Complex Numbers

2026 Syllabus Objectives

By the end of this topic, you should be able to:

Understand the idea of a complex number, recall the meaning of the terms real part, imaginary part, modulus, argument, conjugate, and use the fact that two complex numbers are equal if and only if both real and imaginary parts are equal
Carry out operations of addition, subtraction, multiplication and division of two complex numbers expressed in Cartesian form x + iy
Use the result that, for a polynomial equation with real coefficients, any non-real roots occur in conjugate pairs
Represent complex numbers geometrically by means of an Argand diagram
Carry out operations of multiplication and division of two complex numbers expressed in polar form r(cos θ + i sin θ)
Find the two square roots of a complex number
Understand in simple terms the geometrical effects of conjugating a complex number and of adding, subtracting, multiplying and dividing two complex numbers
Illustrate simple equations and inequalities involving complex numbers by means of loci in an Argand diagram

A complex number is a number that has two parts: a real part and an imaginary part.

The imaginary unit is written as i, and it is defined by the property:

i² = -1

This means that i = √(-1), which is not a real number. By using i, we can work with square roots of negative numbers.

Every complex number can be written in the form:

z = x + iy

where:

Examples:

z = 3 + 4i has Re z = 3 and Im z = 4
z = -2 - 5i has Re z = -2 and Im z = -5
z = 7 (a real number) can be written as 7 + 0i, so Re z = 7 and Im z = 0
z = 2i (a purely imaginary number) can be written as 0 + 2i, so Re z = 0 and Im z = 2

Important: The imaginary part is just the number y, NOT yi. So for z = 3 + 4i, we write Im z = 4, not Im z = 4i.

Two complex numbers are equal if and only if:

Example: If 2 + 3i = a + bi, then a = 2 and b = 3.

If (x + 1) + (2y - 3)i = 4 + 5i, then:

This property is very useful for solving equations involving complex numbers.