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:
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:
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.