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
7
Example 1:
2
7
5
7
5
35
3 7
3 7
3 7
49
15
49
15
49 15
6
35
35
21
21
2
4
1
1
.
4
i
i
i
i
i
i
i
i
i
i
i
i
Division of complex numbers
Division of complex numbers is much like rationalizing the denominator of a
rational expression. For the complex number
z
a b
i
, we define its
complex
conjugate
to be
z
a b
i
. To simplify the quotient of complex number such
as
1
2
a
z
i
d
z
b
i
c
, we have to multiply the numerator and the denominator by the conjugate
of denominator.
1
2
1
2
2
2
2
2
2
2
( )
i
i
i
i
i
i
i
a b
c d
c
d
c
d
c
d
c
d
c d
c
d
c
d
d
c
c
d
a b
a
a
b
b
a
b
a
b
a
a
b
c
d
c
d
c
d
i
i
z
i
i
z
b
i
Example 2:
Let
given two complex number as
1
7 5
z
i
and
2
2 5
z
i
the find the ratio of
these complex number
1
2
7 5
?
2 5
z
z
i
i
Solution:
To solve this expression, we will simplify the complex ratio by multiplying
numerator and denominator by the conjugate of denominator
2
2 5
z
i
, namely
2
2 5
z
i
:
1
2
2
7 2
7
5
5
2 5
5
7 5
2 5
2 5
2 5
2 2 2
5
5
2 5
5
14 35
10
25
14 35
10
25
4 25
29
25 14
35 10
11 25
11
25
29
.
29
29
29
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
z
z
If
r
is negative, then the
principal
square root
of
r
is
2
i
r
i
r
r
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
8
There two
square roots
of
r
are
i
r
and
r
i
. With this convrention, the
usual derivation and formula for the roots of the quadratic equation such as
2
0
x
x
a
b
c
are valid even when
2
0
4
b
ac
:
2 4
2
b
b
ac
a
x
.
Example 3:
Find the roots of the equation
2
1
0
x
x
Solution:
Using the quadratic formula, we have
2
1
1
1
1
1
1
4
3
3
2
2
2
i
x
.
We observe that the solution of the equation in Example 3 are
complex conjugates of each
other. In general, the solutions of any quadratic equation
2
0
x
x
a
b
c
with real coefficients
,
a b
and
c
are always complex conjugates. (If
z
is real,
z
z
, so
z
is its own conjugate.)
Example 4:
Solve
2
x
x
2
2
0
Solution:
Using the quadratic formula
2
2
x
x
4
2
2
2
4
1
2
2 1
2
4
8
2
4
b
2
x
b
c
a
2
a
But
4
4
1
4
2
i
1
2
1
(using
the
definition
of
i
).
Therefore
2
4
2
2
1
2
2
2
2
1
2
2
2
2
i
i
x
x
2
i
. Therefore, the two solutions are
x
i
1
and
x
i
1
.
We next need to address an issue on dealing with square roots of negative numbers. From the
section on radicals, we know that we can do the following.
6
36
4 9
4
9
2 3 6
.
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
9
In
other words, we can break up products under a square root into a product of square roots
provided both numbers are positive. It turns out that we can actually do the same thing if
one
of the
numbers is negative. For instance,
6
36
4
9
4
9
2
3
6
i
i
i
.
As well as complex number use within mathematics, complex numbers have practical
applications in many fields, including physics,
chemistry, biology, economics, electrical engineering,
and statistics.
Any complex number is then an expression of the form
a
i
i
b
x
y
, where
a
and
b
are old-fashioned real numbers. The number
a
is
called the real part
of
a
b
i
, and
b
is
called its
imaginary par
t
.
Example 5.
Find
,
x y
if
2
3
4
2
x
y
i
x
i
i
y
.
Solution:
Left hand side (LHS):
2
2
3
4
2
9 12 2
16
2
2
9
24
16
2
2
7
2
.
24
2
i
i
i
i
i
i
x
y
x
y
x
y
S
x
y
i
LH
i
So, from this we have that:
The real part is
and
Imaginer part is
7
2
2
7
7
3
.
x
x
x
x
x
24
2
2
24
24
24
.
y
y
y
y
y
y
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
10
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