Geometric Interpretation
of the Arithmetic Operations
Addition and Subtraction
Geometrically, addition of two complex numbers
The easiest way to represent the difference
Use the following applet to explore this geometric interpretation. Activate the boxes below to
show the addition or subtraction. You can also drag the points
Exercise 1: Can you think about a geometric interpretation of the addition
of three complex numbers? In general, what would be a geometric interpretation of the addition
of
Multiplication
In the previous section we defined the multiplication of two complex numbers
In the following applet, you can appreciate what happens to the argument of the product.
Drag the points
Exercise 2:
Consider now
Multiplication of complex numbers as stretching (squeezing) and rotation
In the applet below a set of points are defined randomly on the complex plane.
Then each point is multiplied by a given complex number
- What happens when
is inside, or outside, the unit circle? - What happens if
moves only around the unit circle?
Multiply by 1/z
.
As you already have noticed, the geometric interpretation of multiplication of complex numbers is stretching (or squeezing) and rotation of vectors in the plane.
In the previous applet, with the option Multiply by z
, set n = 1
by
dragging the
slider to the left side.
In this case, the applet shows the three complex numbers
- the modulus of
is equal to and - the argument of
is equal to
Exercise 3:
Use the same applet, with the option Multiply by 1/z
, to investigate
what happens when we multiply by n = 1
by dragging the slider to the left side to
show the three complex numbers