Express complex numbers in modulus-argument form and use it to multiply and divide, interpreting these operations geometrically as scaling and rotation

What this dot point is asking

NESA wants you to express a complex number by its distance from the origin and the angle it makes with the positive real axis, the modulus-argument (polar) form. You must convert between Cartesian and polar form, identify the principal argument, and use polar form to multiply and divide complex numbers, understanding these as scaling and rotation in the Argand plane.

Modulus and argument

For $z = x + iy$, the modulus is the distance from the origin,

and the argument $\theta = \arg z$ is the angle measured anticlockwise from the positive real axis to the ray from the origin to $z$. The principal argument is the value in the interval $(-\pi, \pi]$. The Cartesian and polar coordinates are linked by

so that

This is often abbreviated $z = r\operatorname{cis}\theta$, where $\operatorname{cis}\theta = \cos\theta + i\sin\theta$.

Finding the argument correctly

The relation $\tan\theta = \dfrac{y}{x}$ does not by itself fix $\theta$, because the calculator inverse tangent only returns angles in $(-\tfrac{\pi}{2}, \tfrac{\pi}{2})$. You must use the signs of $x$ and $y$ to place $z$ in the correct quadrant. For example, $z = -1 + i$ lies in the second quadrant, so its argument is $\dfrac{3\pi}{4}$, not $-\dfrac{\pi}{4}$. Always sketch the point in the Argand plane before committing to an argument.

Multiplication: scaling and rotation

If $z_1 = r_1\operatorname{cis}\theta_1$ and $z_2 = r_2\operatorname{cis}\theta_2$, then using the angle-sum identities,

So multiplying by $z_2$ scales the modulus by $r_2$ and rotates the point anticlockwise by $\theta_2$. In particular, multiplying by $i = \operatorname{cis}\tfrac{\pi}{2}$ is a rotation by $90^\circ$ with no change in length, and multiplying by a real $k > 0$ is a pure scaling.

Division: subtraction of arguments

Dividing reverses the process:

The modulus of the quotient is the ratio of moduli, and the argument is the difference of arguments. These rules make repeated multiplication and powers far easier than expanding in Cartesian form, and they lead directly into de Moivre's theorem.