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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
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