Complex Numbers

i = √−1, so i² = −1. A complex number is z = x + yi: x real part, y imaginary part.

Add real to real and imaginary to imaginary.

Multiply out like brackets, then replace every i² with −1 and tidy.

The conjugate of x + yi is x − yi. z times its conjugate is always real.

Multiply top and bottom by the conjugate of the denominator; the bottom becomes real.

Plot z = x + yi as (x, y): real axis across, imaginary axis up.

|z| = √(x² + y²), the distance from the origin.

Two complex numbers are equal when real = real and imaginary = imaginary — often gives simultaneous equations.

If a quadratic has complex roots they come in conjugate pairs: x + yi and x − yi. To prove a root, sub it in and show you get 0.