Represent complex numbers on the Argand plane and find modulus and argument, converting to polar form

What this dot point is asking

SCSA wants you to plot complex numbers, compute modulus and argument correctly with attention to quadrant, and convert between Cartesian and polar form.

The Argand plane

The Argand plane represents $z = x + iy$ as the point $(x, y)$, with the horizontal axis the real axis and the vertical axis the imaginary axis. Addition corresponds to vector addition and conjugation corresponds to reflection in the real axis. This geometric picture is what makes polar form and loci natural.

Modulus

The modulus is multiplicative: $|zw| = |z|\,|w|$ and $\left|\tfrac{z}{w}\right| = \tfrac{|z|}{|w|}$. It also satisfies $|z|^2 = z\bar z$ and the triangle inequality $|z + w| \le |z| + |w|$.

Argument

The argument of $z$ is the angle $\theta$, measured anticlockwise from the positive real axis, such that $x = |z|\cos\theta$ and $y = |z|\sin\theta$. Because adding $2\pi$ returns the same point, the argument is only defined up to multiples of $2\pi$. The principal argument $\operatorname{Arg} z$ is the unique value in $(-\pi, \pi]$.

To find it, compute the reference angle $\alpha = \tan^{-1}\!\left|\tfrac{y}{x}\right|$ and then place it in the correct quadrant by inspecting the signs of $x$ and $y$. A sketch settles the sign every time.

Conversion to polar form

Once you have $r = |z|$ and $\theta = \arg z$, the polar (modulus-argument) form is

Going the other way, $x = r\cos\theta$ and $y = r\sin\theta$ recovers Cartesian form.