A
complex number is a two-dimensional number and as such needs two coordinates to describe it. We usually use its

,

coordinates, where

represents its real component, and y represents its imaginary component. When expressed this way a complex number looks like this:

.
There is another method that is more natural for understanding how complex numbers multiply. You can 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

.
So, now when we multiply two complex number
s together we get a third complex number whose argument is just the sum of the two original arguments. Drag the green or blue complex number
s around and notice how their product, represented by the red dot, has an argument equal to the sum of the green dot's angle and the blue dot's angle.