Game plan 1: Complex numbers | Specialist external game plans

Part 1

Converting form

Moving between $a+bi$ and $r\,\text{cis}\,\theta$ comes up in nearly every complex numbers question.

Game plan 1.1

Converting between Cartesian and polar form

1Plot the number first. The quadrant decides the argument, and skipping the sketch is the most common mistake.
2Find the modulus: $r=\sqrt{a^2+b^2}$.
3Find the reference angle from $\tan\theta=\left|\frac ba\right|$, then place it in the correct quadrant so that $-\pi<\theta\le\pi$.
4Write $z=r\,\text{cis}\,\theta$. To go the other way, expand $r(\cos\theta+i\sin\theta)$ using exact values.

The reference angle is always acute. The quadrant then decides where the argument is.

Quadrant	Argument
First	$\theta=\text{ref}$
Second	$\theta=\pi-\text{ref}$
Third	$\theta=\text{ref}-\pi$
Fourth	$\theta=-\text{ref}$

In the Topic Guides

Unit 3 Topic 1.1 · The Unit 2 basics you need

Part 2

Arithmetic: which form?

Each operation is easiest in one form. Choose the form before you start.

Game plan 1.2

Choosing the form for the arithmetic

1Adding or subtracting: use Cartesian form and collect the real and imaginary parts.
2Multiplying, dividing or raising to a power: use polar form. Multiply or divide the moduli, add or subtract the arguments, and use De Moivre’s theorem for powers.
3Dividing in Cartesian form, for small cases: multiply the top and bottom by the conjugate of the denominator.
4After polar arithmetic, bring the argument back into $(-\pi,\pi]$ if the question asks for the principal argument.

In the Topic Guides

Unit 3 Topic 1.1 · De Moivre’s theorem

Part 3

nth roots of a complex number

Find one root, then rotate. The picture is the quickest check.

Game plan 1.3

Finding all the roots of $z^n=w$

1Expect $n$ roots, spaced $\frac{2\pi}{n}$ apart.
2Start with $z_0$. For $w=r\,\text{cis}\,\theta$, $z_0=r^{1/n}\,\text{cis}\left(\frac{\theta}{n}\right)$.
3Add $\frac{2\pi}{n}$ to the argument of $z_0$ to get $z_1$, and keep going until you have $n$ roots.
4Plot them. They sit on a circle of radius $r^{1/n}$ and form a regular $n$-gon. For roots of unity the circle is the unit circle and the roots add to zero.

What the picture checks

If your roots aren’t evenly spaced, you either divided the argument by something other than $n$, or you didn’t add a full $\frac{2\pi}{n}$ each time.

All $n$ roots sit on one circle, $\frac{2\pi}{n}$ apart.

Try it

Change $n$ and $w$

Number of roots Modulus of Argument of

The ringed point is $z_0$. The circle is drawn the same size each time. Read its radius from the box.

In the Topic Guides

Part 4

Factorising polynomials over $\mathbb C$

Look at the coefficients before anything else. They decide which tools you are allowed to use.

Game plan 1.4

Factorising a polynomial

1Check the coefficients. With real coefficients, the conjugate root theorem applies and the roots come in conjugate pairs. With an imaginary coefficient it doesn’t, so use the factor theorem and inspection.
2Given one complex root $\alpha$ of a polynomial with real coefficients, $\bar\alpha$ is also a root. Multiply $(z-\alpha)(z-\bar\alpha)$ to get a real quadratic factor.
3Divide the polynomial by that quadratic, by long division or by equating coefficients, to find the other factors.
4Solve any quadratic that is left over $\mathbb C$. A negative discriminant gives a pair of complex roots.

Watch for

The conjugate root theorem doesn’t work when any coefficient is imaginary. The QCAA sets these polynomials on purpose to catch you out.

In the Topic Guides

Part 5

Proving modulus and argument identities

A proof is marked on its set-up as much as its algebra. The form you choose in the first line decides how hard the rest is.

Game plan 1.5

Writing the proof

1State your form in the first line. Cartesian: “Let $z=a+bi$, where $a,b\in\mathbb R$.” Polar: “Let $z=r\,\text{cis}\,\theta$, where $r=|z|$ and $\theta=\arg z$.”
2Never start from what you are proving. Work one side into the other, or reduce each side to the same expression.
3Never use the identity you are proving as a step in its own proof.
4Choose the form from the type of identity, using the table below.
5Finish with a sentence saying the identity has been shown.

Type	You are asked to prove	Use
A sum or difference inside a modulus	$\|z\pm w\|$, $\|z\pm w\|^2$	Cartesian, always. Substitute the components, expand and regroup.
Products, quotients, powers and arguments	$\|z_1z_2\|$, $\left\|\frac{z_1}{z_2}\right\|$, $\|z^n\|$, $\arg(z_1z_2)$	Polar. Let $z_1=r_1\,\text{cis}\,\theta_1$ and $z_2=r_2\,\text{cis}\,\theta_2$, multiply, then read off the modulus and the argument. One set-up proves both.
Conjugate facts	$z\bar z=\|z\|^2$, $z+\bar z=2\,\text{Re}(z)$	Cartesian. These are statements about components, so substitute and expand.

New in 2025

Argument identities are new to the 2025 syllabus and haven’t appeared in a past paper yet, so expect them.

In the Topic Guides

Unit 3 Topic 1.1 · Proofs of complex number identities

Formulas

Formula panel

Each formula is labelled with how to treat it in the exam.

$z=a+bi=r(\cos\theta+i\sin\theta)=r\,\text{cis}\,\theta$

On the sheet

$|z|=r=\sqrt{a^2+b^2}$, $\ \text{Arg}(z)=\theta$, $\ -\pi<\theta\le\pi$

On the sheet

$z_1z_2=r_1r_2\,\text{cis}(\theta_1+\theta_2)$, $\ \dfrac{z_1}{z_2}=\dfrac{r_1}{r_2}\,\text{cis}(\theta_1-\theta_2)$

On the sheet

De Moivre’s theorem: $z^n=r^n\,\text{cis}(n\theta)$

On the sheet

Factor theorem: $(z-a)$ is a factor of $P(z)$ if and only if $P(a)=0$

Memorise

Remainder theorem: the remainder of $P(z)\div(z-a)$ is $P(a)$

Memorise

Conjugate root theorem, for real coefficients only

Memorise

$z\bar z=|z|^2$, $\ z+\bar z=2\,\text{Re}(z)$, $\ |\bar z|=|z|$, $\ \arg(\bar z)=-\arg(z)$

Memorise

Exact values of $\sin$, $\cos$ and $\tan$ at $0$, $\frac\pi6$, $\frac\pi4$, $\frac\pi3$ and $\frac\pi2$

Memorise

$z=r^{1/n}\,\text{cis}\left(\dfrac{\theta+2k\pi}{n}\right)$, $\ k=0,1,\dots,n-1$. This is Game plan 1.3 written as one formula.

Derive

QCAA language

What the question says, and what it means

Question type	What you’ll see	What it means
Converting form	“Express in the form $a+bi$” or “in the form $r\,\text{cis}\,\theta$”	Game plan 1.1, with exact values
Polar arithmetic	Powers, products and quotients of $\text{cis}$	Game plan 1.2. Use polar form
Roots	$z^n=w$, “roots of unity”, “solutions on the unit circle”	Game plan 1.3. Find $z_0$, then rotate by $\frac{2\pi}{n}$
Factorising	“$P(z)$ has a factor…”, “given that $z=\alpha$ is a root”	Game plan 1.4. Check the coefficients first
Proof	“Show that $\|z_1z_2\|=\|z_1\|\|z_2\|$”	Game plan 1.5. State your form first

Habit

Plot before you find any argument. Then use every condition in the question, such as $w\notin\mathbb R$ or $a,b\in\mathbb Z^+$. Every complex numbers question since 2020 has had one, and it is there to decide something.