The complex plane

Just as real numbers are often represented as points on a line, complex numbers can be represented by points on a plane. z=x+iy can be represented by the point (x,y). Addition and subtraction of complex numbers are analogous to addition and subtraction of 2D vectors.

In this representation, |z| is just the distance of z from the point 0.

Multiplication is particularly interesting. The product of two complex numbers has a magnitude which is the product of their magnitudes, and the angle that it makes with the x axis is the sum of the angles of the two numbers.