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Multiplying Complex Numbers — Definition, Formula & Examples

Multiplying complex numbers means applying the distributive property (FOIL) to two expressions of the form a+bia + bi, then simplifying using the fact that i2=−1i^2 = -1.

Given two complex numbers z1=a+biz_1 = a + bi and z2=c+diz_2 = c + di where a,b,c,d∈Ra, b, c, d \in \mathbb{R}, their product is defined as z1⋅z2=(ac−bd)+(ad+bc)iz_1 \cdot z_2 = (ac - bd) + (ad + bc)i, derived from the distributive law and the identity i2=−1i^2 = -1.

Key Formula

(a+bi)(c+di)=(ac−bd)+(ad+bc)i(a + bi)(c + di) = (ac - bd) + (ad + bc)i
Where:
  • a,ba, b = Real and imaginary parts of the first complex number
  • c,dc, d = Real and imaginary parts of the second complex number
  • ii = The imaginary unit, where i² = −1

How It Works

Treat each complex number like a binomial and multiply using FOIL (First, Outer, Inner, Last). You will get four terms, one of which contains i2i^2. Replace i2i^2 with −1-1, then combine the real parts and the imaginary parts separately. The result is a new complex number in standard form a+bia + bi.

Worked Example

Problem: Multiply (3+2i)(1+4i)(3 + 2i)(1 + 4i).
FOIL: Multiply each pair of terms:
3⋅1+3⋅4i+2i⋅1+2i⋅4i=3+12i+2i+8i23 \cdot 1 + 3 \cdot 4i + 2i \cdot 1 + 2i \cdot 4i = 3 + 12i + 2i + 8i^2
Replace i²: Since i2=−1i^2 = -1, replace 8i28i^2 with −8-8:
3+12i+2i+(−8)=3+12i+2i−83 + 12i + 2i + (-8) = 3 + 12i + 2i - 8
Combine like terms: Group real parts and imaginary parts:
(3−8)+(12+2)i=−5+14i(3 - 8) + (12 + 2)i = -5 + 14i
Answer: −5+14i-5 + 14i

Why It Matters

Multiplying complex numbers is essential in Algebra 2 and Precalculus when solving quadratic equations with no real roots. It also appears in electrical engineering (AC circuit analysis) and signal processing, where complex multiplication models phase shifts and rotations.

Common Mistakes

Mistake: Forgetting that i2=−1i^2 = -1 and leaving i2i^2 in the answer.
Correction: Always replace i2i^2 with −1-1 before combining terms. The final answer should have no powers of ii higher than 1.