# Complex plane

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 .

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