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Chapter 39
4/30/2017
By Chtan FYHS-Kulai
1
4  2,
- 4 ?
4/30/2017
By Chtan FYHS-Kulai
2
 : real numbers
consist of all positive
and negative integers,
all rational numbers
and irrational numbers.
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By Chtan FYHS-Kulai
3
Rational numbers are of
the form p/q, where p, q
are integers.
Irrational numbers are
3
 , e, 2 , 4,

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By Chtan FYHS-Kulai
4
4/30/2017
By Chtan FYHS-Kulai
5
Algebra form
z  a  bi
Notation
Trigonometric form
z  r cos  i sin  
index form
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By Chtan FYHS-Kulai
z  re
i
6
Complex numbers
are defined as numbers of the
form :
a  b 1
or
4/30/2017
a  ib
By Chtan FYHS-Kulai
7
-1 is represented by i
a,b are real numbers.
A complex number consists of
2 parts. Real part and
imaginary part.
4/30/2017
By Chtan FYHS-Kulai
8
Notes:
a  ib
1.When b=0, the complex number
is Real
2.When a=0, the complex number
is imaginary
3.The complex number is zero iff
a=b=0
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By Chtan FYHS-Kulai
9
VENN DIAGRAM
Representation
• All numbers belong to the Complex number
field, C. The Real numbers, R, and the
imaginary numbers, i, are subsets of C as
illustrated below.
Complex Numbers
a + bi
Real Numbers
a + 0i
Imaginary Numbers
0 + bi
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By Chtan FYHS-Kulai
11
Conjugate complex numbers
The complex numbers
and
a  ib
a  ib are called
conjugate numbers.
4/30/2017
By Chtan FYHS-Kulai
12
z  a  ib
z  a  ib
z is conjugate of z .
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By Chtan FYHS-Kulai
13
e.g. 1
Solve the quadratic equation
x  x 1  0
2
Soln:
4/30/2017
1  1  4 1   3
x

2
2
 1  3i

2
By Chtan FYHS-Kulai
14
e.g. 2
Factorise
x  y 
2
z .
2
Soln:
x  y 
2
 z  x  y   iz 
2
2
2
 x  y  iz x  y  iz 
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By Chtan FYHS-Kulai
15
Representation
of complex
number in an
Argand diagram
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By Chtan FYHS-Kulai
16
(Im) y
P(a,b)
a  ib
x (Re)
0
 a  ib
P’(-a,-b)
P a,b  a  ib
Argand diagram
4/30/2017
By Chtan FYHS-Kulai
17
e.g. 3
If P, Q represent the complex
numbers 2+i, 4-3i in the Argand
diagram, what complex number
is represented by the mid-point
of PQ?
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By Chtan FYHS-Kulai
18
y (Im)
Soln:
P(2,1)
x (Re)
0
Q(4,-3)
Mid-point of PQ is (3,-1)
 3  i is the complex number.
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By Chtan FYHS-Kulai
19
i  1
i  1 i  1
3
i  i
i  i
i  i
i 1
i 1
i 1
i i
i i
i i
2
4
5
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6
7
8
9
By Chtan FYHS-Kulai
10
11
12
13
20
Do pg.272 Ex 20a
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By Chtan FYHS-Kulai
21
Equality of
complex
numbers
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By Chtan FYHS-Kulai
22
The complex numbers
a  ib and c  id are said to
be equal if, and only if,
a=c and b=d.
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By Chtan FYHS-Kulai
23
e.g. 4
Find the values of x and y if
(x+2y)+i(x-y)=1+4i.
Soln:
x+2y=1; x-y=4
2y+y=1-4; 3y=-3, y=-1
x-(-1)=4, x=4-1=3
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By Chtan FYHS-Kulai
24
Addition of
complex
numbers
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By Chtan FYHS-Kulai
25
z1  a  ib ; z2  c  id
If
then
z1  z2  a  c  ib  d 
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By Chtan FYHS-Kulai
26
Subtraction
of complex
numbers
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By Chtan FYHS-Kulai
27
z1  a  ib ; z2  c  id
If
then
z1  z2  a  c  ib  d 
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By Chtan FYHS-Kulai
28
Do pg.274 Ex 20b
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By Chtan FYHS-Kulai
29
Multiplication
of complex
numbers
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By Chtan FYHS-Kulai
30
e.g. 5
If z  3  i, find the values of
2
(i) z
(ii) z z
Soln:
(i)
z  3  i 3  i   9  1  6i  8  6i
2
(ii) z z  3  i 3  i   9  1  10
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By Chtan FYHS-Kulai
31
z1  a  ib ; z2  c  id
If
then
z1  z2  a  ib c  id 
 ac  bd   iad  bc 
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By Chtan FYHS-Kulai
32
Division
of complex
numbers
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By Chtan FYHS-Kulai
33
If
then
z1  a  ib ; z2  c  id

z1
a  ib 

c  id 
z2

a  ib  c  id 

c  id  c  id 

ac  bd   i bc  ad 

c
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2
By Chtan FYHS-Kulai
d
2

34
e.g. 6
Express
2i
3i
in the form a  ib .
Soln:
2  i 2  i 3  i 6  1  5i


3i 3i 3i
9 1
5  5i 1

 1  i 
10
2
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By Chtan FYHS-Kulai
35
e.g. 7
If  x  iy 2  i   3  i, find the
values of x and y.
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By Chtan FYHS-Kulai
36
e.g. 8
If z=1+2i is a solution of the equation
2
z  az  b  0 where a, b are real,
find the values of a and b and verify
that z=1-2i is also a solution of the
equation.
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By Chtan FYHS-Kulai
37
The cube
roots of
unity
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By Chtan FYHS-Kulai
38
If
z
is a cube root of 1,
z 1
or
3
z 1  0
3
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By Chtan FYHS-Kulai
39
z 1  0
3
z 1z
2

 z 1  0
 z  1 ; z  z 1  0
2
 1  3i
z 
2
The cube roots of unity are
1
1
1,  1  i 3 ,  1  i 3
2
2

4/30/2017
 
By Chtan FYHS-Kulai

40
Notice that the complex roots
have the property that one is the
square of the other,


2


2

 


 

1
1
1

 2  1  i 3   4 1  i 2 3  3  2  1  i 3
1
1
1

 2  1  i 3   4 1  i 2 3  3  2  1  i 3
4/30/2017
By Chtan FYHS-Kulai
41
let
 1  3i

2
So the cube roots of unity
can be expressed as
1,  , 
4/30/2017
2
By Chtan FYHS-Kulai
42
If we take
then
1
3
  i
2 2
1
3
  i
2 2
or vice versa.
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By Chtan FYHS-Kulai
43
(1) As z   is a solution of
  1
z 1  0
3
3
(2) As z   is a solution of
z  z 1  0
2
    1  0
2
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By Chtan FYHS-Kulai
44
    1
2
(3)
(4)
(5)
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
3n 1
 
  
3n
… etc

2 3n 1

2 3n
By Chtan FYHS-Kulai
 1
2
45
e.g. 9
Solve the equation  z  1  1 .
3
4/30/2017
By Chtan FYHS-Kulai
46
e.g. 10
If  is a cube root of unity, show that
    1
4
4/30/2017
2
By Chtan FYHS-Kulai
47
Soln:
We have ,     1  0
2
     1  0
4
3
2
    0
2
2
    
4
2
    1
4
4/30/2017
2
By Chtan FYHS-Kulai
3
48
Do pg.277 Ex 20c
4/30/2017
By Chtan FYHS-Kulai
49
4/30/2017
By Chtan FYHS-Kulai
50
z  x  iy
y
0
P(x,y)

x
y
    
Is called the
principal value
x
Argand diagram
r is called the modulus of z, θ(in radians)
is called the argument of z.
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By Chtan FYHS-Kulai
51
From the Argand diagram,
x  r cos  , y  r sin 
z  x  iy
 r cos   ir sin 
 r cos   i sin  
This is called the (r,θ) or modulus-argument
form of the complex number
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By Chtan FYHS-Kulai
52
z  x  iy
 r cos   ir sin 
 r cos   i sin  
This is called the (r,θ) or modulus-argument
form of the complex number
or modulus-amplitude form of the complex
number
4/30/2017
By Chtan FYHS-Kulai
53
x  r cos  , y  r sin 
r z 
x y
2
2
 y
  arg z  tan  
x
1
modulus
4/30/2017
argument
(or amplitude)
By Chtan FYHS-Kulai
54
 y
  arg z  tan  
x
1
OR
 y
  am z  tan  
x
1
4/30/2017
By Chtan FYHS-Kulai
55
One important formulae :
z  z  zz
2
2
Refer to Example 15 below
4/30/2017
By Chtan FYHS-Kulai
56
Representation of cube roots of
unity in Argand diagram
B
y
 1 3
 ,

 2 2 


0
 1
3
C   , 
2
2 

4/30/2017
A1,0
2

3
By Chtan FYHS-Kulai
x
ABC
is an equilateral
triangle.
57
Geometrically, if P1, P2, P3
represent the number z1, z2 and
z1+z2. Then, you see the following
diagram :
y
z1
z1+z2
0
4/30/2017
z2
By Chtan FYHS-Kulai
x
58
Multiplication and
division of two
complex numbers
(in modulusargument form)
4/30/2017
By Chtan FYHS-Kulai
59
If
z1  r1 cos1  i sin 1 
z2  r2 cos2  i sin 2 
then
z1  z2  r1r2 cos1  2   i sin 1  2 
4/30/2017
By Chtan FYHS-Kulai
60
z1 r1
 cos1   2   i sin 1   2 
z 2 r2
4/30/2017
By Chtan FYHS-Kulai
61
e.g. 11
If
find
4/30/2017
z  cos  i sin  
1
?
z
By Chtan FYHS-Kulai
62
e.g. 12
If
4i  log 3 m  5 , find
the value of m .
4/30/2017
By Chtan FYHS-Kulai
63
e.g. 13
If
4/30/2017
1 z
i,
1 z
then 1  z  ?
By Chtan FYHS-Kulai
64
e.g. 14
If
4/30/2017
z
i
1
i
 1 i
, then z  ?
By Chtan FYHS-Kulai
65
Miscellaneous
examples
4/30/2017
By Chtan FYHS-Kulai
66
e.g. 15
Evaluate
4  i 6  2i 
5
7
ans : 22 14i
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By Chtan FYHS-Kulai
67
e.g. 16
z  8  6i
Given
find
4/30/2017
2
100
z  16 z 
z
3
By Chtan FYHS-Kulai
,
.
68
e.g. 17
z  5 , 3  4i z
is an imaginary
number, z  ?
4/30/2017
By Chtan FYHS-Kulai
69
e.g. 18
If
1
3
  i
2 2
, then
1     
2
4/30/2017
By Chtan FYHS-Kulai
13
?
70
e.g. 19
Prove that
  
3n
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
2 3n
By Chtan FYHS-Kulai
2
71
e.g. 20
Find the value
1     1     
2
2
ans : 4
4/30/2017
By Chtan FYHS-Kulai
72
e.g. 21
Prove that

4/30/2017
3n 1
 

2 3n 1
By Chtan FYHS-Kulai
 1
73
e.g. 22
Simplify
cos   i sin  
3
cos 2  i sin 2
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By Chtan FYHS-Kulai
74
e.g. 23
If
z1  r1 cos1  i sin 1 
z2  r2 cos2  i sin 2 
Show that
z1 z2  r1r2
4/30/2017
By Chtan FYHS-Kulai
75
e.g. 24
If
z  cos  i sin    
Prove that
1 z


i
tan

2
1 z
2
4/30/2017
By Chtan FYHS-Kulai
76
e.g. 25
If
z  cos  i sin    
Prove that
4/30/2017

1 


1 

By Chtan FYHS-Kulai
1
2 
z 
 i tan 
1
2 
z 
77
Addendum
4/30/2017
By Chtan FYHS-Kulai
78
(1)
In general, z is a complex
number then,
z a
represent a circle with
centre at (0,0) and radius
“a”.
4/30/2017
By Chtan FYHS-Kulai
79
(2)
amz   
represent a straight line
with gradient=tanθ.
4/30/2017
By Chtan FYHS-Kulai
80
(3)
If 4 points P1, P2, P3, P4 are
concyclic, then
z3  z1
z 4  z1
am
 am
z3  z 2
z4  z2
4/30/2017
By Chtan FYHS-Kulai
81
y
P2
P1
P3
0
P4
x
P1, P2, P3, P4 are concyclic
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By Chtan FYHS-Kulai
82
(4)
If 3 points P1, P2, P3 formed
an equilateral triangle,
z2  z3   z3  z1   z1  z2 
2
4/30/2017
2
By Chtan FYHS-Kulai
2
0
83
Do pg.280 Ex 20d
& Misc 20
4/30/2017
By Chtan FYHS-Kulai
84
Do pg.127 Ex 6a
Pg. 130 Ex 6b
Pg. 138 Ex 6d q1-q10, q12,
q14, q16
No need to do
Pg. 135 Ex 6c, pg. 139 Misc
4/30/2017
By Chtan FYHS-Kulai
85
The
end
4/30/2017
By Chtan FYHS-Kulai
86
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