Key Idea: A complex number is z = a + bi, where i² = −1. You add, multiply and divide these numbers, plot them as points, and find their size. In context, the size is often a signal's amplitude.
| Move | How |
|---|---|
| Add or subtract | Combine real parts and imaginary parts separately |
| Multiply | Expand, then replace i² with −1 |
| Divide | Multiply top and bottom by the conjugate of the bottom, z* = a − bi |
| Argand diagram | z is the point (a, b) |
| Modulus | |z| = √(a² + b²), the distance from the origin |
Example: z = (1 + 2i)(3 − i)
- Expand — 3 − i + 6i − 2i²
- Use i² = −1 — 3 + 5i + 2 = 5 + 5i
- Modulus — |z| = √(5² + 5²) = 5√2 ≈ 7.07
Expand first, and only then turn i² into −1.
What is the conjugate of 4 − 7i? 4 + 7i. Only the sign of the imaginary part changes.
Where is 3 − 2i on an Argand diagram? At the point (3, −2).
Find |3 + 4i|. √(9 + 16) = 5.
Find (2 + i) + (1 − 3i). 3 − 2i.