
A complex number can be written as the ordered pair
of real numbers. Therefore, to the complex numbers we can join points in the coordinate plane. The
axis of the coordinate plane is called the real axis and it contains only real numbers. The
axis is called the imaginary axis and it contains only imaginary numbers. The coordinate plane which contains all complex numbers is called the complex plane or
-plane.
Every complex number corresponds to the point
.
The modulus of the complex number is the positive real number
. Geometrically, the modulus of the complex number
we interpret as the distance from the point
to the origin:
Let and
are the given complex numbers. Then is valid:
The expression above we recognized as the formula for distance between two points in the coordinate plane. Therefore, is the distance between points
and
in the complex plane:
Properties of the modulus of complex numbers
Example 1. Specify the set of all complex numbers in the complex plane for which
is valid, where
.
Solution:
Let be the solution. Include
in the equality
:
This means that the distance from the point to the point
is equal to
. A set of points
which are from the point
remote for
is the circle with the center at point
and with the radius
.
Example 2.
In the complex plane sketch all complex numbers for which is valid:
Solution:
Let be the solution.
Then
It follows that the imaginary part of a complex number is
Now we have inequalities:
and
This means that we have parts of a plane for which is valid
and
.
Example 3. Determine all numbers for which is:
Solution:
Let be the solution. Then
We have two inequalities:
and
that is, from the definition of the modulus of complex numbers, we have
and
We obtained the annulus between two concentric circles with the center at point and radii
and
Example 4.
Show in the complex plane all complex numbers for which the following is valid:
Solution:
We need to find all complex numbers for which is the sum of distances from the points which are join to numbers
and
is constant and equal to
.
Without calculating, we can see that this would be an ellipse with focii and
and the length of the semi-major axis
.
Analytically:
Let be the solution. Then
We obtained an equation of the ellipse which the length of the semi-major axis is equal to , and the length of the semi-minor axis is equal to
.