Introduction to Complex Numbers – Pure Mathematics 3 (9709) — Mathematics

Introduction

Complex numbers extend the real number system by introducing a number whose square is $-1$ . This is necessary because no real number satisfies $x^2 = -1$ , so equations such as $x^2 + 1 = 0$ have no real solutions. In 9709 Paper 3, complex numbers appear in a dedicated section of questions and underpin topics such as solving polynomial equations, loci in the Argand diagram, and transformations. Mastery of the vocabulary and notation introduced here is essential — examiners frequently award method marks that depend on correctly identifying real and imaginary parts, computing arguments, or using conjugates.

Core Concept

The Imaginary Unit and the Complex Number

We define the imaginary unit $\mathrm{i}$ by

\mathrm{i}^2 = -1 \quad \Longleftrightarrow \quad \mathrm{i} = \sqrt{-1}.

A complex number $z$ is any number of the form

z = x + \mathrm{i}y, \qquad x, y \in \mathbb{R}.

The real part of $z$ is $\operatorname{Re} z = x$ .
The imaginary part of $z$ is $\operatorname{Im} z = y$ (this is a real number — it is the coefficient of $\mathrm{i}$ , not $\mathrm{i}y$ itself).

The set of all complex numbers is denoted $\mathbb{C}$ . Every real number is a complex number with imaginary part $0$ .

The Complex Conjugate

The conjugate of $z = x + \mathrm{i}y$ is obtained by negating the imaginary part:

z^* = x - \mathrm{i}y.

Key property: $z \cdot z^* = x^2 + y^2$ , a non-negative real number.

Modulus

The modulus of $z = x + \mathrm{i}y$ is the non-negative real number

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

Geometrically, in the Argand diagram (the complex plane with the real part on the horizontal axis and imaginary part on the vertical axis), $|z|$ is the distance from the origin to the point $(x, y)$ .

Argument

The argument of $z$ , written $\arg z$ , is the angle $\theta$ (in radians) that the line from the origin to $z$ makes with the positive real axis. The principal argument satisfies

-\pi < \theta \leq \pi.

It is computed using the inverse tangent, taking care to place $\theta$ in the correct quadrant:

\arg z = \theta, \quad \text{where } \cos\theta = \frac{x}{|z|},\; \sin\theta = \frac{y}{|z|}.

In practice, find $\alpha = \arctan\!\left(\dfrac{|y|}{|x|}\right)$ (the reference angle), then adjust the sign and quadrant according to the signs of $x$ and $y$ .

Equality of Complex Numbers

Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal:

a + \mathrm{i}b = c + \mathrm{i}d \iff a = c \text{ and } b = d.

This principle — "equating real and imaginary parts" — is used repeatedly to solve equations involving unknown real quantities.

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