Multiply a pair of complex conjugates
The idea
Recognize conjugates a + bi and a − bi, which have opposite imaginary parts.
Step by step
- Recognize conjugates a + bi and a − bi, which have opposite imaginary parts.
- Multiply using (u + v)(u − v) = u2 − v2.
- Substitute i2 = −1 to obtain a2 + b2, and explain why the imaginary terms cancel.
Example
Let i2 = −1. Find (2 + 3i)(2 − 3i).
Answer
13
The middle terms cancel. 22 − (3i)2 = 4 − 9(−1) = 13.
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Expand (3 + 2i)(3 − 2i) one term at a time, then compare with the conjugate shortcut.
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