Foundations: What It Is, and What It Is Not

Complex numbers as rotation and phase

Magnitude, angle, and that multiplying rotates

~11 minLesson 4 of 24Written by Claude Fable 5

The math track on this site never covered complex numbers, for a simple reason: nothing there needed them. Quantum computing does, but only as geometry. A quantum gate turns out to be a rotation, and rotating arrows on a plane is exactly what complex numbers are built to do. You need just three facts, and this lesson is the only place they are taught.

Nothing imaginary about it

The name is the whole problem. Calling

i

an "imaginary" number makes it sound like a ghost that does not really exist. Forget that. A complex number is nothing more exotic than a point on a flat plane - a pair of coordinates, one measured rightward and one measured upward. The number

i

is simply the point one unit up from the origin. That is it. Everything spooky about it dissolves the moment you draw it.^[1]

A complex number is a point

Write a complex number as $z = x + i y$ and read it as the point $(x, y)$ : go $x$ to the right along the real axis and $y$ up along the imaginary axis. Two numbers describe that point just as well, and they are the two that matter for us. Its is its distance from the origin, and its (or phase) is the angle it makes with the rightward axis. Distance and direction - that is the whole description:

z = r (cos θ + i sin θ), r = ∣ z ∣, θ = ar g z Polar form: a point described by how far out and which way around.

There is a famous shorthand for the part in parentheses, and we will just state it - packs the whole "point at angle $θ$ on the unit circle" into one symbol:

e^{i θ} = cos θ + i sin θ, z = r e^{i θ}

You do not need to derive Euler's formula or even fully trust it yet. Read $e^{i θ}$ as a name for "the point on the unit circle at angle $θ$ ." Because it sits on the unit circle, its magnitude is always exactly one: $∣ e^{i θ} ∣ = 1$ .

The one rule that matters: multiplying rotates

Here is the single fact everything downstream rests on. When you multiply two complex numbers, their magnitudes multiply and their angles add:

(r_{1} e^{i θ_{1}}) (r_{2} e^{i θ_{2}}) = r_{1} r_{2} e^{i (θ_{1} + θ_{2})} Multiplication in polar form: multiply the lengths, add the angles.

Drag the two points below and watch it happen. The product $z_{1} z_{2}$ always lands at the summed angle, with the multiplied length. Then turn on phase only, which pins both points to the unit circle so every magnitude is one.

Two points on the plane, z1 and z2, and their product. Drag either point: the product lands at the sum of the angles, with magnitudes multiplied.

phase only (lock both magnitudes to 1: multiplying is a pure rotation)

z1 angle = 30°z1 magnitude = 1.00

z2 angle = 60°z2 magnitude = 1.00

z1 · z2 magnitude = 1.00 × 1.00 = 1.00

z1 · z2 angle = 30° + 60° = 90°

With both magnitudes equal to one, the lengths multiply to $1 \times 1 = 1$ and nothing about the size changes - multiplying only turns the arrow. That is the punchline: multiplying by $e^{i θ}$ is a pure rotation by $θ$ , never a stretch. Physicists call that angle the phase, and turning it is all a phase does.

Why a qubit cares

A quantum amplitude is a complex number, so it too has a magnitude and an angle. Its magnitude squared is the probability you already met, and its angle is a phase you cannot see directly - but which decides how amplitudes add. Two arrows of length one pointing the same way add to a long arrow; pointing opposite ways they cancel. Rotating amplitudes by phases and then adding them is precisely how interference is built, and it is why gates, coming up when they become matrices that rotate a qubit, are rotations and nothing more frightening.

import numpy as np

z1 = 1.0 * np.exp(1j * np.deg2rad(30))   # length 1, angle 30 deg
z2 = 1.0 * np.exp(1j * np.deg2rad(60))   # length 1, angle 60 deg

prod = z1 * z2
print(round(abs(prod), 3))               # 1.0   -> lengths multiplied: 1 * 1
print(round(np.rad2deg(np.angle(prod)))) # 90    -> angles added: 30 + 60
# multiplying by a unit-length e^{i theta} only rotates, never rescales

Multiplying complex numbers: the product's magnitude is the product of magnitudes, its angle the sum of angles.

Lock it in

A complex number is just a point on a plane: its magnitude is the distance from the origin, its argument (phase) is the angle. Nothing is imaginary about a point.
Polar form writes it as $r e^{i θ}$ , where $e^{i θ}$ is the unit-circle point at angle theta and has magnitude 1.
Multiplying two complex numbers multiplies their magnitudes and adds their angles.
Multiplying by $e^{i θ}$ is a pure rotation by theta, which is exactly why quantum gates are rotations and phases drive interference.

Check yourself

What does multiplying a complex number by e^{i theta} do to it, geometrically?

You multiply z1 (angle 40 degrees) by z2 (angle 25 degrees). What is the angle of the product?

Recall: the polar multiplication rule, and what e^{i theta} does geometrically.

One rule for products, one picture for phase. Try to state it, then check.

Match each symbol to what it means on the plane.

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the distance from the origin to the point

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the angle the point makes with the real axis

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a pure rotation: the unit-circle point at angle theta

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magnitudes multiply and angles add

Primary source

3Blue1Brown - Complex numbers

For the visual intuition behind every claim here - points on a plane, multiplication as rotation, and where Euler's formula comes from - watch Grant Sanderson build it from the ground up. It is the fastest way to make $e^{i θ}$ feel obvious rather than mystical.

Sources