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
16
called the
imaginary unit
. Thus, associated with each complex number
z
i
a
b
is a unique
ordered pair of real numbers
a
b
, and vice versa. For example,
3
i
5
corresponds to
,
3
5
.
According these ordered pairs of real numbers with points in a rectangular coordinate system, we
obtain a complex plane (
See Figure 8
).
Example 9
(
Plotting in the Complex Plane
)
:
Plot the following complex numbers in a complex plane:
1
2
3
4
i
i
2
3
=
,
=
,
=
3
4
,
z
z
z
z
=
i
,
5
3
Solution:
Polar (Trigonometric form) form of
Consider the complex number
z
i
a
b
as represented on an Argand diagram. The
position of
A
can
be expressed as coordinates
a
b
, the Cartesian form, or in terms of the length and
direction of
OA
. Using Pythagoras’ theorem, the length of
2
2
O
z
A
=
r
=
a
b
N
is read as the
modulus
or
absolute value
of
z
.
The angle that
OA
makes with the positive real axis is
1
tan
b
a
.
The
argument
(or
phase
) of
z
is
, let us try to express it as a function of
a
,
b
.
From
trigonometry one
sees that for any complex number
z
i
a
b
one has
cos
z
a
and
sin
z
b
so that,
cos
sin
cos
sin
z
z
z
z
i
i
and
x
y
3
2
0
2
3
A
i
x
y
3
5
0
3
5
B
i
4
0
C
i
x
y
0
3
D
i
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
17
sin
tan
cos
b
a
.
The complex number
a
b
i
can be
represented geometrically by the point
a
b
. This
point can also be expressed in terms of
polar
coordinate
r
, where
0
r
, see
Figure 9
.
Definition 5
(Trigonometric form)
:
The trigonometric representation of the complex number
Z
:
cos
sin
cos
sin
z
z
z
z
i
b
i
i
a
where
, called the
argument
is given by
sin
2
2
z
b
b
a
b
+
,
cos
2
2
z
a
a
a
b
+
(1)
The representation is often referred to as the polar – coordinate form.
Therefore
1
g
tan
Ar
z
a
b
where the range of
1
tan
is
,
, and one has to be
careful that there are special cases depending on the sign of
a
,
b
:
1
1
1
tan
;
tan
,
;
ta
0
0
0
.
0
0
rg
n
A
,
a
a
b
a
b
z
a
a
b
b
a
b
and
Arg
indete
0
0
0
,
;
,
rminate
0
,
0
.
0
;
a
b
a
z
b
a
b
2
2
Note that the argument
is not uniquely defined: If
first in (1), then
2
k
,
,
,
1 2
k
also
fit. A convention is to choose
to be in region
180
180
. If
90
90
2
2
we find
as
tan
acr
b
a
, since
sin
tan
cos
b
a
.
Example 10:
Given complex number as
z
3
3
i
3
then
find its modulus, argument (phase) and polar
form?
Figure 9
Im
Re
a
b
,
a
b
0
a
b
i
r
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
18
Solution:
First norm of
z
3
3
i
3
is
2
2
3
3 3
9
9 3
9
27
3
r
z
6
6
,
second the Argument (phase) is
1
1
tan
Arg
tan
3 3
3
3
3
z
,
and last polar form of
z
3
3
i
3
is
cos
sin
i
3
6
6
3
e
i
=
z
3
.
Example 11:
Write
1
z
i
in polar form.
Solution:
2
1
2
,
,
tan
cos
Arg
s
1
1
1
1
2
1
2
i
4
4
4
n
z
=
b
r
z
,
i
a
Relationship between Rectangular and Polar forms
Trigonometric Representations:
Let us recall the polar coordinates
a
r
= =
x
cos
and
r
=
b
=
y
sin
. Using this representation, we have that
=
=
= c
z
a
r
r
i
x
o
i
s
y
b
sin
i
+
.
Thus, any complex number
z
b
a
i
can
be written in
polar form
:
a
r
r
= cos +
sin =
r
c
b
i
i
i
s + sin
z
o
(3)
Where
2
2
z
r
a
b
, and
tan
a
=
b
. The angle
is called an
argument
of complex number
z
i
a
b
and is denoted by
1
( ) = tan
Arg
b
z
a
=
. Observe that
Arg
( )
Z
is not unique.
Adding or subtracting any integer multiple of
2
0
2
-
<
gives another argument of
z
.
However, there is only one argument
that satisfies
<
. This is called the
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