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Mathematics

8

Based on the latest syllabus and

textbook(s) issued by CBSE/NCERT

ByVinay Sharma

Sudhansu S. Swain

Edited byPooja JainVipul Jain

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SYLLABUSMATHEMATICS (CLASS–VIII)

NUMBER SYSTEM (50 hrs)

(i) Rational Numbers: • Propertiesofrationalnumbers(includingidentities).Usinggeneralformofexpressiontodescribeproperties.

• Consolidationofoperationsonrationalnumbers.

• Representationofrationalnumbersonthenumberline.

• Betweenanytworationalnumbersthereliesanotherrationalnumber(Makingchildrenseethatifwetaketworationalnumbersthenunlikeforwholenumbers,inthiscaseyoucankeepfindingmoreandmorenumbersthatliebetweenthem.)

• Worldproblem(higherlogic,twooperations,includingideaslikearea)

(ii) Powers: • Integersasexponents.

• Lawsofexponentswithintegralpowers.

(iii) Squares and Square roots, Cubes and Cube roots: • SquaresandSquareroots.

• Squarerootsusingfactormethodanddivisionmethodfornumberscontaining(a)nomorethantotal4digitsand(b)nomorethan2decimalplaces.

• Cubesandcuberoots(onlyfactormethodfornumberscontainingatmost3digits.)

• Estimatingsquarerootsandcuberoots.Learningtheprocessofmovingnearertotherequirednumber.

(iv) Playing with Numbers: • Writingandunderstandinga2and3digitnumberingeneralizedform(100a+10b + c,wherea,b,ccanbeonlydigit

0–9)andengagingwithvariouspuzzlesconcerningthis.(Likefindingthemissingnumeralsrepresentedbyalphabetsinsumsinvolvinganyofthefouroperations.)

• Childrentosolveandcreateproblemandpuzzles.

• Numberpuzzlesandgames.

• Deducingthedivisibilitytestrulesof2,3,5,9,10foratwoorthree-digitnumberexpressedinthegeneralform.

ALGEBRA (20 hrs)

Algebraic Expressions: • Multiplicationanddivisionofalgebraicexp.(Coefficientshouldbeintegers).

• Somecommonerrors(e.g.2+x ≠2x,7x + y ≠7xy). • Identities(a ± b)2 = a2±2ab + b2,a2–b2=(a + b)(a–b).

Factorisation(simplecasesonly)asexamplesthefollowingtypesa(x + y),(x ± y)2,a2–b2,(x + a).(x + b)

• Solving linear equations in one variable in contextual problems involvingmultiplication anddivision (wordproblems)(avoidcomplexcoefficientintheequations).

RATIO AND PROPORTION (25 hrs)

• Slightlyadvancedproblemsinvolvingapplicationsonpercentages,profit&loss,overheadexpenses,Discount,tax.

• Differencebetweensimpleandcompoundinterest(compoundedyearlyupto3yearsorhalf-yearlyupto3stepsonly),arrivingattheformulaforcompoundinterestthroughpatternsandusingitforsimpleproblems.

Prelims_VIII.INDD 3 11/25/2015 11:33:43 AM

• Directvariation–Simpleanddirectwordproblems.

• Inversevariation–Simpleanddirectwordproblems.

• Timeandworkproblems:Simpleanddirectwordproblems.

GEOMETRY (60 hrs)

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• Propertiesofparallelogram(Byverification) (i) Oppositesidesofaparallelogramareequal,

(ii) Oppositeanglesofaparallelogramareequal,

(iii) Diagonalsofaparallelogrambisecteachother.

[Why(iv),(v)and(vi)followfrom(ii)]

(iv) Diagonalsofarectangleareequalandbisecteachother.

(v) Diagonalsofarhombusbisecteachotheratrightangles.

(vi) Diagonalsofsquareareequalandbisecteachotheratrightangles.

(ii) Representing 3D in 2D: • IdentifyandMatchpictureswithobjects[morecomplicatede.g.nested,joint2Dand3Dshapes

(notmorethan2]

• Drawing2-Drepresentationof3-Dobjects(Continuedandextended.)

• Countingvertices,edges&faces&verifyingEuler’srelationfor3-Dfigureswithflatfaces(cubes,cuboids,tetrahedrons,prismsandpyramids.)

(iii) Construction : ConstructionofQuadrilaterals:

• Givenfoursidesandonediagonal.

• Threesidesandtwodiagonals.

• Threesidesandtwoincludedangles.

• Twoadjacentsidesandthreeangles.

MENSURATION (15 hrs)

• Areaofatrapeziumandapolygon.

• Conceptofvolume,measurementofvolumeusingabasicunit,volumeofacube,cuboidandcylinder.

• Volumeandcapacity(measurementofcapacity).

• Surfaceareaofcube,cuboid,cylinder.

DATA HANDLING (15 hrs)

(i) Readingbar-graphs,ungroupeddata,arrangingitintogroups,representationofgroupeddatathroughbar-graphs,con-structingandinterpretingbar-graphs.

(ii) SimplePiechartswithreasonabledatanumbers.

(iii) Consolidatingandgeneralisingthenotionofchanceineventsliketossingcoins,diceetc.Relatingittochanceinlifeevents.Visualrepresentationoffrequencyoutcomesofrepeatedthrowsofthesamekindofcoinsordice.

(iv) Throwingalargenumberofidenticaldice/coinstogetherandaggregatingtheresultofthethrowstogetlargenumberofindividualevents.Observingtheaggregatingnumbersoveralargenumberofrepeatedevents.Comparingwithdataforacoin.Observingstringsofthrows,notionofrandomness.


INTRODUCTION TO GRAPHS (15 hrs)

Preliminaries:

(i) Axes(Sameunits),CartesianPlane

(ii) Plottingpointsfordifferentkindofsituations(perimetervslengthforsquares,areaasafunctionofsideofasquare,plottingofmultiplesofdifferentnumbers,simpleinterestvsnumberofyearsetc.)

(iii) Readingofffromthegraphs.

• Readingoflineargraphs.

• Readingofdistancevstimegraph.


1 CONCEPTS Rationa

l numbers and the

ir properties

Representation o

f rational numbers

on the

number line

Rational number

s between two rat

ional

numbers

RATIONAL NUM

BERS AND THEIR

PROPERTIES

The numbers

of the form

pq

, where p and q

are integers and (

q ≠ 0), are called

rational numbers

.

For example,

34

53

29

611, , ,− etc.

Standard

Form of a Rationa

l Number: A ration

al number pq i

s said to be in standa

rd form if p and q ar

e integers

having no common

divisor other than 1 a

nd p is positive.

Notes:

(i) Every po

sitive rational num

ber is greater than

0.

(ii) Every ne

gative rational num

ber is less than 0.

(iii) Rational n

umbers are closed

under addition, su

btraction, multiplic

ation and division (

provided divisor

is not zero).

(iv) Commut

ativity of addition

is true for natural

numbers, whole n

umbers and intege

rs. It is also true

for rational numbe

rs.

(v) Associati

vity of addition is t

rue for natural num

bers, whole numbe

rs and integers. It

is also true for

rational numbers.

Properties of

Rational Numbers

Additive Iden

tity Element: Zero

is the identity elem

ent for addition an

d subtraction of na

tural numbers,

whole numbers, in

tegers and rationa

l numbers.

For examples,

(i) 3 + 0 = 0

+ 3 = 3

(ii) 0 + 5 = 5

+ 0 = 5

(iii) 34

+ 0 = 0 + 34

= 34

etc.

Multiplicativ

e Identity Element

: One is the multipl

icative identity for

natural numbers, w

hole numbers,

integers and ration

al numbers.

Rational Numbe

rs

CONCEPT IN A N

UTSHELL

1

MBD_SUPR_R

FR_MATH_G8_

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17:42 PM

Super RefresherAll chapters as per NCERTSyllabus and Textbook

Every chapter divided into Sub-topics

Concept in a Nutshellprovides a complete and comprehensive summary of the concept

Highlights essential information which must be remembered

Rational Numbers 3

Numbers Associative for

Addition SubtractionMultiplication Division

Rational Numbers Yes No Yes No

Integers Yes No Yes No

Whole Numbers Yes No Yes No

Natural Numbers Yes Yes Yes No

Try These [Textbook Page 13] Q. 1. Find using distributivity

(i) 75

× 312

+ 75

× 512

−

(ii) 916

× 412

+ 916

× 39

−

Sol. (i) 75

312

75

512

×−

+ ×

= 75

312

512

×−

+

[By distributivity property]

= 75

3 512

×− +

= 75

212

1460

730

× = =

(ii) 9

164

12916

39

×

+ ×

−

= 9

164

123

9× +

−

= 9

164

1239

× −

= 9

1612 12

36×

−

= 9

16036

× = 0576

= 0

TEXTBOOK EXERCISE 1.1 Q. 1. Usingappropriatepropertiesfind:

(i) − −23

× 35

+ 52

35

× 16

(ii) 25

× 37

16

× 32

+ 114

× 25

−

−

Sol. (i) We have: − × + − ×23

35

52

35

16

Try These [Textbook Page 6]

Q. 1. Complete the following table:

Numbers Commutative for


Rational Numbers Yes … … …

Integers … No … …

Whole Numbers … … Yes …

Natural Numbers … … … No

Sol.






Natural Numbers Yes No Yes No

Think, Discuss and Write [Textbook Page 11] Q. 1. If a property holds for rational numbers, will

it also hold for integers? For whole numbers? Which will? Which will not?

Sol. Try yourself.





Rational Numbers … … … No

Integers … … Yes …

Whole Numbers Yes … … …

Natural Numbers … Yes … …

MBD_SUPR_RFR_MATH_G8_C01.indd 3 11/10/2015 1:17:45 PM

Rational Number

s

5

Q. 10. Write:

(i) The r

ational number th

at does not have a

reciprocal.

(ii) The

rational numbers

that are equal to

their reciprocals.

(iii) The r

ational numbers tha

t is equal to their

negative.

Sol. (i)

01

(ii) 1 and

(–1). (iii) Zero.

Q. 11. Fill in the

blanks:

(i) Zero

has reciproc

al.

(ii) The n

umbers and

are their

own reciprocals.

(iii) The

reciprocal of –5 is

.

(iv) Recip

rocal of 1 ,x

where x≠0is

.

(v) The

product of two rat

ional numbers is

always a .

(vi) The r

eciprocal of a positi

ve rational number

is .

Sol. (i) no

(ii) 1 and –1

(iii)

−15

(iv) x

(v) ratio

nal number (vi)

positive

SELF PRACTICE

1.1

1. Find using

distributivity:

(i)

−

− −

3

4×

23

+34

×56

(ii)

−

−

2

3×

56

+23

×72

2. Using appr

opriate properties

find:

23

×37

114

37

×35− − −

3. Find the

additive inverse

of each of the

following:

(i)

13

(ii) 239

(iii) −311

(iv) −−87

4. Verify that

: –(–x) = x for

(i) x =

1317

(ii) x =

− 2131

5. Find the m

ultiplicative invers

e of the following:

(i) 12

(ii) –8 (iii)

516

(iv) −1417

(vi) −1

\ Mult

iplicative inverse o

f −1 is 1

1−,

i.e.,

−11

= −1

Q. 5. Name the

property under

multiplication

used in each of th

e following:

(i)

−45

× 1 = 1 × −4

5 = −

45

(ii)

− −1317

×27

= − −27

×1317

(iii)

−−

1929

×29

19 = 1

Sol. (i) Mult

iplicative identity.

(ii) Com

mutative property

of multiplication.

(iii) Mult

iplicative inverse.

Q. 6. Multiply

613

by the reciprocal o

f −716

.

Sol. 613

716

×

−

multiplication inv

erse of

=

613

167× −

= 96

91

9691−

= −

Q. 7. Tell what

property allows y

ou to compute:

13

× 6×43

as

13

×6 ×43

.

Sol. Associativi

ty of multiplicatio

n.

Q. 8. Is

89

the multiplicative

inverse of –1

18

. Why

or why not?

Sol. −1

18

= −98

\ Multip

licative inverse of

−98

is 8

9−. i.e.,

−89

.

\

89

is not the multiplic

ative inverse of −1

18

.

Q. 9. Is 0.3 the

multiplicative inv

erse of 3

13

. Why

or why not?

Sol. 3

13

= 10

3

\ The m

ultiplicative inver

se of 10

3 is

310

.

i.e., 0.3.

\ Yes, 0.

3 is the multiplicat

ive inverse of 3

13

.

MBD_SUPR_R

FR_MATH_G8_

C01.indd 5

11/10/2015 1:

17:57 PM

Important Questions fromexamination point of viewto ensure passing marks

Self Practice questions forconsolidation of each concept

Try These and Do Thiswith page numbersfully solved to helpthe learners

NCERT Textbook Exerciseswith detailed solution


Time – 2 Hours

Cl

ass VII

Max. Marks

– 50

General Instructi

ons:

● All question

s are compulsory.

● Section A com

prises of 5 questions

carrying 1 mark ea

ch.

● Section B com


carrying 2 marks ea

ch.

● Section C com


carrying 3 marks ea

ch.

● Section D com


carrying 4 marks ea

ch.

SecTion A

1. How many

angles are formed

when 2 lines inter

sect?

2. Evaluate: (2

0)2 + (31)0 + 4

0

3. If the circum

ference of a circula

r sheet is 154 m, fi

nd its radius.

4. Find third

angle of the triang

le which have two

angles as 30° and 8

0°.

5. Find the w

hole quantity if 10%

of its is 7.

SecTion B

6. Raju has sol

ved 24

part of an exercise

while Sameer solv

ed 12

part of it. Who ha

s solved more?

7. In the figur

e below, DCDE ≅D

QPR. What is m∠D

?

8. Find the m

ode and median of t

he data:

13, 16, 12, 1

4, 19, 12, 14, 13, 14

9. Verify that

a ÷ (b + c) ≠ (a ÷ b)

+ (a ÷ c) for the va

lues of a, b and c as

a = 12, b = −4 and

c = 2.

10. ABC is a tr

iangle right-angled

at C. If AB = 25 cm

and AC = 7 cm, fi

nd BC.

SecTion c

11. If Meenaks

hi gives an interest

of `45 for one yea

r at 9% rate p.a. W

hat is the sum she

has borrowed?

12. Write the fo

llowing numbers in

the expanded form

:

(i) 279404

(ii) 20068

(iii) 2806196

13. A picture is

painted on a cardb

oard 8 cm long and

5 cm wide such tha

t there is a margin

of 1.5 cm along eac

h

of its side. Find the

total area of the m

argin.

14. Construct Δ

PQR is PQ = 5 cm, m

∠PQR = 105°, m∠

QRP = 40°.

Sample Paper - i

338

Brain Teasers.indd

338

11/10/2015 2:36:20

AM

MBD Super Refresher Mathematics-VIII

8

i.e., 35

2020

× = 60100 and

34

2525

× = 75100

\ Ten rational numbers between 60

100 and 75

100 can be any of these: 61100

62100

63100

64100

65100

, , , , , …, 70100 ,

71100 ,

72100 ,

73100 ,

74100 .

SELF PRACTICE 1.2 1. Represent the following numbers on the

number line: (i) −13 (ii) 2

7 (iii) 72 (iv) −3

7 .

2. Find a rational number lying between 13 and

12 .

3. Find three rational numbers lying between 3

and 4. 4. Find three rational numbers lying between 23

and 34 . 5. Find ten rational numbers between −5

6 and 5

8 .

6. Find three rational numbers between 14 and

12 .

7. Find three rational numbers between —2 and 0.

8. Find two rational numbers between 15 and

12 .

9. Find seven rational numbers between 13 and

12 . 1. Which of the following statement is false?

(a) Rational numbers are not closed under

addition. (b) Whole numbers are closed under addition.

(c) Integers are closed under addition.

(d) Natural numbers are closed under addition.

2. Which of the following statement is false?

(a) Rational numbers are commutative for

addition. (b) Integers are not commutative for addition.

(c) Natural numbers are commutative for addition.

(d) Whole numbers are commutative for addition.

3. Which of the following statement is true?

(a) Integers are associative for subtraction.

(b) Natural numbers are associative for subtraction.

(c) Whole numbers are not associative for

subtraction. (d) Rational numbers are associative for

subtraction. 4. Which of the following statement is true?

(a) Rational numbers are not associative for

multiplication. (b) Integers are associative for multiplication.

(c) Whole numbers are not associative for

multiplication. (d) Natural numbers are not associative for

multiplication.

MULTIPLE CHOICE QUESTIONS (MCQs)

In each of the following questions four options are given. Choose the correct answer. 5. Which of the following statement is false?

(a) Rational numbers are closed under

subtraction. (b) Integers are closed under subtraction.

(c) Natural numbers are closed under subtraction.

(d) Whole numbers are not closed under

subtraction. 6. Which of the following statement is true?

(a) Rational numbers are not commutative for

subtraction. (b) Natural numbers are commutative for

subtraction. (c) Whole numbers are commutative for

subtraction. (d) Integers are commutative for subtraction.

7. Which of the following statement is true?

(a) Whole numbers are not closed under

multiplication. (b) Integers are not closed under multiplication.

(c) Rational numbers are not closed under

multiplication. (d) Natural numbers are closed under

multiplication. 8. Which of the following statement is false?

(a) Integers are not commutative for

multiplication? (b) Rational numbers are commutative for multi-

plication.MBD_SUPR_RFR_MATH_G8_C01.indd 8

11/10/2015 1:18:12 PM

MBD Super Refresher Mathematics-VIII

10

Q. 8. The rational number 10.11 in the form

pq

is

_____________.

Q. 9. The two rational numbers lying between −2

and −5 with denominator as 1 are ________

and ________.

ANSWERS

6. Positive rational number 7. Opposite

8. 1011100

9. –3, –4

True/FalseIn questions 10 to 13, state wh

ether the given statements

are true (T) or false (F).

Q. 10. 56

lies between 23

and 1.

Q. 11. If xy

is the additive inverse of, cd

thenxy

cd

− = 0.

Q. 12. The negative of the negative of any rational

number is the number itself.

Q. 13. The rational number −−83

lies neither to the

right nor to the left of zero on the number line.

ANSWERS

10. True 11. False 12. True 13. False

Short Answer Type Questions

Q. 14. The cost of 194

metres of wire is `171

2. Find

the cost of one metre of the wire.

Sol. Cost of 194

m of wire = `171

2

Cost of 1 m of wire = `

1712

194

÷

=

1712

419

× = 18

Cost of 1 m wire = `18

Q. 15. 711

of all the money in Hamid’s bank account

is `77,000. How much money does Hamid

have in his bank account?

Sol. Let the total amount in Hamid’s bank account

= `x

As per question, 711

of x = 77,000

NCERT EXEMPLAR QUESTIONS (SOLVED)

Multiple Choice Questions (MCQs)

In questions 1 to 5, out of the four options only one is

correct. Write the correct answer.

Q. 1. The numerical expression

38

57

+−( ) =

−1956

shows that

(a) Rational numbers are closed under

addition.

(b) Rational numbers are not closed under

addition.

(c) Rational numbers are closed under

multiplication.

(d) Addition of rational numbers is not

commutative.

Q. 2. The multiplicative inverse of −1

17

is

(a) 87

(b) −87

(c) 78

(d) 78−

Q. 3. If y be the reciprocal of rational number x,

then the reciprocal of y will be

(a) x (b) y (c)

xy

(d) yx

Q. 4. Between two given rational numbers, we can

find

(a) One and only one rational number.

(b) Only two rational numbers.

(c) Only ten rational numbers.

(d) Infinitely many rational numbers.

Q. 5. x y+

2 is a rational number

(a) Between x and y.

(b) Less than x and y both.

(c) Greater than x and y both.

(d) Less than x but greater than y.

ANSWERS

1. (a) 2. (d) 3. (a) 4. (d) 5. (a)

Fill in the Blanks

In questions 6 to 9, fill in the blanks to make the

statement true.

Q. 6. The reciprocal of a positive rational number is

a _______________.

Q. 7. The rational numbers

13

and −13

are on the

_________ sides of zero on the number line.

MBD_SUPR_RFR_MATH_G8_C01.indd 10

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Rational Number

s

13

Now, the

rational numbers

lying between

them will be −

991000

, −98

1000, ...,

01000

,

11000

,2

1000, ...,

991000

.

Thus, w

e conclude that

rational numbers

lying between two g

iven rational numb

ers are

uncountable.

Q. 2. Find a ra

tional number bet

ween (a + b)–

1 and

(a–1 + b–1), given t

hat a = 13

, =27b

.

Sol. a =

13

, b = 27

a +

b = 13

27+

= 7 6

21+ =

1321

(a + b)

–1 = 1321

–1

= 2113

a

–1 + b–1 =

13

27

11

+

−−

...(i)

= 31

72+

= 6 7

2+ =

132

...(ii)

We now

need a rational nu

mber between

2113

and 132

.

Mean of

2113

and 132

= 12

2113

132+

=

12

42 16926+

=

12

21126

=

21152

Hence, th

e required rationa

l number is

21152

.

Q. 3. If the pr

ice of 12 tables i

s `360025

and the

price of 6 chairs

is `300034

, find the total

price of 4 tables an

d 4 chairs.

Sol. Price

of 12 tables = `360

025

= `18002

5

\ Pric

e of 1 table = `

180025

÷ 12

= `18002

5 ×

112

= ̀9001

30

\ Pri

ce of 4 tables = 4 ×

9001

30 = `

1800215

d

Pri

ce of 6 chairs = ` 3

00034

= `12003

4

\ Pr

ice of 1 chair = `

120034

÷ 6

= 12003

4 ×

16

= `4001

8

\ Pric

e of 4 chair = `4 ×

4001

8 = `

40012

Hence, to

tal price of 4 tables

and 4 chairs.

= `

1800215

40012+

= `

36004 60015

30+

= `

9601930

= `1930

VALUE BASED Q

UESTIONS (VBQs

)

Q. 1. Two stu

dents Shrey and

Hitesh gave the

following stateme

nts, respectively.

(a) If a nu

mber is divisible b

y 3 it will also be

divisible by 9.

(b) If a nu

mber is divisible b

y 9 it will also be

divisible by 3.

Who is te

lling a lie? What

is the importance

of truth in life?

Sol. Shrey is

telling a lie. It is i

mportant to speak

the truth in life as

it developes faith,

love and

transparency in the

minds of other peo

ple and

keeps the person,

who speaks the tru

th, calm

and relaxed.

CHAPTER ASSES

SMENT

1. Choose

the correct option

in each of the

following:

(i) A nu

mber of the form

pq

is said to be

rational number, if

(a) p

and q are integers.

(b) p

and q are integers

and q ≠ 0.

(c) p

and q are integers

and p ≠ 0.

(d) p

and q are integers

and p ≠ 0 also q ≠ 0

.

(ii) The

additive inverse o

f 247

is

(a)

187

(b)

−718

(c) −18

7

(d) 718

MBD_SUPR_R

FR_MATH_G8_

C01.indd 13

11/10/2015 1:

18:30 PM

Mathematics

Value-Based Questionsto apply mathematical conceptsto real life situations with stress onsocial values

Four Sample Papersof 50 marks each

Multiple Choice Questions (MCQs)for testing conceptual skills of students

NCERT Exemplar Problemswith complete solution tosupplement the NCERTsupport material

Chapter Assessment with answers at the end of each chapter


1 Rational Numbers 1–14

2 Linear Equations in One Variable 15–34

3 Understanding Quadrilaterals 35–58

4 Practical Geometry 59–68

5 Data Handling 69–88

6 Squares and Square Roots 89–112

7 Cubes and Cube Roots 113–126

8 Comparing Quantities 127–150

9 Algebraic Expressions and Identities 151–172

10 Visualising Solid Shapes 173–185

11 Mensuration 186–215

12 Exponents and Powers 216–226

13 Direct and Inverse Proportions 227–243

14 Factorisation 244–259

15 Introduction to Graphs 260–278

16 Playing with Numbers 279–292

Sample Papers (1–4) 293–300

CONTENTS


1CONCEPTS

Rational numbers and their properties Representation of rational numbers on the

number line Rational numbers between two rational

numbers

RATIONAL NUMBERS AND THEIR PROPERTIES

The numbers of the form pq , where p and q are integers and (q ≠ 0), are called rational numbers.

For example, 34

53

29

611

, , , − etc.

Standard Form of a Rational Number: A rational number pq is said to be in standard form if p and q are

integers having no common divisor other than 1 and p is positive.

Notes: (i) Every positive rational number is greater than 0. (ii) Every negative rational number is less than 0. (iii) Rational numbers are closed under addition, subtraction, multiplication and division (provided divisor

is not zero). (iv) Commutativity of addition is true for natural numbers, whole numbers and integers. It is also true

for rational numbers. (v) Associativity of addition is true for natural numbers, whole numbers and integers. It is also true for

rational numbers.

Properties of Rational Numbers Additive Identity Element: Zero is the identity element for addition and subtraction of natural numbers,

whole numbers, integers and rational numbers. For examples, (i) 3 + 0 = 0 + 3 = 3 (ii) 0 + 5 = 5 + 0 = 5

(iii) 34

+ 0 = 0 + 34

= 34

etc.

Multiplicative Identity Element: One is the multiplicative identity for natural numbers, whole numbers, integers and rational numbers.

Rational Numbers

CONCEPT IN A NUTSHELL

1


MBD Super Refresher Mathematics-VIII2

For example,

(i) 6 × 1 = 1 × 6 = 6

(ii) (– 7) × 1 = 1 × (– 7) = – 7

(iii) 53

× 1 = 1 × 53

= 53

etc.

Additive inverse: For every rational number pq

, there exists a rational number pq

pq

+

−

such that;

pq

pq

+

−

= 0 and similarly,

−

+

=

pq

pq

0.

Then, −pq

is called the additive inverse of pq

.

Multiplicative inverse (Reciprocal): Every non-zero rational number pq

has its multiplicative inverse

pq . For example

pqqp×

=

qp

pq×

= 1

\ qp

is called the reciprocal of pq

.

Note:

(i) Zero has no reciprocal (ii) Reciprocal of 1 is 1 (iii) Reciprocal of –1 is –1

Distributive law of multiplication over addition: For any three rational numbers abcd

, and ef

, we ave:

ab

cd

ef

× +

=

abcd

ab

ef

×

+ ×

.

NCERT TEXTBOOK EXERCISE (SOLVED)


Q. 1. Fill in the blanks in the following table:

Numbers Closed Under


Rational Numbers Yes Yes … No

Integers … Yes … No


Natural Numbers … No … …

Sol.

Numbers Closed Under


Rational Numbers Yes Yes Yes No

Integers Yes Yes Yes No




Rational Numbers 3






Natural Numbers Yes Yes Yes No

Try These [Textbook Page 13] Q. 1. Find using distributivity

(i) 75

× 312

+ 75

× 512

−

(ii) 916

× 412

+ 916

× 39

−

Sol. (i) 75

312

75

512

×−

+ ×

= 75

312

512

×−

+

[By distributivity property]

= 75

3 512

×− +

= 75

212

1460

730

× = =

(ii) 916

412

916

39

×

+ ×

−

= 916

412

39

× +−

= 916

412

39

× −

= 916

12 1236

×−

= 916

036

× = 0576

= 0

TEXTBOOK EXERCISE 1.1 Q. 1. Usingappropriatepropertiesfind:

(i) − −23

× 35

+ 52

35

× 16

(ii) 25

× 37

16

× 32

+ 114

× 25

−

−

Sol. (i) We have: − × + − ×23

35

52

35

16





Rational Numbers Yes … … …

Integers … No … …


Natural Numbers … … … No

Sol.







Think, Discuss and Write [Textbook Page 11] Q. 1. If a property holds for rational numbers, will

it also hold for integers? For whole numbers? Which will? Which will not?

Sol. Try yourself.





Rational Numbers … … … No

Integers … … Yes …

Whole Numbers Yes … … …

Natural Numbers … Yes … …



= − × − × +23

35

35

16

52

(By commutativity)

= 35

23

16

52

−−

+ (By distributivity)

= 35

4 16

52

− −

+ =

35

56

52

×−

+

= − +12

52

= − +1 5

2

= 42

= 2

(ii) We have: 25

37

16

32

114

25

× −

− × + ×

= 25

37

114

25

16

32

× −

+ × − ×

(By commutativity)

= 25

37

114

14

− +

− =

25

6 114

14

− +

−

= 25

514

14

×−

−

= − − = − − = −17

14

4 728

1128

Q. 2. Write the additive inverse of each of the following:

(i) 28

(ii) −59

(iii) −−

65

(iv) 29−

(v) 196−

Sol. (i) Additive inverse of 28

is −28

.

(ii) Additive inverse of −59

is 59

because

= −

+5

959

= − +5 5

9=

09

= 0

(iii) We may write; −−

65

= ( ) ( )( ) ( )− × −− × −

6 15 1

= 65

\ Additive inverse of 65

is −65

because

−65

+ 65

= − +6 6

5 =

05

= 0

(iv) In standard form, we write; 29−

as − 29

.

\ Additive inverse of − 29

is + 29

because − +29

29

= − +2 29

= 09

= 0

(v) In standard form, we write; 19

6− as −

196

.

\ Additive inverse of – 196

is 196

because −

+196

196

= − +19 19

6 = 06

= 0

Q. 3. Verify that: −(−x) = x for:

(i) x = 1115

(ii) x = −1317

Sol. (i) For x = 1115

⇒ –(–x) = − −

=( )1115

1115

= x

Thus; –(–x) = x is verified.

(ii) For x = −1317

⇒ – (–x) = – −−

( )1317 =

−1317 = x

Thus –(–x) = x is verified.

Q. 4. Find the multiplicative inverse of the following:

(i) –13 (ii) −1319

(iii) 15

(iv) − −58

× 37

(v) –1 × −25

(vi) –1

Sol. (i) –13 \ Multiplicative inverse of −13 is

113−

,

i.e., −113

.

(ii) −1319

\ Multiplicative inverse of −1319

is 1913−

,

i.e., − 1913

.

(iii) 15

\ Multiplicative inverse of 15

is 51

, i.e., 5.

(iv) −

×−5

83

7 =

( ) ( )− × −×

5 38 7

= 1556


is 5615

.

(v) − ×−1 25

= ( ) ( )− × −1 2

5 =

25


is 52

.


Rational Numbers 5

Q. 10. Write: (i) The rational number that does not have a

reciprocal. (ii) The rational numbers that are equal to

their reciprocals. (iii) The rational numbers that is equal to their

negative.

Sol. (i) 01

(ii) 1 and (–1). (iii) Zero.

Q. 11. Fill in the blanks: (i) Zero has reciprocal. (ii) The numbers and are their

own reciprocals. (iii) The reciprocal of –5 is .

(iv) Reciprocal of 1 ,x

where x≠0is .

(v) The product of two rational numbers is always a .

(vi) The reciprocal of a positive rational number is .

Sol. (i) no (ii) 1 and –1

(iii) −15

(iv) x

(v) rational number (vi) positive

SELF PRACTICE 1.1 1. Find using distributivity:

(i) −

− −

34

× 23

+ 34

× 56

(ii) −

−

23

× 56

+ 23

× 72

2. Using appropriate properties find:

23

× 37

114

37

× 35

−− −

3. Find the additive inverse of each of the following:

(i) 13

(ii) 239

(iii) −311

(iv) −−

87

4. Verify that: –(–x) = x for

(i) x = 1317

(ii) x = − 2131

5. Find the multiplicative inverse of the following:

(i) 12 (ii) –8 (iii) 516

(iv) −1417

(vi) −1

\ Multiplicative inverse of −1 is 11−

,

i.e., −11

= −1

Q. 5. Name the property under multiplication used in each of the following:

(i) −45

× 1 = 1 × −45

= −45

(ii) − −1317

× 27

= − −27

× 1317

(iii) −

−19

29× 29

19 = 1

Sol. (i) Multiplicative identity. (ii) Commutative property of multiplication. (iii) Multiplicative inverse.

Q. 6. Multiply 613

by the reciprocal of −716

.

Sol. 613

716

×−

multiplication inverse of

= 613

167

×−

= 9691

9691−

= −

Q. 7. Tell what property allows you to compute:

13

× 6× 43

as

13

×6 × 43

.

Sol. Associativity of multiplication.

Q. 8. Is 89

the multiplicative inverse of –1 18

. Why

or why not?

Sol. −118

= −98

\ Multiplicative inverse of −98

is 89−

. i.e., −89

.

\89

is not the multiplicative inverse of −118

.

Q. 9. Is 0.3 the multiplicative inverse of 313

. Why

or why not?

Sol. 313

= 103

\ The multiplicative inverse of 103

is 3

10.

i.e., 0.3.

\ Yes, 0.3 is the multiplicative inverse of 313

.



Sol. (i) A = 15

, B = 45

, C = 55

= 1, D = 85

, E = 95

.

(ii) J = −116

, I = −86

, H = −76

, G = −56

, F = −26


RATIONAL NUMBERS BETWEEN TWO RATIONAL NUMBERS

If x and y be two rational numbers, such that x < y,

then, 12

(x + y) is a rational number between x and y.

For example, between 13

and 12

, the required rational

number is = 12

13

12

+

=

12

2 36+

=

12

56

× = 5

12

Hence, 512

is a rational number lying between 13

and12

.

Let us see, whether, we are able to say like this in the

case of numbers like 3

10 and

710

.

You might have thought that they are 4

105

10, , and

610

.

You can also write 3

10 as

30100

and 710

as 70100

.

Now, the numbers, 31

10032100

33100

, , ,…, 68

10069100

,

all between 3

10 and

710

. The number of these

rational numbers is 39. This is called the density property of rational numbers.

TEXTBOOK EXERCISE 1.2

Q. 1. Represent these numbers on the number line:

(i) 74

(ii) −56

Sol. (i)

(ii) –12__

6

–11__

6

–10__

6

__

6

–9 __

6

–8 __

6

–7 __

6

–6 __

6

–5 __

6

–4 __

6

–3 __

6

–2 __

6

–1 0

–2 –1

6. Name the property under multiplication used in each of the following:

(i) −

316

× 815

= 815

× 316−

(ii) 23

× 67

× 1415

= 23

× 67

× 1415

−

−

(iii) 56

× 45

710

= 56

× 45

56

× 710

−+

−

−

+

−

7. Multiply −719

by the reciprocal of 513

.

8. Tell what property allows you to compute:

34

× 8× 25

as

34

×8 × 25


REPRESENTATION OF RATIONAL NUMBERS ON THE NUMBER LINEYou have learnt to represent natural numbers, whole numbers, integers and rational number on a number line. We shall revise them.

(i) Natural numbers: e.g.,

1 2 3 4 5 6 7

(ii) Whole numbers: e.g.,

(iii) Integers: e.g.,

(iv) Rational numbers: e.g.,

(a)

(b)

(c)

Try These [Textbook Page 17]Write the rational number for each point labelled with a letter.

(i)

(ii)


MBD Super Refresher MathematicsClass-VIII CBSE /NCERT

Publisher : MBD GroupPublishers

ISBN : 9789385905230Author : Vinay Sharma,Sudhansu S. Swain

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