What is not multiplying by the conjugate to divide?

A quotient needs the denominator rationalised by its conjugate to reach a + bi form.

Write z = 4e^iπ/2 in Cartesian form. [1 mark]

SEAB Original-style practice: Given z_1 = 3 + 2i and z_2 = 1 - i, find z_1 z_2 and z_1z_2 in the form a + bi.

Product: z_1 z_2 = (3 + 2i)(1 - i) = 3 - 3i + 2i - 2i^2 = 3 - i + 2 = 5 - i (using i^2 = -1). Quotient: multiply numerator and denominator by the conjugate of z_2, namely 1 + i: $(z_1)/(z_2) = ((3 + 2i)(1 + i))/((1 - i)(1 + i)) = (3 + 3i + 2i + 2i^2)/(1 + 1) = (3 + 5i - 2)/(2) = (1 + 5i)/(2) = 12 + 52i.$ Markers reward the product with i^2 = -1 applied, multiplying by the conjugate to rationalise the denominator, and both answers in a + bi form.

SingaporeFurther MathsSyllabus dot point

How do we represent complex numbers in Cartesian, polar and exponential form, and what does the Argand diagram show?

Represent complex numbers in Cartesian, polar and exponential form, perform arithmetic, and interpret them on the Argand diagram

A focused answer to the H2 Further Mathematics outcome on complex numbers. Cartesian, polar (modulus-argument) and exponential forms, conjugates, arithmetic, the geometry of the Argand diagram, and converting between forms.

Generated by Claude Opus 4.811 min answerUpdated 2026-06-06

Reviewed by: AI editorial process; not yet individually human-reviewed

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What this dot point is asking
The answer
Examples in context
Try this

What this dot point is asking

SEAB wants you to represent a complex number in three equivalent forms, move freely between them, carry out arithmetic, and interpret a complex number geometrically as a point or vector on the Argand diagram. The forms each suit different tasks: Cartesian for addition, polar and exponential for multiplication and powers.

The answer

Cartesian form

A complex number is $z = x + iy$ where $x = \operatorname{Re}(z)$ , $y = \operatorname{Im}(z)$ , and $i^2 = -1$ . Addition and subtraction are componentwise. The conjugate is $\bar{z} = x - iy$ , and $z\bar{z} = x^2 + y^2 = |z|^2$ , which is why multiplying by the conjugate rationalises a denominator.

The modulus and argument

The modulus is the distance from the origin,

|z| = \sqrt{x^2 + y^2},

and the argument $\arg z = \theta$ is the angle the point makes with the positive real axis, taken in the principal range $-\pi < \theta \leq \pi$ . Always check the quadrant before quoting $\theta$ , since $\tan\theta$ alone is ambiguous.

Polar (modulus-argument) form

z = r(\cos\theta + i\sin\theta), \qquad r = |z|, \quad \theta = \arg z.

Exponential form

Euler's relation $\mathrm{e}^{i\theta} = \cos\theta + i\sin\theta$ gives the compact

z = r\,\mathrm{e}^{i\theta}.

This form makes multiplication and powers immediate: $r_1\mathrm{e}^{i\theta_1}\cdot r_2\mathrm{e}^{i\theta_2} = r_1 r_2\,\mathrm{e}^{i(\theta_1 + \theta_2)}$ .

The Argand diagram

Plotting $z = x + iy$ at the point $(x, y)$ turns algebra into geometry. The modulus is the length of the position vector, the argument is its angle, addition is the parallelogram (vector) sum, and multiplication scales the length by $|z|$ and rotates by $\arg z$ .

Worked example

Express $z = -2 + 2i$ in exponential form and use it to find $z^2$ in Cartesian form.

Step 1: Find the modulus

|z| = \sqrt{(-2)^2 + 2^2} = \sqrt{8} = 2\sqrt{2}.

Step 2: Find the argument with a quadrant check

$z$ has negative real and positive imaginary parts, so it is in the second quadrant. The reference angle has $\tan\alpha = \dfrac{2}{2} = 1$ , so $\alpha = \dfrac{\pi}{4}$ ; the actual argument is $\theta = \pi - \dfrac{\pi}{4} = \dfrac{3\pi}{4}$ .

Step 3: Write the exponential form

z = 2\sqrt{2}\,\mathrm{e}^{i\,3\pi/4}.

Step 4: Square using the exponential form

z^2 = (2\sqrt{2})^2\,\mathrm{e}^{i\,3\pi/2} = 8\,\mathrm{e}^{i\,3\pi/2}.

Step 5: Convert back to Cartesian

$\mathrm{e}^{i\,3\pi/2} = \cos\dfrac{3\pi}{2} + i\sin\dfrac{3\pi}{2} = 0 - i$ , so $z^2 = 8(0 - i) = -8i$ .

Examples in context

Example 1. Alternating-current circuits. Voltages and currents that oscillate sinusoidally are represented as complex phasors $r\mathrm{e}^{i\theta}$ , where the modulus is the amplitude and the argument the phase. Adding signals becomes adding complex numbers, the foundation of AC analysis.

Example 2. Rotations in the plane. Multiplying a complex number by $\mathrm{e}^{i\theta}$ rotates its point about the origin by angle $\theta$ without changing its length, so complex multiplication is a clean algebraic encoding of plane rotation.

Try this

Q1. Find the modulus and argument of $z = -1 - i$ . [2 marks]

Cue. $|z| = \sqrt{2}$ ; third quadrant with reference angle $\tfrac{\pi}{4}$ , so $\arg z = -\dfrac{3\pi}{4}$ .

Q2. Write $z = 4\mathrm{e}^{i\pi/2}$ in Cartesian form. [1 mark]

Cue. $4(\cos\tfrac{\pi}{2} + i\sin\tfrac{\pi}{2}) = 4(0 + i) = 4i$ .

Q3. Evaluate $(2 + i)(2 - i)$ . [1 mark]

Cue. This is $z\bar{z} = 2^2 + 1^2 = 5$ .

Exam-style practice questions

Practice questions written in the style of SEAB exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.

Original5 marksExpress

z = 1 + i\sqrt{3}

in modulus-argument form and in exponential form, giving the argument in radians.

Show worked answer →

Modulus: $|z| = \sqrt{1^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = 2$ .

Argument: $z$ lies in the first quadrant (positive real and imaginary parts), and $\tan\theta = \dfrac{\sqrt{3}}{1} = \sqrt{3}$ , so $\theta = \dfrac{\pi}{3}$ .

Modulus-argument form: $z = 2\left(\cos\dfrac{\pi}{3} + i\sin\dfrac{\pi}{3}\right)$ .

Exponential form: $z = 2\,\mathrm{e}^{i\pi/3}$ .

Markers reward the modulus $2$ , the correct argument $\tfrac{\pi}{3}$ (with quadrant checked), and both required forms.

Original5 marksGiven

z_1 = 3 + 2i

and

z_2 = 1 - i

, find

z_1 z_2

and

\dfrac{z_1}{z_2}

in the form

a + bi