The complex plane, modulus and argument (WACE Year 12 Mathematics Specialist Unit 3)

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.

Quadrant-by-quadrant argument rules

It pays to know the adjustment in each quadrant once you have the reference angle $\alpha = \tan^{-1}\!\left|\dfrac{y}{x}\right|$. In the first quadrant ($x > 0$, $y > 0$) the argument is $\alpha$. In the second quadrant ($x < 0$, $y > 0$) it is $\pi - \alpha$. In the third quadrant ($x < 0$, $y < 0$) it is $-(\pi - \alpha)$, i.e. $\alpha - \pi$. In the fourth quadrant ($x > 0$, $y < 0$) it is $-\alpha$. On the axes, read the angle directly: $\arg(\text{positive real}) = 0$, $\arg(\text{positive imaginary}) = \dfrac{\pi}{2}$, $\arg(\text{negative real}) = \pi$, and $\arg(\text{negative imaginary}) = -\dfrac{\pi}{2}$.

Modulus as a distance and the triangle inequality

Because $|z - w|$ is the distance between the points $z$ and $w$ in the Argand plane, the modulus is the bridge between algebra and geometry that powers the loci topic. The triangle inequality $|z + w| \le |z| + |w|$ is the statement that one side of a triangle cannot exceed the sum of the other two, with equality only when $z$ and $w$ point the same way. The reverse triangle inequality $\big||z| - |w|\big| \le |z - w|$ is also worth knowing for bounding arguments. These distance interpretations recur whenever a question mixes modulus conditions with geometry.