Arithmetic of Complex Numbers in Cartesian Form — Mathematics

Introduction

Complex numbers in Cartesian form $x + iy$ can be combined using the four standard arithmetic operations — addition, subtraction, multiplication, and division. This is a core skill in the 9709 Paper 3 syllabus: every subsequent topic (modulus–argument form, loci, polynomial roots) depends on fluent arithmetic with complex numbers. The examiner specifically requires full working to be shown for multiplication and division, so marks are available for method even if a slip occurs.

Core Concept

A complex number in Cartesian form is written $z = x + iy$ , where $x = \operatorname{Re}(z)$ is the real part and $y = \operatorname{Im}(z)$ is the imaginary part, both real numbers, and $i = \sqrt{-1}$ so that $i^2 = -1$ .

Let $z_1 = a + ib$ and $z_2 = c + id$ where $a, b, c, d \in \mathbb{R}$ .

Addition and Subtraction — combine real parts and imaginary parts separately:

z_1 + z_2 = (a + c) + i(b + d)

z_1 - z_2 = (a - c) + i(b - d)

Multiplication — expand the brackets exactly as for algebraic expressions, then replace $i^2$ with $-1$ :

z_1 z_2 = (a + ib)(c + id) = ac + iad + ibc + i^2 bd = (ac - bd) + i(ad + bc)

Division — multiply numerator and denominator by the complex conjugate of the denominator to make the denominator real:

\frac{z_1}{z_2} = \frac{a + ib}{c + id} = \frac{(a + ib)(c - id)}{(c + id)(c - id)}

The denominator $(c + id)(c - id) = c^2 + d^2$ , which is a positive real number (provided $z_2 \neq 0$ ). Then separate the result into real and imaginary parts.

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Introduction

Core Concept

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