, . Complex numbers
1
Definition numbers have the form
1 .
1: Complex :
Xtiy
where X ,
y
ER and i is the
imaginary unit defined
,
to
satisfy :
i = -
1 .
i3 i . =
-i
ii ? j = 1
j5 it i =
.
= i
For complex number where is called the real part of 2 and
a
Extiy ,
YeR
X, X
y
is called the
imaginary part of 2 ,
written
Rez Imz
x=
y
=
,
x 5-2i Rex 5
eg
= = [mx = -2
defined and the usual rules of
Operations of addition ,
multiplication, subtraction and division are
algebra apply .
b
eg
a = 2+ 3i and =
-
1 + 2i
ab = (2+ 3i)( -
1 + 2i) = -
2 -
3i + 4i + 6i2
=
- 2+ i =
6 =
-
8+i
a /b = 2 + 3i = 2 +3i .
(1 -
2i) =
-
2 3i -
- 4i -
6i2
- 1 + 2i
-
1 +2i( -
1 -
2i) ( 12
-
= (2i) 2
47i = Ei
= + -
Definition 13: The complex of 2 where is defined be
conjugate iy yer
.
= X +
X to
, ,
xoiy
E
it is denoted
by
.
,
We O is the of it is
say argument
2 ,
denoted We
by argz . can represent
-----iz = -
2- i in the polar form: Z = ros Of irsing
= r(coso + isind
The principal value of the Lis definition the value O is
by satisfying
the of : TOLI
argument
I
argument
of -
and is denoted
by Arg2
.
Example 1 6 Find the principal
.
% value of the
argument
of Z of 21-i. .
Arg =
Arg(1 i= Arg(1-i) π/4 23
+ =
-
=
-1
,
i
-
Arg(25) Arg(-i)
=
-
/z
-
5
,Definition 1 7 The modulus complex
of number
zxxiy EyeR
is the
length of the
corresponding i e
:
. a vector , .
,
(2) = (x +
iy) = x2 +
y2
Example 1. 9 : Find and the modulus of 1. Write1 t polar form
argument
the in .
N
-----
i
r= 1 +1 = 2 0 = π/
I
p Y
1 1 + i =(cos + isin() .
Properties : 1) /zwl = /zllw)
al(z + w((z) + Iw) (the
triangle inequality .
Properties of I : 1) +We + w
2) w = z
3) zz =
1213
4)2 =
7212 ,
20
Example 3-4i Find 121 and write Iz1 and 1 form What is ZI ?
xtily XyeR
1 13
. : 2= in the .
.
,
(2) =
32+ 42
= 25 = 5
z = 3 + 4i
-
(z =
=
/1212 3 3/25 "e5i +
22 =
(2) = 25
Multiplication and Division in Polar Form
2 = r1(cos On + Isino)
ra(cos02 + IsinGal
on a
22 =
where P1 O2fIR ,
and re, re 30
3122
excata
Then : 2122 = rera(cos con + Gel + isinContoe)
2/22 = /ra(cosco-8) + isin(on-or ,
za o
Powers Let r(coso + isino
Integer z= .
Then
zr(Cosno + isinno).
Example 1 . 16 : Find (1 + il
| + il = E ,
0 = /4.
1+i =
E(cosT + isinπ/)
Ce + i)" =
2 4)cs ·
4 . T + isin 4 .
T
= CostT + isinit) =
-
4 .
, Roots 1 .
17 : If z = wh
,
n = 1 2 3
,
. ..
,
then to each value w
of there Corresponds one value of 2
. Conversely ,
to a
given
2to there correspond
precisely
n distinct values otw . Each of these values is called an th root of 2 , and we write
n2
w = .
Let z= r(coso + isind #0. Then the n values of "I can be obtained as follows :
=(cos Ottignot). , where 0,
Example 121
1 18 : Take
. 2 = 1. Then = 1 and
Argz = o
,
and so
= cos2 Yn + isin 2T/n
*
for K = 0
,
1
, ..., n - 1 .
(3) cos(23 ) + isin (20)
%
for = 0 : = = coso + isino -
,
= 1. (1) cos/) isin(1)= =
+
2
(T) cos(2 2) isin(at)= -
: =
3
·
+
x = 3 :
(3M) 2s(2.3) isin(23) 12 = + = - i
A
k = 1
-2π/3
-
& >
1 k = 0
2π/3
-
2
k= 2
1
Definition numbers have the form
1 .
1: Complex :
Xtiy
where X ,
y
ER and i is the
imaginary unit defined
,
to
satisfy :
i = -
1 .
i3 i . =
-i
ii ? j = 1
j5 it i =
.
= i
For complex number where is called the real part of 2 and
a
Extiy ,
YeR
X, X
y
is called the
imaginary part of 2 ,
written
Rez Imz
x=
y
=
,
x 5-2i Rex 5
eg
= = [mx = -2
defined and the usual rules of
Operations of addition ,
multiplication, subtraction and division are
algebra apply .
b
eg
a = 2+ 3i and =
-
1 + 2i
ab = (2+ 3i)( -
1 + 2i) = -
2 -
3i + 4i + 6i2
=
- 2+ i =
6 =
-
8+i
a /b = 2 + 3i = 2 +3i .
(1 -
2i) =
-
2 3i -
- 4i -
6i2
- 1 + 2i
-
1 +2i( -
1 -
2i) ( 12
-
= (2i) 2
47i = Ei
= + -
Definition 13: The complex of 2 where is defined be
conjugate iy yer
.
= X +
X to
, ,
xoiy
E
it is denoted
by
.
,
We O is the of it is
say argument
2 ,
denoted We
by argz . can represent
-----iz = -
2- i in the polar form: Z = ros Of irsing
= r(coso + isind
The principal value of the Lis definition the value O is
by satisfying
the of : TOLI
argument
I
argument
of -
and is denoted
by Arg2
.
Example 1 6 Find the principal
.
% value of the
argument
of Z of 21-i. .
Arg =
Arg(1 i= Arg(1-i) π/4 23
+ =
-
=
-1
,
i
-
Arg(25) Arg(-i)
=
-
/z
-
5
,Definition 1 7 The modulus complex
of number
zxxiy EyeR
is the
length of the
corresponding i e
:
. a vector , .
,
(2) = (x +
iy) = x2 +
y2
Example 1. 9 : Find and the modulus of 1. Write1 t polar form
argument
the in .
N
-----
i
r= 1 +1 = 2 0 = π/
I
p Y
1 1 + i =(cos + isin() .
Properties : 1) /zwl = /zllw)
al(z + w((z) + Iw) (the
triangle inequality .
Properties of I : 1) +We + w
2) w = z
3) zz =
1213
4)2 =
7212 ,
20
Example 3-4i Find 121 and write Iz1 and 1 form What is ZI ?
xtily XyeR
1 13
. : 2= in the .
.
,
(2) =
32+ 42
= 25 = 5
z = 3 + 4i
-
(z =
=
/1212 3 3/25 "e5i +
22 =
(2) = 25
Multiplication and Division in Polar Form
2 = r1(cos On + Isino)
ra(cos02 + IsinGal
on a
22 =
where P1 O2fIR ,
and re, re 30
3122
excata
Then : 2122 = rera(cos con + Gel + isinContoe)
2/22 = /ra(cosco-8) + isin(on-or ,
za o
Powers Let r(coso + isino
Integer z= .
Then
zr(Cosno + isinno).
Example 1 . 16 : Find (1 + il
| + il = E ,
0 = /4.
1+i =
E(cosT + isinπ/)
Ce + i)" =
2 4)cs ·
4 . T + isin 4 .
T
= CostT + isinit) =
-
4 .
, Roots 1 .
17 : If z = wh
,
n = 1 2 3
,
. ..
,
then to each value w
of there Corresponds one value of 2
. Conversely ,
to a
given
2to there correspond
precisely
n distinct values otw . Each of these values is called an th root of 2 , and we write
n2
w = .
Let z= r(coso + isind #0. Then the n values of "I can be obtained as follows :
=(cos Ottignot). , where 0,
Example 121
1 18 : Take
. 2 = 1. Then = 1 and
Argz = o
,
and so
= cos2 Yn + isin 2T/n
*
for K = 0
,
1
, ..., n - 1 .
(3) cos(23 ) + isin (20)
%
for = 0 : = = coso + isino -
,
= 1. (1) cos/) isin(1)= =
+
2
(T) cos(2 2) isin(at)= -
: =
3
·
+
x = 3 :
(3M) 2s(2.3) isin(23) 12 = + = - i
A
k = 1
-2π/3
-
& >
1 k = 0
2π/3
-
2
k= 2
