Complex Multiplication Is Rotation

On the complex plane, adding complex numbers is just adding arrows. But multiplication hides a small miracle — and it is the whole reason complex numbers are the natural language of rotation. The secret is to write the numbers in polar form z = r(\cos\theta + i\sin\theta) and watch what a product does to the modulus r and the angle \theta.

Multiplying in polar form

Take two numbers in polar form, z_1 = r_1(\cos\theta_1 + i\sin\theta_1) and z_2 = r_2(\cos\theta_2 + i\sin\theta_2), and multiply them out.

Step 1 — pull the moduli to the front. The real factors r_1 and r_2 just multiply:

z_1 z_2 = r_1 r_2 (\cos\theta_1 + i\sin\theta_1)(\cos\theta_2 + i\sin\theta_2).

Step 2 — expand the two brackets (every term times every term):

= r_1 r_2\big(\cos\theta_1\cos\theta_2 + i\cos\theta_1\sin\theta_2 + i\sin\theta_1\cos\theta_2 + i^2\sin\theta_1\sin\theta_2\big).

Step 3 — use i^2 = -1 on the last term, turning +i^2\sin\theta_1\sin\theta_2 into -\sin\theta_1\sin\theta_2:

= r_1 r_2\big[(\cos\theta_1\cos\theta_2 - \sin\theta_1\sin\theta_2) + i(\sin\theta_1\cos\theta_2 + \cos\theta_1\sin\theta_2)\big].

Step 4 — recognise the angle-sum identities. Those two bracketed groups are exactly \cos(\theta_1 + \theta_2) and \sin(\theta_1 + \theta_2):

\cos\theta_1\cos\theta_2 - \sin\theta_1\sin\theta_2 = \cos(\theta_1 + \theta_2), \qquad \sin\theta_1\cos\theta_2 + \cos\theta_1\sin\theta_2 = \sin(\theta_1 + \theta_2).

Step 5 — read off the result. The product is back in polar form:

z_1 z_2 = r_1 r_2\big(\cos(\theta_1 + \theta_2) + i\sin(\theta_1 + \theta_2)\big).

Stare at that for a moment. To multiply two complex numbers, you multiply the moduli and add the angles. Lengths multiply; directions add.

So multiplication is "rotate and scale"

Now specialise. Suppose z_2 has modulus 1 — a unit-modulus number \cos\theta + i\sin\theta sitting on the unit circle. Multiplying any z by it leaves |z| unchanged (since r_2 = 1) and adds \theta to the argument:

z \cdot (\cos\theta + i\sin\theta) \;=\; \text{rotate } z \text{ by the angle } \theta.

Multiplying by a unit-modulus number rotates the plane. The cleanest example is i itself: since i = \cos 90° + i\sin 90°, multiplying by i is a quarter turn. Check it directly on z = a + bi:

i(a + bi) = ai + bi^2 = -b + ai,

which sends the point (a, b) to (-b, a) — precisely a 90° anticlockwise rotation. Multiply by i four times and you turn 360° back to the start, matching i^4 = 1.

Write z_1 = r_1(\cos\theta_1 + i\sin\theta_1) and z_2 = r_2(\cos\theta_2 + i\sin\theta_2). Then:

The rule "multiply by \cos\theta + i\sin\theta to rotate" is the complex twin of the rotation matrix. Write z = a + bi as the column \begin{pmatrix} a \\ b \end{pmatrix} and multiply out (\cos\theta + i\sin\theta)(a + bi); its real and imaginary parts are

(a\cos\theta - b\sin\theta) + i(a\sin\theta + b\cos\theta),

which is exactly

\begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}\begin{pmatrix} a \\ b \end{pmatrix}.

One complex multiplication packages the entire 2 \times 2 rotation matrix into a single number on the unit circle. That compression is why complex numbers feel so effortless for anything spinning.

A complex number rotates the plane with a single multiplication. The obvious question is whether the same trick works in three dimensions — and it does, but you need more imaginary units. The quaternions add three of them, i, j, k, all squaring to -1, and multiplying by a unit quaternion rotates 3-D space the way multiplying by \cos\theta + i\sin\theta rotates the plane. Game engines and spacecraft use them for exactly this reason: the rotation lives in one tidy product, with no axes to flip or gimbals to lock.

Spin a number with the angle slider

The blue arrow is a fixed number z. The orange arrow is the product z \cdot (\cos\theta + i\sin\theta). Drag the \theta slider and watch the product swing around the origin: its length never changes (multiplying by a unit-modulus number doesn't scale), only its direction — the angle \theta is being added to the argument. Set \theta = 90° to see the quarter-turn of multiplying by i.