Checkpoint: Complex Number Workout

This checkpoint pulls together every complex-number skill from Module 1. Each sub-problem below targets one concept that quantum mechanics uses daily — read the derivations, then put them into code.

(A) Modulus and the Born rule

The modulus of $z = a + bi$ is the distance from the origin in the complex plane:

$|z| = \sqrt{a^2 + b^2}.$

For $z = 3 + 4i$ : $|z| = \sqrt{9 + 16} = \sqrt{25} = 5$ .

The Born rule ties modulus to physics: if a quantum amplitude is $z$ , the probability of that outcome is $|z|^2$ . Fluency with moduli is therefore fluency with quantum measurement.

(B) Complex multiplication and conjugates

Multiplying two complex numbers $(a + bi)(c + di) = (ac - bd) + (ad + bc)i$ follows from treating $i^2 = -1$ . Squaring $2 + i$ :

$(2 + i)^2 = 2^2 + 2\cdot 2\cdot i + i^2 = 4 + 4i - 1 = 3 + 4i.$

The real part is $3$ .

The complex conjugate $z^* = a - bi$ flips the imaginary sign. A key identity is $z \cdot z^* = |z|^2$ , which is why inner products in quantum mechanics always pair a bra with a ket: $\langle\psi|\phi\rangle$ implicitly conjugates one factor.

(C) Euler's formula and phases

Euler's formula is the bridge between exponentials and trigonometry:

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

So $e^{i\theta}$ has real part $\cos\theta$ and imaginary part $\sin\theta$ . For $\theta = \pi/3$ :

$\operatorname{Re}\!\left(e^{i\pi/3}\right) = \cos\!\left(\frac{\pi}{3}\right) = \frac{1}{2}.$

Quantum states carry phase factors of the form $e^{i\theta}$ . Because $|e^{i\theta}|^2 = \cos^2\theta + \sin^2\theta = 1$ , a global phase never changes measurement probabilities — only relative phases between terms matter.

(D) Roots of unity and the unit circle

The $n$ -th roots of unity solve $z^n = 1$ . There are exactly $n$ of them:

$\omega_k = e^{2\pi i k/n}, \quad k = 0, 1, \ldots, n-1.$

They are equally spaced on the unit circle, so every root of unity has modulus 1. For $n = 5$ the primitive root is $\omega = e^{2\pi i/5}$ , and

$|\omega| = \left|e^{2\pi i/5}\right| = \sqrt{\cos^2\!\left(\tfrac{2\pi}{5}\right) + \sin^2\!\left(\tfrac{2\pi}{5}\right)} = 1.$

Roots of unity appear in the quantum Fourier transform, the backbone of Shor's factoring algorithm.

Try it

Compute the four answers and return their sum. The four values are: (A) $|3+4i|$ , (B) $\operatorname{Re}((2+i)^2)$ , (C) $\operatorname{Re}(e^{i\pi/3})$ , and (D) $|e^{2\pi i/5}|$ .

Run your code to see the quantum state.