Download - Complex Numbers (KDU)
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Complex NumbersComplex Numbers
Slide 1
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Content of the sessionContent of the session
Complex numbers, Algebra of complex
numbers, De Moivres theorem, Roots of
complex numbers, Solving Complex equations,Modules, Argument, Polar form, Argand
diagram.
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INTENDED LEARNING OUTCOMESINTENDED LEARNING OUTCOMES
y Find the complex solution ofequations and
solve problems by usingdeMoivre'sT
heorem
Identify the complex number system and perform
thealgebra operations of complex numbers
Identify algebraic andGeometric properties of
complex numbers
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We know that ifx20, such numbers are calledas real
numbers.
But we already met equation such as x2 = -1, whose roots are
clearly not real. To work with this type ofequations we need
another category of numbers, namely the set of numbers whose
squares are negative real numbers.
Members of these sets are calledimaginary numbers.
Ex-
.20,7,1
Generally memes of this set is denoted as
real.iswhere,2 nn
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.1,
1
1n
2
22
!!
v!
v!
iwhereni
n
nThen ,
So every imaginary numbers can be represent in the form
ni,w
here n is real and .1!i
Properties of imaginary numbers
a. Imaginary numbers can beaddedand subtracted from
anotherimaginary numbers.
Ex- (i).
(ii).
iii 752 !
iii)17(7
!
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b. The product orquotient of tow imaginary numbers is
real.
)1(,6623 22 !!!v iwhereiii
4416 !z ii
Ex- (i).
(ii).
c. Power ofI can be simplified.
i
i
i
i
i
iiiii
ii
iiii
!!!
!!!
!!!
!!
2
1
45
2224
23
1
.1.
1)1()(
.
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Definition ofa Complex Number
Where a real numbers and an imaginary numbers are added or
subtracted, the expression which cannot be simplified is called a
complex number.
iiiiEx 52,32,34,23:
In general complex numbers can be written in the form a + ib and
denoted by z = a + ib. where a and b are any real number
including zero.
Ifb =0, the numbera + bi = a is areal number.
If a=0, the numbera + bi is calledan imaginary number.
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Real numbers and imaginary numbers are
subsets of the set ofcomplex numbers.
ComplexNumbers
RealNumbersImaginaryNumbers
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Operations ofcomplex numbers
Addition and Subtraction of Complex Numbers
Sum:
Difference:
i)db()ca()dic()bia( !
i)db()ca()dic()bia( !
If a + bi and c +di are two complex numbers written in
standard form,their sum and difference are defined as follows
Ex- Perform the subtraction and write theanswerinstandard form.
1. ( 3 + 2i ) (6 + 13i )
= 3 + 2i 6 13i
= 3 11i Slide 9
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234188).2 i
4
234238
234298
!
!
v!
ii
ii
Multiplying Complex Numbers
Multiplying complex numbers is similar to multiplying
polynomials and combining like terms.
Ex-01): Perform the operation and write theresult in standardform.
(6 2i )(2 3i )
=12 18i 4i + 6i2
=12 22i + 6(-1)
=6 22i Slide 10
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02). ( 3 + 2i )( 3 2i )
= 9 6i + 6i 4i2
= 9 4(-1)
= 13
Complex Conjugates
Any pair of complex numbers of the form a ib havea
product which is real.
.
))((
22
22
ba
ibaibaiba
!
!
Such complex numbers are said to be conjugateandeach is
the conjugate of the other.
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Direct division by a complex number cannot be carried out
because the denominator is made up of two independent
terms. This difficulty can be overcome by making the
denominator as real, a process is called realizing the
denominator.
Division
To find the quotient of two complex numbers multiply the
numerator and denominator by the conjugate of thedenominator.
dicbia
dic
dic
dic
bia
y
!
22
2
dc
bdibciadiac
!
22 dc
iadbcbdac
!
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Ex:- Perform the operation and write theresult in standard
form.
i
i
21
76
i
i
i
i
21
21
21
76
v
!
22
2
21
147126
!
iii
41
5146
!
i
5
520 i!
5
5
5
20 i! i! 4
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Ex:- Perform the operation and write theresult in standard form.
ii
i
4
31 i
i
ii
i
i
i
v
v
!
44
431
222
2
14
312
!
i
i
ii
116
312
1
1
!
ii
ii
17
3
17
121 ! ii
17
3
17
121 !
i17
317
17
1217
! i
17
14
17
5!
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The imaginarypart ofz is denoted by
Im(z)=Im(a+ib)=b
Note: 1). If z=a+ ib, therealpart ofz is denoted by
Re(Z)=Re(a+ib)=aand
2). Thezero complex number
Acomplex numberis zero its realand imaginary term areeach zero
i.e a + ib =0 a=0 and b =0.
Slide 15
3). Equal complex numbers:
Let z1 =a1 + ib1 and z2 =a2 + ib2.
If z1 = z2 a1 + ib1 = a2 + ib2.
( a1
- a2
)+ i(b1
- b2) = 0
( a1 - a2 ) = 0 and (b1 - b2) = 0
a1
= a2
and b1
= b2