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: .[more]
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 numbers together we get a third complex number whose argument is just the sum of the two original arguments. Drag the green or blue complex numbers 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.[less]
Snapshot 1: It's not so obvious how green times blue equals red.
Snapshot 2: But green and blue have angles associated with them…
Snapshot 3: … and red's angle is just the sum of green's angle and blue's angle…
Snapshot 4: … and furthermore, this is true regardless of the order.