Chapter 1: complex number

1.
 
Definition and Properties of CN 
We start the development of the complex number system by defining a complex number and 
several special types of complex numbers. We then define 
equality
, 
addition
, and 
multiplication
in 
this system, and from these definitions the 
important special properties
and 
operational rules
for 
addition
, 
subtraction
, 
multiplication
, and 
division
will follow. 
Definition 1 
(Definition of Complex number 
z
)
: 
A complex number is a number of the form 
a
b
i

where 
a
and 
b
are real numbers. If 
z
i
a
b
 
then 
a
is known as real part of 
z
and 
b
as the imaginary part. We write 
 
e
z
R
a

 and 
 
m
z
I
b

. 
Define an 
imaginary unit
i
such that 
2
i
1
 
. 
Definition 2 
(Definition of imaginary unit 
i
)
: 
There is no real number 
z
 such that 
2
z
1
 
. Defined 
an imaginary unit
 
i
 (denoted also 
j
) 
such that 
2
i
1
 
(that is) 
i
1
 
. Defined 
an imaginary number
 
z
, to be of the form 
1
=
i
y
y
z
   
for 
y
any real number. Then the 
imaginary number
 
z
is such that 
2
2
z
y
 
. 
The imaginary unit 
i
introduced in 
Definition 2
is not a real number. It is a special symbol 
used in the representation of the elements in this new complex number system. Note that if 
Z
=
a
b
i
+
then 
( )
a
=
Re
z
and 
( )
b
=
Im
z
. 
Geometric Representation is: 
The complex number 
z
i
a
b
 
can be represented by the order pair 


a b

and plotted as a point in the plane called complex plane (Argand 
plane) where 
x

axis is called a real axis for the real part of 
z
and 
y

axis is called the imaginary axis for the imaginary part of 
z
, 
Figure 2
. 
y 
x 
Figure 2
( , )
x y
0


---

IIUM, Faculty of Engineering, 
 
Department Engineering in Science 
Engineering Mathematics I
 
Semester 1, 2021/2022
 
Chapter I:
 
Complex Numbers
 
Lecturer
Associate Professor Dr. Abdurahim Okhunov
11
Powers of 
i
: 
The natural number power of imaginary unit 
i
 taken on particularly simple forms as follow
:
 
 
  
  
  
  
;
;
;
;
;
;
;
;
.
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
1
1
1
1
1
1
1
1
1
1
i
i
1
1
1
1
i
i
i
i
i







   


 



  

  













0
1
2
3
2
4
2
2
5
4
6
4
2
7
4
3
8
4
4
Some important identities involve the powers of 
i
: 
1
5
9
2
6
10
3
7
11
4
8
12
=
=
=
=
=
=
=
=
=
=
i
i
i
i
i
i
i
i
i
i
i
=
=
=
=
i
1
1
i
=
=
i



Note that
, when the powers of i are simplified, they cycle in steps of four. 
In general, what are the possible values for 
n
i
, 
n
a natural number? Explain how you could 
easily evaluate 
n
i
for any natural number 
n
. 
Exercises 1
 
(Powers of Complex number)
: 
Then evaluate each of the following: 
a)
17
i
, b) 
24
i
,
c) 
38
i
,
d) 
47
i
Properties of Complex Number 
In this section we will discuss about two types of CN: 
Algebraic properties of 
 
Some examples of complex numbers are 
1
2
5
1
3
5
0
0
3
5
0
2
3
0
i
i
i
i
i
i






;
;
;
;
;
Particular kinds of complex numbers are given special names as follows: 
Imaginary Unit
i
; 
Complex Number
a
b
i

, 
a
and 
b
real numbers; 
Pure Imaginary number
0
i
i
b
b


, 
0
b

; 
Real number
0
a
a
i


, 
0
b

; 
Zero 
0
0
0
i
 
, 
0
b

; 

---

IIUM, Faculty of Engineering, 
 
Department Engineering in Science 
Engineering Mathematics I
 
Semester 1, 2021/2022
 
Chapter I:
 
Complex Numbers
 
Lecturer
Associate Professor Dr. Abdurahim Okhunov
12
Example 6
 
(
Special types of Complex Numbers
)
:
 
Given the list of complex numbers: 
1
2
5
1
3
5
0
0
3
5
0
2
3
0
i
i
i
i
i
i






;
;
;
;
;
List all the imaginary numbers, pure imaginary numbers, Real numbers, and Zero. 
Solution: 
Imaginary numbers:
1
3
5
1
2
3
0
2
5
3
b
i
i
i
i




;
;
;
;
Pure imaginary numbers:
0
3
3
i
i


=
Real numbers;
5
5
0
0
0
0
i
i



=
=
Zero:
0
0
0
i

=
In 
Definition 1
, notice that we identify a complex number of the form 
0
a
i

with the real 
number 
a
, a complex number of the form 
0
b
i

, 
0
b

, with the Pure 
imaginary number
i
b
, and 
the complex number 
0
0
i

with the real number 
0
. Thus, a real number is also a complex number; 
just a rational number is also a real number. 
Any complex number that is not a real number is called 
an imaginary number.
Geometric properties of 
(
z
)
 
Complex numbers can be represented as points in the plane, using the correspondence 


,
y
i
x
x
y
  
. The 
real complex numbers lie on the
x

axis, 
which is then called the real axis
, 
while the 
imaginary numbers lie on the
y

axis, 
which is known as the imaginary axis
. The complex 
numbers with positive imaginary part lie in the upper half plane, while those with negative imaginary 
part lie in the lower half plane. 
Because 
of 
the 
equation 

 
 



1
1
2
2
1
2
1
2
x
y
x
y
x
x
y
y
i
i
i
 

 


 

complex 
numbers add vectorially, using the parallelogram law. Similarly, the 
complex number 
1
2
Z
Z

can be represented by the vector from 


,
2
2
x y
to 


,
1
1
x y
, where 
1
1
1
=
x
Z
i
y

+
and 
2
2
2
=
x
Z
i
y

+
(
See Figure 4
). 
The geometrical representation of complex numbers can be very useful 
when complex number methods are used to investigate properties of 
triangles and circles. It is very important in the branch of calculus known as Complex Function theory, 
where geometric methods play an important role. 
0 
2
Z
 
y 
x 
1
Z
 
1
2
Z
Z

 
1
2
Z
Z

 
Figure 4

---

IIUM, Faculty of Engineering, 
 
Department Engineering in Science 
Engineering Mathematics I
 
Semester 1, 2021/2022
 
Chapter I:
 
Complex Numbers
 
Lecturer
Associate Professor Dr. Abdurahim Okhunov
13