Introduction
with natural
started
Mathematics
then extended to integers rationals
but there were a problem
Some equations like
1
x
no
To solve such equations
a new unit
51
i
Any number of the form
at
a
where
a
reals
0
real solution
squared equals to 1
have
numbers
because
no
real
mathematicians
number
introduced
1
bi
be R is called a complex number
a
real part Re 12
b
imaginary part
Im
2
Signal Processing
Control System
Impedance
Reactance
Power Calculations
AC Circuit Analysis
Electromagnetic fields
Rotating Machines
Communication System
Equality
a
ib and 22 astiba
a
Complex numbers
are equal
if real and imaginary parts are equal
2
22
7
An
Az and
bi
be
N½Z½Q½R½C
Zero The complex number whose real part is 0
and imaginary part is 0
0 0 0
The
additive identity in the complex
is the
zero
number system
bi
at
0
2
0
i
ato
0
b 0
bi
a
2
Unity The complex
whose
number
real
1 and imaginary part is 0
The unity is the multiplicative identity
complex
system
1
2
bi
a
1
0
part is
of
the
2
Conjugate If z is a complex number the number
obtained by changing the sign of its imaginary
part is called the complex conjugate or simply
conjugate of 2
It is denoted by Z or 2T
If
If
z
is
then
bi
a
z
a
Further
he
2.22
Tize
I
real then
22
IT
Et
bi
a
I
22
21
2
Ti
In
22
2
La
In
general
2 I
Someone
a
and
227
Division
If
their
bi
be
a
prove two useful formulas
can
Relz
bi
at
Imla
and 22 are
2
III
two complex numbers
EE
with real and imaginary parts
let
a
2
50
a
EI
bi
and
bi
e
e
d
22
we
c
can
write
di
dil
We don't leave i in the denominator not because
but because it's non
it's mathematically
wrong
standard inconvenient for further operations
By using the definition of equality of
equation for a a bi
complex numbers solve
EX
1.1
37
let
2
91
22
i
2
z
a
bi
then
2
9i
a
bi
2A
29
9
a
2b
2
b
1
92
Thus
43
9
21
2b
z
92
Eliminate
one
10
i
102
2
2
1
i
2
3
5
22 222
3
5
both equations above
By equating
2
variable
complex
izz
10
2
1
i
10
2
izz
ith
31
7
22
5
22
2
132
Now
z
1 1
In
7
Z
17
3 58
7 932
55
131 72 13 3
11
31
5
Z
Complex Plane The complex plane is a way to
visualize complex numbers as points for vectors
on a flat 2D coordinate system
Horizontal axis x axis represents the real part
Vertical axis y axis represents the imaginary part
A complex number
by an ordered pair
z
of
iy is uniquely determined
x
real numbers
x y
Modulus
It is the length of the
vector from the origin 10,0
x
to the point
y in the
complex plane
Alaways
a
non
negative
Works just like a distance formula from
coordinate geometry
The modulus also called
represented by
some
121
properties
1712
ZEE
x
absolute value
y
and 121
LEE
the
and
12111221
171 21
also
and
1211 12
1212
1221
The
distance between two points in a complex
plane is the same as the distance between
the origin and the point x2 x1 y yr
and
be represented
can
by
122 2,1
122 no
y
y
Example Express the quantity in terms of x and y
let
2
1
12
1
I
x
3i
50
12 1
3i1
3112
x
iy
then
iy
1
x
1
3i
y 3
x
1
ly 3 i
Example Find the modulus of complex number
1
2i
1 2
2 i
9 i
We can find modulus if the given complex
member is in standard form
1
i
2
2
12 92
32
similarly
2
32
Now
t
12
1
i
3
11 it
12
3
1mF
3
1957
For each pair compare squared distances
90 81
99 Gi
vs
To Origin
110
191
8il
611
1 i
1 10 81
1191 Gi
To
Thus
102
82
199
1 61
1 i 1
9 i 12
964
157
19 7112 81 49
100 49
110 7i1
90 81 is closer to
let z
i x
iz
Im iz
x
iy
iy
a
closer to the origin
then
in i
y
1
730
149
i
I
x
ytix
iy
5
Im
iz
2
x
2
2
12
the equality is true for arbitrary y
is an equation
of two vertical lines
Since
2
12
Therefore
the complex
Im ix̅
2
2
12 yi
Polar Form
The relation
related
by
can
of
written
or
2
between
rose
my
x
iy
as
2
r
us
or
isino
r
and
I
to express
reoso
cost
52 yi
are
rsinD
y
enable
as
Numbers
number
complex
satisfying
then
Complex
These equations
2
number
i rsiero
a
nonzero
we see that r can
as
the
be
interpreted
distance from the origin
0
121
r
The angle 0 of inclination of the
vector 2 which will always be
measured in radians from the positive real axis
D is
when
measured counterclockwise
ve when measured
clockwise
the
D is
The angle 0 is
is denoted by
called
0
arg
To find the argeal
periodic
some
A calculator
will
that is
and
tano
care
must
be
give only angles satisfying
tan
Yx
angles in the
quadrants
of a
a
use
it
2 2
argument
we
As tand is
exercised
11
an
11 2
first
and
fourth
NOTE We have to choose 0 consistent with
the quadrant in which a is located
Principal Argument
The principal argument of a is unique and
is represented
the symbol Arg Cal that is
by
IT C
Arg z
L IT
Multiplication and Division
The polar form of a complex
is
convenient when multiplying or dividing
complex numbers
especially
two
number
Suppose
cos D
E
72
82
cos02
isino
isin 02
and
where 0
and
Z
and Oz
2
are
respectively
the arguments
Then
Zizz
i
E
Sino sino
sino cos 02 cos 0 sin 02
cosDicos02
ride
8182
2
02
0
cos
0
isin
02
similarly
cos
21
Further
arg z 22
arg
EI
0
02
arg
org Zi
isin 70
targ
02
Zz
arg
2
of
Integer
Powers of Z
82
z
8
73
so
0 0
0 0
isin
isin 20
cos 20
cos
Z
83
23
continue
r
7
de Moivre's
in this
manner
cosno
isin no
Formula
The complex number
z
can
also written as
reise
rlcos0 isino
Z
Ex 1.3 P
isierno
cosno
isino
coso
is in 30
cos 30
Write each complex number in
polar form Finally write the polar form in
the
21
3
let
atib
form of
z
3
3
3
5
5
and
51
72
5
5132
r
13
372
19T
312
82
1512 155312
25751
10
Or
tan
tan
33
1
513
is
0
3
02
tain
02
13
Z
312 cis
2
z
53
z2
E Zz
Fits
I
312 cis
3 2
3012
1
10
3012 cis
cos
22
10 cis
10 cis
I
and
cis
13
12
E
isin
12
II
7,72
305
5211
15 1
13
z
Coso
r
F
Ek
k
isino
s 0
0
1,2
There
are
n
exactly
th
then
Roots
nth roots
isin
2 1
n
5
12
1151 1 13
General Formula for
If
i
of
are
0121
1
n
distinct
roots
of
a
complex number
They are spaced equally around a circle in
the complex plane each separated by an
angle of
21
Example
Solve
3
We write
1
in
polar form
so
1
the
1
1 cos 0
i sin 0
roots can be written
as
Ek
cos
21231
Zo
1
isin
12
21 1
0 1,2
Theis
21
1132
i 3
72
21
These
are
placed
forming
an
equilateral triangle
the
on
unit
circle
Example Use de Moivre's formula with n 2
to
find trigonometric identities for cos20 and sin20
Solution For any
cost
For
n
2
DER
cosno
isino
we
can
write
isunno
have
cost isino
cos 0
2isinocoso
050
Lisino cost
By comparing real
and
we
cos20 isier 20
isin20
is into
cos 20
cos 20
isin 20
sin'D
and imaginary parts
sin'D
20520
costel
sin 20
25hr0coso