Multiplying Complex Numbers
Simple Explanation
Multiply complex numbers using the distributive property (FOIL), exactly like multiplying two binomials — but then replace every i² that appears with −1, and combine like terms.
Why Do We Need It?
Multiplication is essential for later topics in this chapter (like De Moivre's Theorem), and appears constantly in applications combining complex quantities like AC circuit impedances.
Formula
Multiplying Complex Numbers
(a+bi)(c+di) = (ac−bd) + (ad+bc)i
Expand using the distributive property (like multiplying two binomials), then replace i² with −1 and combine like terms.
- a+bi, c+di
- — the two complex numbers being multiplied
When to use it: Whenever two complex numbers are multiplied together.
Worked Example
Multiply two complex numbers
Multiply (2+3i)(1−4i).
Why Does This Work?
FOIL is just the distributive property applied twice, which works for ANY two binomial-shaped expressions — the only genuinely new step for complex numbers is substituting i²=−1 afterward, which is exactly what makes the result "collapse" back into standard a+bi form instead of staying as an expression with i².
Real-Life Example
Combining impedances in parallel circuits
Calculating combined impedance for components in parallel in an AC circuit requires multiplying (and dividing) complex impedance values.
Correctly multiplying complex numbers — including handling i² — is essential to getting the right combined impedance value.
Practice
Multiply (1+i)(1−i).
MediumCommon mistake
Forgetting to replace i² with −1 after expanding — leaving i² sitting in the final answer is incomplete; it must always be simplified.
Quick Review
- (a+bi)(c+di) = (ac−bd) + (ad+bc)i.
- Expand with FOIL, then replace every i² with −1.
- A complex number times its own conjugate always gives a purely real result.