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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)
(i) Understanding Shapes: • Propertiesofquadrilaterals–Sumofanglesofaquadrilateralisequalto360°.(Byverification)
• 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.
Prelims_VIII.INDD 4 11/25/2015 11:33:43 AM
INTRODUCTION TO GRAPHS (15 hrs)
Preliminaries:
(i) Axes(Sameunits),CartesianPlane
(ii) Plottingpointsfordifferentkindofsituations(perimetervslengthforsquares,areaasafunctionofsideofasquare,plottingofmultiplesofdifferentnumbers,simpleinterestvsnumberofyearsetc.)
(iii) Readingofffromthegraphs.
• Readingoflineargraphs.
• Readingofdistancevstimegraph.
Prelims_VIII.INDD 5 11/25/2015 11:33:43 AM
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_
C01.indd 1
11/10/2015 1:
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
Addition SubtractionMultiplication Division
Rational Numbers Yes … … …
Integers … No … …
Whole Numbers … … Yes …
Natural Numbers … … … No
Sol.
Numbers Commutative for
Addition SubtractionMultiplication Division
Rational Numbers Yes No Yes No
Integers Yes No Yes No
Whole Numbers Yes No Yes No
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.
Try These [Textbook Page 9]
Q. 1. Complete the following table:
Numbers Associative for
Addition SubtractionMultiplication Division
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
Prelims_VIII.INDD 6 11/25/2015 11:33:44 AM
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
prises of 5 questions
carrying 2 marks ea
ch.
● Section C com
prises of 5 questions
carrying 3 marks ea
ch.
● Section D com
prises of 5 questions
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
Prelims_VIII.INDD 7 11/25/2015 11:33:45 AM
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
Prelims_VIII.INDD 8 11/25/2015 11:33:45 AM
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_SUPR_RFR_MATH_G8_C01.indd 1 11/23/2015 4:35:08 PM
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)
Try These [Textbook Page 4]
Q. 1. Fill in the blanks in the following table:
Numbers Closed Under
Addition SubtractionMultiplication Division
Rational Numbers Yes Yes … No
Integers … Yes … No
Whole Numbers … … Yes …
Natural Numbers … No … …
Sol.
Numbers Closed Under
Addition SubtractionMultiplication Division
Rational Numbers Yes Yes Yes No
Integers Yes Yes Yes No
Whole Numbers Yes No Yes No
Natural Numbers Yes No Yes No
MBD_SUPR_RFR_MATH_G8_C01.indd 2 11/23/2015 4:35:10 PM
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) 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
Try These [Textbook Page 6]
Q. 1. Complete the following table:
Numbers Commutative for
Addition SubtractionMultiplication Division
Rational Numbers Yes … … …
Integers … No … …
Whole Numbers … … Yes …
Natural Numbers … … … No
Sol.
Numbers Commutative for
Addition SubtractionMultiplication Division
Rational Numbers Yes No Yes No
Integers Yes No Yes No
Whole Numbers Yes No Yes No
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.
Try These [Textbook Page 9]
Q. 1. Complete the following table:
Numbers Associative for
Addition SubtractionMultiplication Division
Rational Numbers … … … No
Integers … … Yes …
Whole Numbers Yes … … …
Natural Numbers … Yes … …
MBD_SUPR_RFR_MATH_G8_C01.indd 3 11/23/2015 4:35:12 PM
MBD Super Refresher Mathematics-VIII4
= − × − × +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
\ Multiplicative inverse of 1556
is 5615
.
(v) − ×−1 25
= ( ) ( )− × −1 2
5 =
25
\ Multiplicative inverse of 25
is 52
.
MBD_SUPR_RFR_MATH_G8_C01.indd 4 11/23/2015 4:35:19 PM
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
.
MBD_SUPR_RFR_MATH_G8_C01.indd 5 11/23/2015 4:35:24 PM
MBD Super Refresher Mathematics-VIII6
Sol. (i) A = 15
, B = 45
, C = 55
= 1, D = 85
, E = 95
.
(ii) J = −116
, I = −86
, H = −76
, G = −56
, F = −26
CONCEPT IN A NUTSHELL
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
CONCEPT IN A NUTSHELL
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_SUPR_RFR_MATH_G8_C01.indd 6 11/23/2015 4:35:29 PM
MBD Super Refresher MathematicsClass-VIII CBSE /NCERT
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