Represent complex numbers in Cartesian and polar form, perform arithmetic, and apply de Moivre's theorem

What this dot point is asking

SCSA wants you to move fluently between Cartesian form $z = x + iy$ and polar (modulus-argument) form, plot numbers on the Argand plane, carry out all four arithmetic operations, use the conjugate, and apply de Moivre's theorem to find powers and the $n$ roots of a complex number.

Cartesian form and arithmetic

Write $z = x + iy$ where $x = \operatorname{Re}(z)$ and $y = \operatorname{Im}(z)$, and $i^2 = -1$. Addition and subtraction act component-wise. Multiplication uses the distributive law with $i^2 = -1$:

The conjugate of $z = x + iy$ is $\bar z = x - iy$. The product $z\bar z = x^2 + y^2 = |z|^2$ is real and non-negative, which is the key to division: multiply numerator and denominator by the conjugate of the denominator.

The Argand plane, modulus and argument

Plot $z = x + iy$ as the point $(x, y)$. The modulus $|z| = \sqrt{x^2 + y^2}$ is the distance from the origin. The argument $\arg z$ is the angle measured anticlockwise from the positive real axis. The principal argument $\operatorname{Arg} z$ lies in $(-\pi, \pi]$. Always check the quadrant: a bare $\tan^{-1}(y/x)$ can land in the wrong half-plane.

Polar form and de Moivre's theorem

In polar form $z = r\,\text{cis}\,\theta$ where $\text{cis}\,\theta = \cos\theta + i\sin\theta$. Multiplication multiplies moduli and adds arguments; division divides moduli and subtracts arguments:

For the $n$ distinct $n$th roots of $w = R\,\text{cis}\,\phi$, solve $z^n = w$:

The roots all have modulus $R^{1/n}$ and are spaced $\tfrac{2\pi}{n}$ apart, so they sit on the vertices of a regular polygon.