HSC Maths Complex Numbers: The Ext 2 Survival Guide

Last updated: 24 July 2026

Complex numbers are the first thing you learn in Extension 2 and they never leave. They turn up in polynomials, vectors, calculus, proofs. If you nail them early, the rest of Ext 2 slots into place.

The big idea

A complex number has a real part and an imaginary part: $z = x + iy$, where $i^2 = -1$.

That $i$ is the whole trick. Every complex number is a point on the Argand diagram — x-axis is real, y-axis is imaginary.

Modulus and argument

The modulus ($|z|$) is the distance from the origin — Pythagoras again. The argument ($\\text{arg } z$) is the angle from the positive real axis.

$|z| = \\sqrt{x^2 + y^2}$ $\\text{arg } z = \\tan^{-1}(y/x)$

These two things let you switch between Cartesian ($x + iy$) and modulus-argument (polar) form. The polar form is:

$z = r(\\cos\\theta + i\\sin\\theta)$

where $r = |z|$ and $\\theta = \\text{arg } z$.

Why polar form matters

Multiplying and dividing complex numbers is tedious in Cartesian. In polar form it's clean:

• Multiply: multiply the moduli, add the arguments
• Divide: divide the moduli, subtract the arguments
• Powers: raise the modulus to the power, multiply the argument

That last one is De Moivre's theorem:

$(\\cos\\theta + i\\sin\\theta)^n = \\cos(n\\theta) + i\\sin(n\\theta)$

De Moivre's theorem — the workhorse

This theorem is how you find powers of complex numbers, roots of complex numbers, and trig identities.

Finding roots: To find the $n$th roots of a complex number, raise the modulus to $1/n$ and divide the argument by $n$. Add $2\\pi$ for each successive root. The roots are equally spaced around the Argand circle.

Trig identities: Expand $(\\cos\\theta + i\\sin\\theta)^n$ using the binomial theorem, equate real and imaginary parts, and you get $\\cos(n\\theta)$ and $\\sin(n\\theta)$ expressed in powers of $\\cos\\theta$ and $\\sin\\theta$.

Conjugates

The conjugate of $x + iy$ is $x - iy$. Visually, it's a reflection across the real axis.

Key facts:

• $z \\times \\overline{z} = |z|^2$
• $\\overline{z_1 + z_2} = \\overline{z_1} + \\overline{z_2}$
• The conjugate of a product is the product of the conjugates
• Real coefficients mean roots come in conjugate pairs

Polynomials with complex roots

If a polynomial has real coefficients and one complex root $a + ib$, then $a - ib$ is also a root. This is why cubics always have at least one real root — the complex ones come in pairs.

You'll use this to factor polynomials that look unfactorable and to find all roots of a quartic when you're given one complex root.

The loci questions

"Sketch the region where $|z - (1 + i)| < 2$" — this is describing a circle centred at $(1, 1)$ with radius 2. Loci questions test whether you see geometry in the algebra.

Common ones:

• $|z - z_0| = r$ — circle centre $z_0$, radius $r$
• $|z - z_1| = |z - z_2|$ — perpendicular bisector of the segment joining $z_1$ and $z_2$
• $\\text{arg}(z - z_0) = \\theta$ — ray from $z_0$ at angle $\\theta$

How to get good

Complex numbers snowball. If you're shaky on polar form, De Moivre's will be miserable. Practise the conversions first, then the theorems, then the loci.

Get topic-by-topic complex number practice with instant solutions on PracticePapers.io, free during beta.

Memory aid

• Cartesian → point on a plane
• Polar → angle and distance from origin
• De Moivre → powers are easy in polar
• Conjugate pairs → real coefficients always means paired roots