Complex Number Multiplication — Definition, Formula & Examples

Complex number multiplication is the process of multiplying two complex numbers by distributing each part and using the fact that

i^2 = -1

to simplify the result into standard

a + bi

form.

Given two complex numbers $z_1 = a + bi$ and $z_2 = c + di$ where $a, b, c, d \in \mathbb{R}$ and $i^2 = -1$ , their product is defined as $z_1 \cdot z_2 = (ac - bd) + (ad + bc)i$ .

Key Formula

(a + bi)(c + di) = (ac - bd) + (ad + bc)i

Where:

$a, b$ = Real and imaginary parts of the first complex number
$c, d$ = Real and imaginary parts of the second complex number
$i$ = The imaginary unit, where i² = −1

How It Works

Multiply two complex numbers the same way you expand

(a + bi)(c + di)

using the distributive property (FOIL). You get four terms:

ac

adi

bci

, and

bdi^2

. Since

i^2 = -1

, the last term becomes

-bd

, which is real. Combine the real parts and the imaginary parts to write your answer in standard form

a + bi

Worked Example

Problem: Multiply (3 + 2i)(1 + 4i).

Distribute (FOIL): Multiply each term in the first factor by each term in the second factor.

(3)(1) + (3)(4i) + (2i)(1) + (2i)(4i) = 3 + 12i + 2i + 8i^2

Replace i² with −1: The term 8i² becomes 8(−1) = −8.

3 + 12i + 2i + (-8) = 3 + 12i + 2i - 8

Combine like terms: Add the real parts (3 and −8) and the imaginary parts (12i and 2i).

(3 - 8) + (12 + 2)i = -5 + 14i

Answer:

-5 + 14i

Why It Matters

Complex number multiplication is essential in Algebra 2 and precalculus when solving polynomial equations with no real roots. It also appears in electrical engineering, where multiplying complex impedances models AC circuit behavior, and in computer graphics for representing rotations.

Common Mistakes

Mistake: Forgetting that i² = −1 and leaving an i² term in the answer.

Correction: Always replace i² with −1 before combining terms. The final answer should have no powers of i higher than 1.