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Geometric Interpretation

of the Arithmetic Operations


Addition and Subtraction

Geometrically, addition of two complex numbers Z1 and Z2 can be visualized as addition of the vectors by using the parallelogram law. The vector sum Z1+Z2 is represented by the diagonal of the parallelogram formed by the two original vectors.

The easiest way to represent the difference Z1Z2 is to think in terms of adding a negative vector Z1+(Z2). The negative vector is the same vector as its positive counterpart, only pointing in the opposite direction.

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 Z1 and Z2 around.

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 n complex numbers?


Multiplication

In the previous section we defined the multiplication of two complex numbers Z1 and Z2 as

Z1Z2=(x1+iy1)(x2+iy2)=(x1x2y1y2)+i(x1y2+x2y1).
In this case, to appreciate what happens geometrically we need to consider the polar form of Z1 and Z2. That is
Z1=r1(cosϕ1+isinϕ1)Z2=r2(cosϕ2+isinϕ2)
Then the product can be written in the form
Z1Z2=r1r2[(cosϕ1cosϕ2sinϕ1sinϕ2)+i(sinϕ1cosϕ2+cosϕ1sinϕ2)].
Now by means of the addition theorems of the sine and cosine this expression can be simplified to
Z1Z2=r1r2[cos(ϕ1+ϕ2)+isin(ϕ1+ϕ2)].
Thus the product Z1Z2 has the modulus r1r2 and the argument ϕ1+ϕ2.

In the following applet, you can appreciate what happens to the argument of the product. Drag the points Z1 and Z2 around and observe the behaviour of the angles. Then drag the slider below.

Exercise 2: Consider now Z1=r1(cosϕ1+isinϕ1)Z2=r2(cosϕ2+isinϕ2) such that Z20. Find the polar representation of Z1/Z2. What is the geometric interpretation of this expression?


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 z. On the right-side screen, drag around the point z and analyze the behaviour of the points (⭕) multiplied by z and try to answer the following questions:

  • What happens when z is inside, or outside, the unit circle?
  • What happens if z moves only around the unit circle?
Note: You can also study the behaviour of the points (⚫) multiplied by 1/z by activating the box 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 z0,z and z1=z0z, represented as vectors. When z0 and z are non zero, then

  • the modulus of z1 is equal to |z0z|, and
  • the argument of z1 is equal to Arg (z0+z).
If |z|>1, we deal with stretching. If |z|<1, it is a case of squeezing.

Exercise 3: Use the same applet, with the option Multiply by 1/z, to investigate what happens when we multiply by 1/z. Set n = 1 by dragging the slider to the left side to show the three complex numbers z0,z and z2=z01z. What happens to the modulus and argument of z2?

The Principal Argument