A Visual, Intuitive Guide to Complex Multiplication

Complex numbers are the sum of a real and an imaginary number, represented as a + bi. It is a two-dimensional number and as such needs two coordinates to describe it. Using the complex plane, we can plot complex numbers similar to how we plot a coordinate on the Cartesian plane:

To multiply complex numbers:

Each part of the first complex number gets multiplied by each part of the second complex number.

Another method that is more natural for understanding how complex numbers multiply, is to represent a complex number by its magnitude—its distance from the origin—and its argument—its angle as measured counterclockwise from the positive real number line. These two numbers taken together uniquely determine every complex number, just as readily as a + bi.