chapter2 number system

8/14/2019 Chapter2 Number System

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Siti Zanariah Satari DCT1043 Chapter 2 Number System

NUMBER SYSTEMNUMBER SYSTEM

Chapter 2Chapter 2

DCT1043DCT1043


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ContentContent

2.1 Real Numbers2.1 Real Numbers

2.2 Indices2.2 Indices

2.3 Logarithm2.3 Logarithm

2.4 Complex Number2.4 Complex Number


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2.1 Real Numbers2.1 Real Numbers

At the end of this topic you should be able toAt the end of this topic you should be able to

Define natural numbers, whole numbers, integers,Define natural numbers, whole numbers, integers,prime numbers, rational numbers and irrationalprime numbers, rational numbers and irrationalnumbersnumbers

Represent rational and irrational numbers in decimalRepresent rational and irrational numbers in decimalformform Represent the relationship of number sets in realRepresent the relationship of number sets in real

number system diagrammaticallynumber system diagrammatically State clearly the properties of real numbers such asState clearly the properties of real numbers such as

closure, commutative, associative, distributive,closure, commutative, associative, distributive,identity and inverse under addition and multiplicationidentity and inverse under addition and multiplication

Represent the open, closed and half-open intervals inRepresent the open, closed and half-open intervals innumber linenumber line


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The Set of Real NumbersThe Set of Real Numbers RR

0

{ }, 2, 1Z = K , 1 andcan be divide 1 and its number only

x x Z xP

x

+ =

Natural/Counting Numbers/ positive integers

{ } { }1, 2, 3, 4, 5, or 1, 2, 3, 4, 5,N Z+

= =K K

Integers

{ }, 2, 1, 0,1, 2,I Z= = K K

Whole Numbers

{ }0,1,2,3,4,5,W = K

Rational Numbers

{ }and are integers, 0aQ a b bb= Irrational Numbers

FractionsTerminating or repeating

decimal numbers

Zero

Negative Integers Prime number

Composite number

{ }andx x Z x P +

{ }H Q x x Q= =

Proper,Improper,

MixedNumber

Nonterminating &

nonrepeating

decimalnumber


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Relationship among various setsRelationship among various sets

of numberof number

Irrational

numbers

Real NumbersR

Rational Numbers Q

IntegersZWhole numbers W

Natural

NumbersN

Irrational

NumbersH

N W Z Q R


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Representing Real Numbers asRepresenting Real Numbers as

DecimalDecimal

Every real number can be written as decimal

Repeating or terminating

Repeating (rational numbers)

0.6666666 , 0.33333 Terminating (rational numbers)

=0.5

Neither Terminate nor Repeat (irrationalnumbers)

( ) ( )0.6 0.3

2 1.41421... , 3.14159...= =


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ExerciseExercise

Given a set of numbersGiven a set of numbers

List the numbers in the set that areList the numbers in the set that are Natural numbersNatural numbers Whole numbersWhole numbers IntegersIntegers Rational numbersRational numbers Irrational NumbersIrrational Numbers Real numbersReal numbers

2110, 7, , ,0,5,6.125125,0.3, 95


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Representing Real Numbers asRepresenting Real Numbers as

Number LineNumber Line

0

x

Origin

Negativedirection

Positivedirection


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ExerciseExercise

Graph the elements of each set on aGraph the elements of each set on anumber linenumber line

3 78,5, 2 ,1.75, 2, 0.6, , , 64

5 2

2110, 7, , ,0,5,6.125125,0.3, 9

5

49, 0,0.025, 3,9.2, 100

5


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Properties of the RealProperties of the Real

Number SystemNumber SystemRules of Operations

( ) ( )

1. Co mmutative law of addition

2. Associative law of addition

3. 0 0 Identit

Under Addition

a b b a

a b c a b c

a a

+ = +

+ + = + +

+ = +

( )

( ) ( )

y law of addition

4. =0 Inverse Law of addition

1. Commutative law of multiplication

2.

a a

Under Multiplication

ab ba

a bc ab c

+

=

= Associa tive law of multiplication

3. 1 1 Identity law of multiplication1

4. =1

a a

aa

=

g g

( )

Inverse Law of multiplication

1. Distributiv e law for multiplication

w

Under Addition and Multiplication

a b c ab bc+ = +

.r.t addition


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Properties of Negatives

( )

( ) ( ) ( )

( ) ( )

( )

1.

2.

3.

4. 1

a a

a b ab a b

a b ab

a a

=

= =

=

=

Properties Involving Zero

1. 0 0

2. If 0 then 0, 0 or both

a

ab a b

== = =

g


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Properties of Quotients

( )

( )

( )

( )

( )

1. if , 0

2. , 0

3. 0

4. , 0

5. , , 0

6.

a cad bc b d

b d

ca ab c

cb b

a a ab

b b b

a c acb d

b d bd

a c a d ad b c d

b d b c bca c ad bc

b d bd

= =

=

= =

=

= =

++ =

g

g

( )

( )

, 0

7. , 0

b d

a c ad bcb d

b d bd

=


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ExerciseExercise

Simplify the following algebraicSimplify the following algebraicexpressionexpression

( ) ( )7 4 3 2 5 x y x y+ + +

( ) ( )3 10 10 16 y z y z + + +

20 3x x +

( ) ( )4 10 8 4 3 5x x +

( ) ( )5 2 9 9 8y y

4 9 2 12 x y x y +

( )5 3 2 1 7x +

( ) ( )2 27 2 1 3p p+ +


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Open & Closed IntervalOpen & Closed Interval

Open Interval

{ } ( )or ,x a x b a b<

=

=

=

( )

fraction eksponents rule

4. fraction ek sponents rule

5. product rule for eksponents

6.

q

qpp

m n m n

mm n

n

a a

Rule for Same Base

a a a

aa

a

+

=

=

=

( )

( )

quotient rule for eksponents

7. power rule for eksponents

( )

5. product rule for e ksponents

6.

nm mn

nn n

nn

n

a a

Rule for Same Index power

a b ab

a a

b b

=

=

= quotient rule for ekspon ents


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ExerciseExercise

Without using a calculator, evaluateWithout using a calculator, evaluatethe followingthe following

( )

2

3

2

. 5

. 121

a

b

( )

1 1

2 2

1

2

1122

9 8.

2

. 32 2

e

f

2

3

3

2

27

. 8

1.9

c

d


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ExerciseExercise

SimplifySimplify

( )

3

1

253

2

4 1 3

. 2 16 4

1.

16

1. 2

4

n n na

b x

c x y xy

( )

42 3

5

3

2 4

2 2

3.

27

.

.

x yd

xy

xye

x y

x yf

x y

+


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ExerciseExercise

IfIf

ExpressExpress

in term ofin term ofyy

3xy =

4 13 and 9x x +


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Radicals

1

2

1If then

2

read 'the square root of ' where

is any real numbers (rational or irrational)

is called the radical signis called radicand

n

a a

a

a

a

=

=

Principle of Square Root

products rule for radicals

quotients rule for radicals

a b ab

a a

bb

=

=


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Surds

the square root of ,

where the roots cannot be evaluated exac tlyThere are irritional numbers as they can not be expressed as fractions

of the form where and are integers

a a

aa b

b

Operation on Surds

( )

( )

( )

2

Multiplication,

Division,

Addition,

Subtraction,

Product, 2

a b ab

aa b

bb a c a b c a

b a c a b c a

a b a b ab

=

=

+ = +

=

= +


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ExerciseExercise

Simplify theSimplify the

followingfollowing. 128

. 3 5 4 10

3 5.2 27

a

b

c

( )

( ) ( )

. 2 13 3 5 500

. 5 5 3 2

. 5 3 2 3 5 2

d

e

f

+

+ ( )2

3 7.

5 63

. 50 18 3 2

. 2 1

g

h

i

+


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Conjugate Surds

and

it can be used to rationalise the denominator

a h b k a h b k +

Rationalized Surds

)

1 1)

1 1)

a a b a bi

bb b b

a b a biia ba b a b a b

a b a biii

a ba b a b a b

= =

= = + +

+ += =

+ +


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ExerciseExercise

Simplify the following bySimplify the following by rationalizingrationalizingthe denominatorsthe denominators

3.

11

5.

3 2

2.

3

a

b

c

1.

2 5

1.

3 5

3 5.

3 5

d

e

f

+

+

( )2

1.

2 1

3 2.

3 1 2 2 1

3 5 2.

2 3 3 1

g

h

i

+ +


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Surds Rules

For , , ,

)

)

)

)

n n

n n n

n

nn

m n mn

a b R m n Z

i a a

ii ab a b

a aiii

b b

iv a a

+ =

=

=

=


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ExerciseExercise

Write the following in term of surdsWrite the following in term of surds

Write the following in term of indexWrite the following in term of index

( )3 1 12

35 3 35. . 4 .i a ii xy iii x y+

( )23 445 9. . 5 . 3i x ii a iii m n


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ExerciseExercise

SimplifySimplify

( )3 2 4

3 25 3 3

412 3

2 7 23

2

.

2

.

i x y x y

a b a

ii c bc

1 1

2 4

1

2

.a a

iii

a

+


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2.3 Logarithm2.3 Logarithm

At the end of this topic you should beAt the end of this topic you should beable toable to

State and use the law of algorithmState and use the law of algorithm

Change the base of logarithmChange the base of logarithm

Understand the meaning of lnUnderstand the meaning of ln MM and logand log

MM Solve equations involving logarithmSolve equations involving logarithm


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What is LogarithmWhat is Logarithm

Logarithm is theLogarithm is the powerpower

WHY?WHY?

For any positive number , except 1,

logx a

a

y a y x= =

Index formIndex form Logarithm formLogarithm form


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ExerciseExercise

Convert the following to logarithmicConvert the following to logarithmic

formform

Convert the following to index formConvert the following to index form

1

2 1 21

. 4 =16 . 2 . 2 2 . 5 1002

xi ii iii iv = = =

2 3

10 8

1

. log 32 5 . log 481

3. log 100 2 . log 16

4

i ii

iii iv

= =

= =


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ExerciseExercise

Without using a calculator, evaluateWithout using a calculator, evaluatethe followingthe following

2 9 1. log 8 . log . log 181

wi ii iii

4 5 10

1. log 64 . log . log 1000

625iv v vi


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Common & NaturalCommon & Natural

AlgorithmsAlgorithms Common Logarithms -Common Logarithms - Logarithms toLogarithms to

base 10base 10

Natural Logarithms -Natural Logarithms - Logarithms toLogarithms to

basebase ee

1010 logx y y x= =

ln

x

y e y x= =

2.718828...e


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ExerciseExercise

Solve the following equationsSolve the following equations

( )2. 6 21 . 5 9 . 1.6 21xx xi ii iii+ = = =

3 4. 9 . 7 . 125 0 x x xiv e v e vi e= = =


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Laws of Logarithms

If , , , , and ,then

1. log 1 , 1

2. log 1 0 , 1

log3. log , 1, 1 chan ge of base law

log

14. log , 1, 1 chan ge of base law

log

5.

a

a

ca

c

a

b

a b c M N R p R

a a

a

bb a c

a

b a ba

+

= =

=

=

log log log , 1 product law

6. log log log , 1 quotient law

7. log log , 1 power l aw

8. log log logar i

a a a

a a a

pa a

a a

MN M N a

M M N aN

M p M a

M N M N

= +

=

=

= = thm equation


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ExerciseExercise

SimplifySimplify

2 23

3. log . log . log

x y x y

i xy ii iiiz z

( )2 32

1. log . log . loga a

a a biv v x y vi

axy

+


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ExerciseExercise

Without using a calculator, evaluate theWithout using a calculator, evaluate thefollowingfollowing

4

10 81 25. 7 log 3log 2log . 3log 4 3log10 2log10

9 80 24

i ii + +

3

2 2 2. log 40 log 0.1 log 0.25 . 3log 2log 1 log 3a a av vi x y+ + + +

( ) 2 2 231 31 1

. log 70 log log 3log 5 . log 9 log 7 log 335 4 2

iii iv + + +


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ExerciseExercise

Evaluate the following by using theEvaluate the following by using thechange of base lawchange of base law

Given that logGiven that logaa2 = 0.301 and log2 = 0.301 and logaa3 =3 =0.477, find0.477, find

5 23 7 5 9

25

log 4 log 10. log 41 . log 5 log 9 log .

log 10

i ii iiig

g g

log 23. log .

4 log 3

aa

a

ai ii


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2.4 Complex Number2.4 Complex Number

At the end of this topic you should be able toAt the end of this topic you should be able to Define complex numberDefine complex number

Represent a complex number in Cartesian formRepresent a complex number in Cartesian form

Define the equality of two complex numberDefine the equality of two complex number

Define the conjugate of a complex numberDefine the conjugate of a complex number

Perform algebraic operations on complex numberPerform algebraic operations on complex number Find the square root of complex numberFind the square root of complex number

Represent the addition and subtraction of complexRepresent the addition and subtraction of complex

number using the Argand diagramnumber using the Argand diagram

Find the modulus and argument of a complex numberFind the modulus and argument of a complex number

Express a complex number in polar formExpress a complex number in polar form


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What is Complex NumberWhat is Complex Number

A number that can be expressed in the formA number that can be expressed in the form a + bia + biwherewhere aa andand bb areare real numbersreal numbers andand ii is theis the

imaginary unit.imaginary unit.

Imaginary unitImaginary unit is the number represented byis the number represented by ii,,

wherewhere

Imaginary numberImaginary number is a number that can beis a number that can be

expressed in the formexpressed in the form bibi, where, where aa andand bb areare realreal

numbersnumbers andand ii is theis the imaginary unit.imaginary unit.

When written in the formWhen written in the form a + bia + bi ,,a complex numbera complex number

is said to be inis said to be in Standard Form.Standard Form.

21 and 1i i= =

Th S f C lTh S t f C l


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The Set of ComplexThe Set of Complex

NumbersNumbers

{ }

( ) ( )

: ,

Re the real parts of while Im the imagina ry parts of

C z z a ib a b R

a z C b z C

= = +

= =

In Cartesian Form;

R

Real NumbersR

Rational Numbers Q

IntegersZ

Whole numbers WNatural

NumbersN

IrrationalNumbers

H

Complex Numbers C

Imaginary

Numbers i


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ExerciseExercise

Write the following in complex numberWrite the following in complex number

formform

(( a +bia +bi))

. 9 . 3 4

. 4 12 . 9 7

i ii

iii iv

+

+ +

O ti C lO ti C l


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Operations on ComplexOperations on Complex

NumbersNumbersFor , then ;For , then ; Adding complex numbersAdding complex numbers

Subtracting complex numbersSubtracting complex numbers

Multiplying complex numbersMultiplying complex numbers

Dividing complex numbersDividing complex numbers

( ) ( )( )

( )

( )

( )

( ) ( )1 2 2 2

ac bd bc ad ia bi c di z z a bi c di

c di c di c d

+ + + = + + = =

+ +

( ) ( ) ( ) ( )1 2 z z a bi c di ac bd ad bc i = + + = + +

( ) ( ) ( ) ( )1 2 z z a bi c di a c b d i+ = + + + = + + +

( ) ( )1 2and z a bi z c di= + = +

( ) ( ) ( ) ( )1 2 z z a bi c di a c b d i = + + = +


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ExerciseExercise

GivenGiven z =z = 22 + i+ i andand ww = 3 + 2= 3 + 2ii. Find. Find

z + wz + w z wz w

z .wz .w

z / wz / w


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Same Complex Numbers

2 complex numbers z1 = a + bi and z2 = c + di are same ifa = c andb = d.

Example:

Given z1 = 2 + (3y+1)i and z2 = 2x + 7i

with z1 = z2 . Find the value ofx andy.


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Conjugate Complex

A complex conjugate of a complexnumber z =a + bi is z* =a bi

Ifz1and z2 are complex numbers, then

( )

( )

( )

* * *

1 2 1 2

* * *

1 2 1 2

**

1 1

*

*

1 1

1.

2.

3.

1 14.

z z z z

z z z z

z z

z z

+ = +

=

=

=


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ExerciseExercise

GivenGiven z =z = 22 + i+ i andand ww = 3 + 2= 3 + 2ii. Find. Find

z* + wz* + w z w*z w*

z* .w*z* .w*

((zz*)* /*)* / w*w*


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ExerciseExercise

GivenGiven zz= 2 + 3= 2 + 3i ,i , FindFind

the value ofthe value ofaa andand bb ifif

2 3 5z z+ +2

*

za ib

z= +


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ExerciseExercise

1.1. GivenGiven (3 +(3 + ii) () (aa + 2+ 2ii) = 2) = 2ii..Find the value ofFind the value ofaa andand bb

2.2. GivenGiven

FindFind z*.z*.Then findThen find z*.zz*.z

21

izi

+=


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Argand DiagramArgand Diagram

Represents any complex numberRepresents any complex numberzz = a + ib= a + ibin terms of its Cartesian coordinate pointin terms of its Cartesian coordinate point

PP((a, ba, b)) or its polar coordinateor its polar coordinater

PP ((a, ba, b))

zz

Real numberReal numberlineline

ImaginaryImaginarynumber linenumber line

PP ((a, ba, b))

rr= |z|= |z|

Real numberReal numberlineline

ImaginaryImaginarynumber linenumber line

bb bb

aa aa


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ExerciseExercise

Illustrate each of the following byIllustrate each of the following byusing Argand diagramusing Argand diagram

zz11 = 2= 2 + i+ i

zz22 = 3= 3ii

zz33 == 22 ii zz44 == 33ii


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Modulus and ArgumentModulus and Argument

ModulusModulus

ArgumentArgument

In polar form,In polar form,

2 2

For complex number , the modulus given by

, z always positive

z a ib

r z a b

= +

= = +

1

For complex number , the argument given by

tan

z a ib

b

a

= +

=

( )cos sin z r i = +


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ExerciseExercise

Find the modulus and argument forFind the modulus and argument for

1

2

3

4

1

5 12

2

1 5

1 1

z i

z i

z i

iz

= +

= +=

+=


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ExerciseExercise

GivenGiven zz11 = 3= 3 ++ 44ii andand zz22 = 2= 2 33ii

Determine |Determine | zz11 zz22 || argumentargument zz11 zz22 Express zExpress z11 zz22 in polar formin polar form


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chapter2 number system

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