WOLFRAM NOTEBOOK

WOLFRAM|DEMONSTRATIONS PROJECT

Complex Multiplication

​
show green's angle (argument)
show blue's angle (argument)
show red's composite angle (argument)
choose order
green × blue
blue × green
●
●
A complex number is a two-dimensional number and as such needs two coordinates to describe it. We usually use its
x
,
y
coordinates, where
x
represents its real component, and y represents its imaginary component. When expressed this way a complex number looks like this:
x+iy
.
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
x+iy
.
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.
Wolfram Cloud

You are using a browser not supported by the Wolfram Cloud

Supported browsers include recent versions of Chrome, Edge, Firefox and Safari.


I understand and wish to continue anyway »

You are using a browser not supported by the Wolfram Cloud. Supported browsers include recent versions of Chrome, Edge, Firefox and Safari.