complex numbers (kdu)

8/3/2019 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

Slide 4


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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

!

Slide 5


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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()(

.

Slide 6


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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.

Slide 7


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Real numbers and imaginary numbers are

subsets of the set ofcomplex numbers.

ComplexNumbers

RealNumbersImaginaryNumbers

Slide 8


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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.

Slide 11


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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

!

Slide 12


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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

Slide 13


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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

complex numbers (kdu)

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