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Chapter 1

Sets and Numbers

 
1.4 Complex Numbers

As it is known, the square of any real number is a non negative number. So, there is not real number a such
as 

We consider a number i with the property . This number, obviously is not real and it is called imaginary unit.

We we append the number i to the set R of real numbers a new set of numbers is created that has the following members:

The operations in R are extended in C with the same rules. The members of the set C are called complex numbers and are of the form .

In the above formula the number  is called real part of the complex number z (denoted by a=Re(z)) and the number  is called imaginary part of the complex number z ( denoted by b=Im(z)).

The complex numbers  and  are called equal if and only if 
and .

The properties in C are defined as following:

                ( a + bi ) + ( c + di ) = a + bi + c + di = a + c + ( b + a )i                 ( a + bi ) - ( c + di ) = a + bi - c - di = a - c + ( d - a )i                 ( a + bi )(c + di ) = a( c + di ) + bi( c + di ) = ac + adi + bic + bd = ac + adi +bci - bd =
                = ( ac - bd ) + ( ad + bc)i
    

 
 

To every complex number z = a + bi corresponds a point M(a,b) of a cartesian plane. On the other hand, to every point M(a,b) of a cartesian plane corresponds a complex number z = a + bi. This way the set of complex numbers is the same with the set of all pairs of real numbers. Now the plane is called the "complex plane", the horizontal axis( which consists of all points (a,0) where ) is called the "real axis" and the verticla axis(which consists of all points (0,b), ) is called the "imaginary axis".

Definition:

If z = a + bi,  then the conjugate of z is defined as  = a - bi
and the modulus  of z is defined as .

It is easy to prove the following properties:
 
1. 

2. 

3. 

4. 

5. 

6. 

7. 

8. 

9. 

10.