Adding and Subtracting Complex Numbers
Simple Explanation
Add or subtract complex numbers by combining their real parts together, and separately combining their imaginary parts together — just like combining like terms in algebra.
Why Do We Need It?
This is the most basic operation on complex numbers, and it is needed constantly whenever complex quantities (like impedances or signal components) are combined.
Formula
Adding and Subtracting Complex Numbers
(a+bi) ± (c+di) = (a±c) + (b±d)i
Combine the two real parts together, and separately combine the two imaginary parts together.
- a+bi, c+di
- — the two complex numbers being added or subtracted
When to use it: Whenever two complex numbers are added or subtracted.
Worked Example
Add two complex numbers
Add (3+4i) and (1−6i).
Why Does This Work?
Since a+bi is really a sum of two DIFFERENT kinds of quantities (a real number and "b copies of i"), adding two complex numbers works exactly like adding two algebraic expressions with different variables (e.g. combining 3x+4y and x−6y) — like terms combine, unlike terms don't mix.
Real-Life Example
Combining two AC circuit impedances in series
Two circuit components, each with their own complex impedance, are connected in series.
The total impedance of the series combination is found by simply adding the two complex impedances — real parts with real parts, imaginary parts with imaginary parts.
Practice
Compute (6−3i) + (−2+5i).
MediumCommon mistake
When subtracting, forgetting to distribute the minus sign to BOTH the real and imaginary parts of the second number — e.g. (5−2i)−(3+7i) needs both the 3 AND the 7i subtracted, not just the 3.
Quick Review
- (a+bi) ± (c+di) = (a±c) + (b±d)i.
- Combine real parts together, and imaginary parts together, separately.
- When subtracting, distribute the minus sign to both parts of the second number.