Just enough complex numbers

In the warmup lesson we introduced complex numbers quickly because we needed them to talk about qubits. This lesson gives them a proper tour. Nothing here is new math — we’re just making sure the basics are solid before they start doing heavy lifting.

A complex number is a point on a plane

Real numbers live on a line: $-1, 0, 1, \pi$. One number, one position.

A complex number lives on a plane. It has two pieces — a real part and an imaginary part — and you can think of it as an arrow from the origin to a point, or equivalently, as the point itself.

$z = a + bi$

Here $a$ is the real part and $b$ is the imaginary part. The $i$ at the end is the imaginary unit, defined by:

$i^2 = -1$

That one rule is the whole of what makes complex numbers different from real numbers. Everything else follows.

Arithmetic

Adding complex numbers is just adding the parts:

$(a + bi) + (c + di) = (a + c) + (b + d)i$

It’s literally adding arrows. Tail-to-head addition, just like vectors in 2D.

Multiplication is where it gets interesting. Using the $i^2 = -1$ rule:

$(a + bi)(c + di) = ac + adi + bci + bdi^2 = (ac - bd) + (ad + bc)i$

Don’t memorize this formula. Memorize the trick — multiply it out like polynomials and then collapse $i^2$ to $-1$.

Polar form: the real lesson

The Cartesian form $a + bi$ is fine for adding. But for quantum mechanics, we care more about the polar form:

$z = r \cdot e^{i\varphi}$

Here:

• $r = |z| = \sqrt{a^2 + b^2}$ is the magnitude (length of the arrow)
• $\varphi$ is the phase (angle the arrow makes with the positive real axis)

In Cartesian form you’re saying “go right $a$ units and up $b$ units.” In polar form you’re saying “go $r$ units at angle $\varphi$.” Same point, different description.

The identity that connects the two is Euler’s formula:

$e^{i\varphi} = \cos\varphi + i\sin\varphi$

So $r \cdot e^{i\varphi} = r\cos\varphi + ir\sin\varphi$, which gives you Cartesian coordinates $(r\cos\varphi, r\sin\varphi)$ — exactly the formula for converting a polar point to Cartesian.

Why polar form matters for quantum computing

In polar form, multiplication becomes a much simpler operation:

$r_1 e^{i\varphi_1} \cdot r_2 e^{i\varphi_2} = (r_1 r_2) e^{i(\varphi_1 + \varphi_2)}$

Magnitudes multiply, phases add. That’s it. Compare this to the Cartesian formula above (which has four separate terms and a sign flip) and you’ll see why physicists prefer polar form when they can.

In particular, multiplying any complex number by $e^{i\theta}$ — a complex number of magnitude $1$ — just rotates it in the plane by angle $\theta$, without changing its magnitude. That’s the key geometric operation behind quantum gates.

The magnitude squared gives you probability

For a complex amplitude $\alpha = a + bi$:

$|\alpha|^2 = \alpha \cdot \bar{\alpha} = a^2 + b^2$

where $\bar\alpha = a - bi$ is the complex conjugate. This is always a non-negative real number, and in the polar form it equals $r^2$.

This is the quantity that becomes a probability via the Born rule. No matter how much you multiply an amplitude by a phase $e^{i\theta}$, the magnitude squared doesn’t change — which is why global phases are invisible to measurements, as we’ll revisit in Lesson 2.4.

That’s it

Addition, multiplication, polar form, $e^{i\varphi}$ is a rotation, $|z|^2$ is a real non-negative number. With those five facts you have everything needed for the rest of the course. We’ll pick up a couple more identities (like $\cos^2 + \sin^2 = 1$ and the Euler formulas) when they appear, but nothing more mysterious than what you just saw.