: e^{iθ} and the Unit Circle — illustration

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The big idea: Euler's formula, e^{iθ} = cos θ + i sin θ, is the bridge between exponential growth and circular motion. It says that feeding the exponential an imaginary input produces a unit-length complex number sitting at angle θ. Once you see this, every complex number has a clean polar form r·e^{iθ}, multiplication turns into 'add the angles and multiply the lengths', and rotation is simply multiplication by a phase e^{iθ}. It also delivers the most famous identity in mathematics, e^{iπ} + 1 = 0.

🎯 By the end, you'll be able to

State Euler's formula and read e^{iθ} as a point on the unit circle
Write a complex number in polar form r·e^{iθ}
Multiply and divide complex numbers by adding/subtracting angles and scaling moduli
Interpret multiplication by e^{iθ} as rotation by θ
Derive Euler's identity e^{iπ} + 1 = 0 as a special case

📎 You should already know

Complex numbers and the Argand plane
Radian angle measure
Sine and cosine on the unit circle

An imaginary exponent

Powers of e normally grow: e¹, e², e³ race off to infinity. So what could e raised to an imaginary power possibly mean? The answer is one of the great surprises of mathematics: it does not grow at all. Instead e^iθ lands on the unit circle, at the point whose angle from the positive real axis is exactly θ.

Feeding the exponential an imaginary input converts growth into turning.

🔑 Euler's formula

For any real angle θ (in radians), e^iθ = cos θ + i sin θ. The real part is the horizontal coordinate, the imaginary part is the vertical one, so e^iθ is the point on the unit circle at angle θ. Its distance from the origin is always |e^iθ| = 1.

\[ e^{i\theta} = \cos\theta + i\sin\theta \]

Euler's formula: the imaginary exponential traces the unit circle as θ increases.

🎮 Euler Rotator LIVE

e^{i theta} rides the unit circle — rotation is just multiplication by a phase.

Polar form: every complex number is a length and an angle

A complex number z sits somewhere in the plane, so it has a distance from the origin (its modulus r = |z|) and a direction (its argument θ). Euler's formula lets you fold both into one compact expression, the polar form z = r e^iθ. The clumsy x + iy is great for adding; the polar form is unbeatable for multiplying.

\[ z = r\,e^{i\theta} = r(\cos\theta + i\sin\theta), \qquad r = |z|,\; \theta = \arg z \]

Polar form packages a complex number's length and angle together.

Multiplication becomes rotation

Because exponents add, multiplying two complex numbers in polar form is effortless: r₁e^iθ₁ · r₂e^iθ₂ = r₁r₂ e^{i(θ₁+θ₂)}. You multiply the lengths and add the angles. In particular, multiplying any number by e^iθ (length 1) leaves its size alone and simply rotates it by θ.

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

Multiply moduli, add arguments. Multiplying by a unit phase e^{iθ} is pure rotation by θ.

📝 Worked example: Write 1 + i in polar form, then square it using Euler's formula.

Modulus: \( r = \sqrt{1^2 + 1^2} = \sqrt{2} \). Argument: the point (1, 1) is at 45°, so \( \theta = \pi/4 \).
Polar form: \( 1 + i = \sqrt{2}\,e^{i\pi/4} \).
Square it: multiply moduli and add angles — \( (\sqrt2)^2\,e^{i(\pi/4 + \pi/4)} = 2\,e^{i\pi/2} \).
Convert back: \( 2\,e^{i\pi/2} = 2(\cos\tfrac{\pi}{2} + i\sin\tfrac{\pi}{2}) = 2i \).

✓ (1 + i)² = 2i — and directly, (1 + i)² = 1 + 2i + i² = 2i, which agrees.

✨ The most beautiful identity

Set θ = π in Euler's formula: e^iπ = cos π + i sin π = −1 + 0. Rearranged, e^iπ + 1 = 0 — a single equation tying together e, i, π, 1 and 0. Geometrically it just says that rotating the number 1 by half a turn (π radians) lands you on −1.

⚠️ θ is in radians, and it wraps

Euler's formula uses radians, not degrees — e^iπ is a half-turn, not a tiny angle. And because a full turn is 2π, the angle is only defined up to multiples of 2π: e^iθ = e^i(θ+2π). The same point on the circle has infinitely many valid arguments.

Check your understanding

1. What is |e^{iθ}| for any real θ?

e^{iθ} = cos θ + i sin θ lies on the unit circle, so its modulus is √(cos²θ + sin²θ) = 1.

2. What does multiplying a complex number by e^{iθ} do to it?

e^{iθ} has modulus 1 and argument θ, so multiplying by it adds θ to the argument while leaving the modulus unchanged — pure rotation.

3. What is the value of e^{iπ}?

e^{iπ} = cos π + i sin π = −1 + 0i = −1, the half-turn of the number 1.

4. In polar form, what is 1 + i?

|1 + i| = √2 and its angle is 45° = π/4, so 1 + i = √2·e^{iπ/4}.

5. The product (2·e^{iπ/6})(3·e^{iπ/3}) equals…

Multiply the moduli (2·3 = 6) and add the arguments (π/6 + π/3 = π/2): 6·e^{iπ/2}.

✅ Key takeaways

Euler's formula e^{iθ} = cos θ + i sin θ places the imaginary exponential on the unit circle at angle θ, with |e^{iθ}| = 1.
Every complex number has a polar form z = r·e^{iθ}, packaging its modulus r and argument θ.
Multiplication multiplies moduli and adds arguments, so multiplying by e^{iθ} is pure rotation by θ.
Setting θ = π gives Euler's identity e^{iπ} + 1 = 0, a half-turn of the number 1.
The angle is measured in radians and is defined only up to multiples of 2π.