introduction operations of complex number the complex plane the modulus & argument the polar...
TRANSCRIPT
• INTRODUCTION• OPERATIONS OF COMPLEX NUMBER
• THE COMPLEX PLANE• THE MODULUS & ARGUMENT
• THE POLAR FORM
CHAPTER 1:COMPLEX NUMBER
CLASSIFICATION
COMPLEX NUMBERS
(C)
REAL NUMBERS
(R)
INTEGERS(Z)
RATIONAL NUMBERS
(Q)
IRRATIONAL
NUMBERS(Q)
WHOLE NUMBERS
(W)
NATURAL NUMBERS
(N)
To solve algebraic equations that do not have real solutions.
To solve Complex number:
Since,
INTRODUCTION TO COMPLEX NUMBERS
4
04
2
04
2
2
x
x
x
xReal solution
No real solution
ix 24
112 ii
Example 1 :Solve
8
5
)
)
ib
ia
Example :Solution
1)1()()
)1()()2428
2225
iib
iiiiia
Definition 1.1If z is a complex number, then the
standard equation of Complex number denoted by:
where a, b Ra – Real part of z (Re z) b – Imaginary part of z (Im z)
biaz
Example 1.2 :Express in the standard form, z:
493)
42)
b
a
Re(z) = 2, Im (z) = -2
Example 1.2 :Solution:
izzb
izza
1023403)
2242)
Re(z) = 3, Im (z) = 2√10
Definition 1.22 complex numbers are said equal if
and only if they have the same real and imaginary parts:
Iff a = c and b = d
dicbia
Example 1.3 :Find x and y if z1 = z2:
iyixb
iyixa
20105)
9432)
Definition 1.3If z1 = a + bi and z2 = c + di, then:
OPERATIONS OF COMPLEX NUMBERS
ibcadbdaczziii
idbcazzii
idbcazzi
)()()
)()()
)()()
21
21
21
Example 1.4 :Given z1 = 2+4i and z2= 1-2i
21
21
21
)
)
)
zzc
zzb
zza
Definition 1.4The conjugate of z = a + bi can be defined as:
***the conjugate of a complex number changes the sign of the imaginary part only!!!
biabiaz
Example 1.5 :Find the conjugate of
izd
zc
izb
iza
10)
10)
23)
2)
The Properties of Conjugate Complex Numbers
)Im(2
)
)Re(2
)
;)
11)
..)
)
)
)
2121
2121
2121
zzz
viii
zzz
vii
nzzvi
zzv
zzzziv
zzzziii
zzzzii
zzi
nn
Definition 1.5 (Division of Complex Numbers)
If z1 = a + bi and z2 = c + di then:
22
2
1
)(
dc
iadbcbdacdic
dic
dic
bia
dic
bia
z
z
Multiply with the conjugate of denominator
Example 1.6 :Simplify and write in standard form, z:
i
ib
i
ia
31
43)
1
2)
The complex number z = a + bi is plotted as a point with coordinates (a,b).Re (z) x – axis Im (z) y – axis
THE COMPLEX PLANEOR
ARGAND DIAGRAM
Im(z)
Re(z)O(0,0)
z(a,b)
a
b
Definition 1.6 (Modulus of Complex Numbers)
The modulus of z is defined by
THE MODULUS & ARGUMENT OF A COMPLEX NUMBER
22 bazr Im(z)
Re(z)O(0,0)
z(a,b)
a
b
r
Example 1.7 :Find the modulus of z:
izb
iza
53)
2)
The Properties of Modulus
2121
22
1
2
1
2121
2
)
)
0,)
)
)
)
zzzzvi
zzv
zz
z
zz
iv
zzzziii
zzzii
zzi
nn
Definition 1.7 (Argument of Complex Numbers)
The argument of the complex number z = a + bi is defined as
a
b1tan
1st QUADRANT
2nd QUADRANT
4th QUADRANT
3rd QUADRANT
900 18090
360270 270180
Example 1.8 :Find the arguments of z:
izd
izc
izb
iza
2)
1)
53)
2)
Based on figure above:
THE POLAR FORM OF COMPLEX NUMBER
b
a
(a,b)
r
Re(z)
Im(z)
a
b
rb
ra
1tan
sin
cos
The polar form is defined by:
Example 1.9:Represent the following complex number in
polar form:
izc
izb
iza
2)
53)
2)
,
@
sincos
rz
irz
Answer 1.9 :Polar form of z:
90sin90cos2)
96.120sin96.120cos34)
43.333sin43.333cos5)
izc
izb
iza
Example 1.10 :Express the following in standard form of
complex number:
270sin270cos2)
)180sin180(cos3)
)45sin45(cos2)
izc
izb
iza
Answer 1.10 :Standard form:
izc
zb
iza
2)
3)
22)
Theorem 1:If z1 and z2 are 2 complex numbers in
polar form where
then,
1111 sincos irz 2222 sincos irz
21212
1
2
1
21212121
sincos)
sincos)
ir
r
z
zii
irrzzi
Example 1.11 :a) If z1 = 4(cos30+isin30) and z2 =
2(cos90+isin90) . Find :
b) If z1 = cos45+isin45 and z2 = 3(cos135+isin135) . Find :
2
1
21
)
)
z
zii
zzi
2
1
21
)
)
z
zii
zzi
Answer 1.11 :
iz
zii
zzi
b
iz
zii
izzi
a
3
1)
3)
)
31)
344)
)
2
1
21
2
1
21
Think of Adam and Eve like an imaginary number, like the square root of minus one: you can never see any concrete proof that it exists, but if you include it in your equations, you can calculate all manner of things that couldn't be
imagined without it.
Philip PullmanIn The Golden Compass (1995, 2001),
372-373.
QUOTES