Key Idea: Complex numbers have the form z = a + bi, where i² = −1. You add, multiply and divide them, and plot them on the Argand diagram. They are on Paper 1, Paper 2 and can appear in Paper 3.
Meet i — adding & multiplying
- Add or subtract: real with real, imaginary with imaginary.
- Multiply: expand like brackets, then turn every i² into −1.
- Powers of i cycle: i, −1, −i, 1, then repeat.
The conjugate & dividing
- Conjugate — z* = a − bi. Flip only the sign of the imaginary part.
- Real product — z × z* = a² + b², always real.
- Divide — Multiply top and bottom by the conjugate of the bottom.
- Split — Write the result as a + bi.
Modulus & the Argand diagram
- z = a + bi is the point (a, b): real across, imaginary up.
- |z| = √(a² + b²), its distance from the origin.
- |z| is never negative, because it is a distance.
i² = −1 is the one rule every calculation uses.
(5 − i)(2 + 3i)? 10 + 15i − 2i − 3i² = 13 + 13i.
Conjugate of 3 + 2i? 3 − 2i.
To divide by 3 + 2i, what do you multiply by? 3 − 2i, on the top and the bottom.
|−5 + 12i|? √(25 + 144) = 13.
What is i⁴? 1.