A number for √(−1): No real number squares to give −1. So we define i with i² = −1. A complex number is then z = a + bi — a real part a and an imaginary part b.
To add or subtract, just combine the real parts and the imaginary parts separately.
IB-style question — add and subtract
Let z₁ = 3 + 2i and z₂ = 1 − 5i.
Find z₁ + z₂.
Step by step
- Add the real parts together, and the imaginary parts together — they don't mix.
- Simplify each.
Final answer
4 − 3i.
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Expand like brackets, then swap i²: Multiply two complex numbers exactly like expanding two brackets (FOIL). The only new step: wherever you get i², replace it with −1.
IB-style question — multiply
Expand (3 + 2i)(1 − 4i).
Step by step
- Expand the brackets (FOIL).
- Replace i² with −1: the −8i² becomes +8.
- Collect real and imaginary parts.
Final answer
11 − 10i.