Ministry of Higher Education
and Scientifics research
University of Babylon
Complex Functions
Lecture 2
Ali Hussein Mahmood Al-Obaidi
1.2 complex number:A complex number
number
.
College of Education for Pure Sciences
Physics Department
Three Stage
ali.alobaidi81@yahoo.com
can be defined as an ordered pair of real
Let
and
then their sum is
be any two complex numbers,
and their product is
Remarks:1. The order pair
is called a pure imaginary numbers.
2. The order pair
is called real numbers and we write it as , so
the set of real numbers is subset of the set of complex numbers .
3. is called real part of complex number
and we denote it
as
.
4. is called imaginary part of complex number
and we
write it as
, so
is real numbers.
5. We can write a complex number
as algebraic (ordinary)
form
where
is called imaginary unit.
6. The algebraic form of complex numbers is a best the Cartesian form
of complex numbers, especially in expression of complex number
and satisfied its properties.
7. By using product of complex numbers, we get
1
Ministry of Higher Education
and Scientifics research
University of Babylon
Complex Functions
Lecture 2
Ali Hussein Mahmood Al-Obaidi
College of Education for Pure Sciences
Physics Department
Three Stage
ali.alobaidi81@yahoo.com
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

In general case, If
is positive integer number, then
Property (1):- the complex number
imaginary parts are zero.
Property (2):- if
EX (1):- Find value of
iff (if and only if) real and
and
and
, then
which satisfy the following equation
Sol:By property (2)
From eq. (1), we get
Put in eq. (2)
So that
2
Ministry of Higher Education
and Scientifics research
University of Babylon
Complex Functions
Lecture 2
Ali Hussein Mahmood Al-Obaidi
College of Education for Pure Sciences
Physics Department
Three Stage
ali.alobaidi81@yahoo.com
Remark:-we rewrite the addition, subtraction, multiplication and division
operations of complex number in a algebraic form (ordinary form) as
follows:If
we suppose
and
, then
, we get
by property (2), we have
We need to find
since we want , that we go to do it.
By Cramer's rule
3
Ministry of Higher Education
and Scientifics research
University of Babylon
Complex Functions
Lecture 2
Ali Hussein Mahmood Al-Obaidi
College of Education for Pure Sciences
Physics Department
Three Stage
ali.alobaidi81@yahoo.com
But this method is difficult, what is easy method, however we see it in
section 4.
4