Ch39 Complex Numbers1

8/8/2019 Ch39 Complex Numbers1

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1/16/2011 1By Chtan FYHS-Kulai

Chapter 39


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!

!

4-

24


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numbersreal:

consist ofall positive

andnegative integers,all rational numbers

andirrational numbers.


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Rational numbers are of

the form p/q, where p,qare integers.

Irrational numbers are

.3 4,,, eT


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1/16/2011 By Chtan FYHS-Kulai 5


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Notation

Algebra form

biaz

Trigonometric form

UU sincos irz !

index formUirez !


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Complex numbers

are defined asnumbersofthe

form : 1 baor

iba


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1- is represented by iba are real numbers.

A complex number consistsof

2 parts. Real partand

imaginary part.


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Notes:

1.When b=0, the complex number

is Real

2.When a=0, the complex numberisimaginary

3.The complex number is zeroiffa=b=0

iba


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VENN DIAGRAMRepresentation

All numbers belong to the Complex numberfield,C. The Real numbers, R, and theimaginary numbers, i, are subsets ofC as

illustrated below.

Real Numbers

a + 0i

Imaginary Numbers

0 + bi

Complex Numbersa + bi


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Conjugate complex numbers

The complex numbers

iba

and are callediba

conjugate numbers.


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ibaz

ibaz !

z isconjugate of .z


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e.g. 1

Solve the quadratic equation

012 ! xx

Soln:

2

31

2

31

2

411

i

x

s!

s!

s!


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e.g. 2

Factorise . 22

zyx

Soln:

2222 izyxzyx !

izyxizyx !


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Representationof complex

number in an

Argand diagram


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x

y

0

P(a,b)

P(-a,-b) baP ,

Argand diagram

iba

iba iba

(Re)

(Im)


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e.g. 3

IfP, Q represent the complex

numbers2+i, 4-3iin the Argand

diagram, what complex numberis represented by the mid-point

ofPQ?


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Soln:

x

y

0

P(2,1)

Q(4,-3)Mid-point ofPQis (3,-1)

i@ 3 is the complex number.

(Re)

(Im)


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12

!i

ii !3

14 !i

ii !5

16 !i

ii !7

18 !i

ii !9

110 !i

ii !11

112 !i

ii !13


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Do pg.272 Ex 20a


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Equality of

complex

numbers


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The complex numbers

and are said tobe equal if, and only if,

a=cand b=d.

iba idc


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e.g. 4

Find the valuesofxandyif(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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Addition of

complex

numbers


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If idczibaz !!21

;

then

dbicazz !21


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Subtraction

of complex

numbers


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If idczibaz !!21

;

then

dbicazz !21


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Do pg.274 Ex 20b


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Multiplication

of complex

numbers


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e.g. 5

If , find the valuesof(i) (ii)

iz ! 32z zz

Soln:(i) iiiiz 6861933

2!!!

(ii) 101933 !!! iizz


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If idczibaz !!21

;

then

bcadibdac

idcibazz

!

!y

21


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Division

of complex

numbers


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If idczibaz !!21

;

then

22

2

1

dc

adbcibdac

idc

idc

idc

ibaidc

iba

z

z

!

!

!


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e.g. 6

Express in the form .ii

3

2

iba

Soln:

ii

i

i

i

i

i

i

i

!

!

!

!

12

1

10

55

19

516

3

3

3

2

3

2


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e.g. 7

.

,32

yandxofvalues

thefindiiiyxIf !


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e.g. 8

Ifz=1+2iis a solutionofthe equationwhere a, b are real,

find the valuesofa and b andverify

that z=1-2iis also a solutionofthe

equation.

02 ! bazz


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The cube

roots of

unity


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If is a cube root of1,z

01

1

3

3

!

!

z

orz


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013 !z

0112

! zzz

01;1 2 !!@ zzz

2

31 iz

s!@

312

1,31

2

1,1

ii

areunityofrootscubeThe

@


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Notice that the complex roots

have the property that one is thesquare ofthe other,

31213321

4

1312

1

2

iii !!-

312

13321

4

131

2

12

iii !!

-


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If we take i23

21 ![

then i2

3

2

1![

or vice versa.


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(1) As is a solutionof[!z

013 !

z1

3 !@[

(2) As is a solutionof[!z

012

! zz01

2 !@ [[


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(3)

1

13213!

nn [[(4)

12

![[

(5) 2323

!n

n [[ etc


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e.g. 9

Solve the equation . 113

!z


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e.g. 10

1

,

24 ![[

[ thatshowunityofrootcubeaisIf


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Soln:

01havee 2 ![[W

0122 ![[[0

234 ! [[[324

[[[!

1

24!@ [[


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Do pg.277 Ex 20c


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i


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x

y

0

P(x,y)

x

y

U

TUT e

Iscalled theprincipal value

iyxz !

Arganddiagram

riscalled the modulus ofz, (in radians)iscalled the argumentofz.


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From the Arganddiagram,

UU sin,cos ryrx !!

UUUU

sincos

sincos

ir

irr

iyxz

!

!

!

Thisiscalled the (r,) or modulus-argument

form of the complexnumber


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UUUU

sincos

sincos

ir

irr

iyxz

!!

!

Thisiscalled the (r,) or modulus-argument

form of the complexnumber

or modulus-amplitude form of the complex

number


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UU sin,cos ryrx !!

22 yxzr !!

!!

x

yz 1tanargU

modulus argument(or amplitude)


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!!

x

yz 1tanargU

OR

!!

x

yzam 1tanU


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One important formulae :

zzzz !!

22

Refer to Example 15 below


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Geometrically, ifP1,P2,P3

represent the numberz

1,z

2 andz1+z2. Then, yousee the following

diagram : y

x0 z2

z1

z1+z2


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Multiplication and

division oftwo

complex numbers

(in modulus-argument form)


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1111 sincos UU irz !

2222sincos UU irz !

If

then

21212121sincos

UUUU ! irrzz


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2121

2

1

2

1sincos UUUU ! i

r

r

z

z

11


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e.g. 11

If UU sincos iz !

find ?1 !z

12


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e.g. 12

If , find

the value ofm .

5log4 3 ! mi

13


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e.g. 13

If , thenizz !

1

1 ?1 ! z

14


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e.g. 14

If , theniiiz ! 1

1?!z


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Miscellaneous

examples

15


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e.g. 15

Evaluate

75

264 ii

? Aians 1422:

16


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e.g. 16

Given ,

find .

iz 682 !

z

zz100

163

17


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e.g. 17

ziz 43,5 !

is animaginarynumber, ?!z

18


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e.g. 18

If , theni2

3

2

1![

!132

1 [[[ .

19


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e.g. 19

Prove that

2323 ! nn [[

20


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e.g. 20

Find the value

22

11 [[[[

? A4

:ans

21


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e.g. 21

Prove that

113

213 ! nn

[[

e 22


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e.g. 22

Simplify

UU

UU

2sin2cos

sincos3

i

i

e g 23


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e.g. 23

If 1111 sincos UU irz ! 2222

sincos UU irz !

Show that

2121 rrzz !

e g 24


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e.g. 24

If UU sincos iz ! U

Prove that

Utan1

1

2

2

iz

z!

e g 25


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e.g. 25

If UU sincos iz ! U

Prove that

Utan1

1

11

2

2

i

z

z!


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Addendum

(1)


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In general, z is a complex

number then,

a

z!

represent a circle with

centre at (0,0) and radius

a.

(1)

(2)


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(2)

U!zam

represent a straight linewith gradient=tan.


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0

y

x

1P 2P

3P 4

P

P1,P2,P3,P4 are concyclic

(4)


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(4)

If3 pointsP1,P2,P3 formed

an equilateral triangle,

0221

2

13

2

32! zzzzzz


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Do pg.280 Ex 20d

& Misc 20


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Do pg.127 Ex 6a

Pg. 130 Ex 6b

Pg. 138 Ex 6d q1-q10, q12,

q14, q16

Noneed todo

Pg. 135 Ex 6c, pg. 139 Misc


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The

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