Modulus and Argument

Every complex number $z = a + bi$ has a modulus (or magnitude)

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

the distance from the origin in the complex plane, and an argument $\arg(z) = \operatorname{atan2}(b, a)$, the angle it makes with the positive real axis. Together they give the polar form $z = |z|\,e^{i\arg(z)}$.

The modulus is exactly what the Born rule needs: if a quantum amplitude is $z$, the corresponding probability is $|z|^2$. So fluency with $|z|$ is fluency with quantum measurement.

Try it

This is a numerical exercise — your code should return a number, not a circuit. Compute the modulus of $z = 3 + 4i$ (the classic 3–4–5 triangle).

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