# Complex numbers

The reason for the introduction of complex numbers is so that every quadratic equation will have a solution. For instance, an equation contains solutions in a set of real numbers, however does not contains solutions in a set of real numbers. Because of these and similar equations, we expand the set of real numbers () to the set in which they will have the solution.

Let be the intended solution to the equation ; therefore .  The number is called the unit imaginary number. The unit imaginary number has the main role in describing a set of complex numbers which will be the extension of a set of real numbers .

The product of any real number and imaginary unit is a complex number. Numbers such as these are called imaginary numbers.

A complex number is the addition of a real and an imaginary number, that is, a complex number is the number of the shape , where and are real numbers. The number is called a real part, and is called an imaginary part of the complex number . We write:

A set of complex numbers is denoted as:

Two complex numbers and are equal if

Example 1. Determine and such that the following is valid:

Solution:

Two complex numbers are equal iff their real and imaginary parts are equal. Therefore, we have:

and

Powers of the imaginary unit

If we count further

we can observe that values of powers are repeated. Therefore,

is valid.

Example 2.

Calculate the following:

Solution:

by dividing with gives the rest , that is

Similarly, we obtain:

Finally, we have:

Example 3. Calculate:

Solution:

Since

then we have

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