![RATIONAL NUMBERS 7
1.2.3 Associativity
(i) Wholenumbers
Recalltheassociativityofthefouroperationsforwholenumbersthroughthistable:
Operation Numbers Remarks
Addition ......... Additionisassociative
Subtraction ......... Subtractionisnotassociative
Multiplication Is 7 × (2 × 5) = (7 × 2) × 5? Multiplicationisassociative
Is 4 × (6 × 0) = (4 × 6) × 0?
For any three whole
numbers a, b and c
a × (b × c) = (a × b) × c
Division ......... Divisionisnotassociative
Fillinthistableandverifytheremarksgiveninthelastcolumn.
Checkforyourselftheassociativityofdifferentoperationsfornaturalnumbers.
(ii) Integers
Associativityofthefouroperationsforintegerscanbeseenfromthistable
Operation Numbers Remarks
Addition Is (–2) + [3 + (– 4)] Additionisassociative
= [(–2) + 3)] + (– 4)?
Is (– 6) + [(– 4) + (–5)]
= [(– 6) +(– 4)] + (–5)?
For any three integers a, b and c
a + (b + c) = (a + b) + c
Subtraction Is 5 – (7 – 3) = (5 – 7) – 3? Subtractionisnotassociative
Multiplication Is 5 × [(–7) × (– 8) Multiplicationisassociative
= [5 × (–7)] × (– 8)?
Is (– 4) × [(– 8) × (–5)]
= [(– 4) × (– 8)] × (–5)?
For any three integers a, b and c
a × (b × c) = (a × b) × c
Division Is [(–10) ÷ 2] ÷ (–5) Divisionisnotassociative
= (–10) ÷ [2 ÷ (– 5)]?](https://image.slidesharecdn.com/8-maths-ncert-chapter-1-181216065336/85/8-maths-ncert-chapter-1-7-320.jpg)
![16 MATHEMATICS
The point to be labelled is twice as far from and to the right of 0 as the point
labelled
1
3
.Soitistwotimes
1
3
,i.e.,
2
3
.Youcancontinuetolabelequally-spacedpointson
thenumberlineinthesameway.Thenextmarkingis1.Youcanseethat1isthesameas
3
3
.
Then comes
4 5 6
, ,
3 3 3
(or 2),
7
3
and so on as shown on the number line (vi)
(vi)
Similarly, to represent
1
8
, the number line may be divided into eight equal parts as
shown:
We use the number
1
8
to name the first point of this division.The second point of
division will be labelled
2
8
, the third point
3
8
, and so on as shown on number
line(vii)
(vii)
Anyrationalnumbercanberepresentedonthenumberlineinthisway.Inarational
number,thenumeralbelowthebar,i.e.,thedenominator,tellsthenumberofequal
parts into which the first unit has been divided. The numeral above the bar i.e., the
numerator, tells ‘how many’of these parts are considered. So, a rational number
such as
4
9
means four of nine equal parts on the right of 0 (number line viii) and
for
7
4
−
, we make 7 markings of distance
1
4
each on the left of zero and starting
from0.Theseventhmarkingis
7
4
−
[numberline(ix)].
(viii)
(ix)](https://image.slidesharecdn.com/8-maths-ncert-chapter-1-181216065336/85/8-maths-ncert-chapter-1-16-320.jpg)
8 maths-ncert-chapter-1
- 1. RATIONAL NUMBERS 1 1.1 Introduction InMathematics,wefrequentlycomeacrosssimpleequationstobesolved.Forexample, theequation x + 2 = 13 (1) is solved when x = 11, because this value of x satisfies the given equation.The solution 11 is a natural number. On the other hand, for the equation x + 5 = 5 (2) the solution gives the whole number 0 (zero). If we consider only natural numbers, equation (2) cannot be solved.To solve equations like (2), we added the number zero to thecollectionofnaturalnumbersandobtainedthewholenumbers.Evenwholenumbers willnotbesufficienttosolveequationsoftype x + 18 = 5 (3) Do you see ‘why’?We require the number –13 which is not a whole number.This led us to think of integers, (positive and negative). Note that the positive integers correspondtonaturalnumbers.Onemaythinkthatwehaveenoughnumberstosolveall simpleequationswiththeavailablelistofintegers.Considertheequations 2x = 3 (4) 5x + 7 = 0 (5) forwhichwecannotfindasolutionfromtheintegers.(Checkthis) Weneedthenumbers 3 2 tosolveequation(4)and 7 5 − tosolve equation(5).Thisleadsustothecollectionofrationalnumbers. We have already seen basic operations on rational numbers.Wenowtrytoexploresomepropertiesofoperations on the different types of numbers seen so far. Rational Numbers CHAPTER 1
- 2. 2 MATHEMATICS 1.2 Properties of Rational Numbers 1.2.1 Closure (i) Wholenumbers Let us revisit the closure property for all the operations on whole numbers in brief. Operation Numbers Remarks Addition 0 + 5 = 5, a whole number Whole numbers are closed 4 + 7 = ... . Is it a whole number? underaddition. In general, a + b is a whole number for any two whole numbers a and b. Subtraction 5 – 7 = – 2, which is not a Whole numbers are not closed wholenumber. undersubtraction. Multiplication 0 × 3 = 0, a whole number Whole numbers are closed 3×7=....Isitawholenumber? undermultiplication. In general, if a and b are any two whole numbers, their productab isawholenumber. Division 5 ÷ 8 = 5 8 , which is not a wholenumber. Check for closure property under all the four operations for natural numbers. (ii) Integers Let us now recall the operations under which integers are closed. Operation Numbers Remarks Addition – 6 + 5 = – 1, an integer Integers are closed under Is – 7 + (–5) an integer? addition. Is 8 + 5 an integer? In general, a + b is an integer for any two integers a and b. Subtraction 7 – 5 = 2, an integer Integers are closed under Is 5 – 7 an integer? subtraction. – 6 – 8 = – 14, an integer Whole numbers are not closed underdivision.
- 3. RATIONAL NUMBERS 3 – 6 – (– 8) = 2, an integer Is 8 – (– 6) an integer? Ingeneral,foranytwointegers a and b, a – b is again an integer. Check if b – a is also an integer. Multiplication 5 × 8 = 40, an integer Integers are closed under Is – 5 × 8 an integer? multiplication. – 5 × (– 8) = 40, an integer Ingeneral,foranytwointegers a and b, a × b is also an integer. Division 5 ÷8 = 5 8 , which is not Integers are not closed aninteger. underdivision. You have seen that whole numbers are closed under addition and multiplication but notundersubtractionanddivision.However,integersareclosedunderaddition,subtraction andmultiplicationbutnotunderdivision. (iii) Rational numbers Recallthatanumberwhichcanbewrittenintheform p q ,wherepandqareintegers and q ≠ 0 is called a rational number. For example, 2 3 − , 6 7 are all rational numbers. Since the numbers 0, –2, 4 can be written in the form p q , they are also rationalnumbers.(Checkit!) (a) You know how to add two rational numbers. Let us add a few pairs. 3 ( 5) 8 7 − + = 21 ( 40) 19 56 56 + − − = (arationalnumber) 3 ( 4) 8 5 − − + = 15 ( 32) ... 40 − + − = Isitarationalnumber? 4 6 7 11 + = ... Isitarationalnumber? We find that sum of two rational numbers is again a rational number. Check it for a few more pairs of rational numbers. We say that rational numbers are closed under addition. That is, for any two rational numbers a and b, a + b is also a rational number. (b) Willthedifferenceoftworationalnumbersbeagainarationalnumber? We have, 5 2 7 3 − − = 5 3 – 2 7 29 21 21 − × × − = (arationalnumber)
- 4. 4 MATHEMATICS TRY THESE 5 4 8 5 − = 25 32 40 − = ... Isitarationalnumber? 3 8 7 5 − − = ... Isitarationalnumber? Trythisforsomemorepairsof rationalnumbers.Wefindthatrationalnumbers are closed under subtraction. That is, for any two rational numbers a and b, a – b is also a rational number. (c) Let us now see the product of two rational numbers. 2 4 3 5 − × = 8 3 2 6 ; 15 7 5 35 − × = (both the products are rational numbers) 4 6 5 11 − − × = ... Isitarationalnumber? Takesomemorepairsofrationalnumbersandcheckthattheirproductisagain arationalnumber. We say that rational numbers are closed under multiplication. That is, for any two rational numbers a and b, a × b is also a rational number. (d) We note that 5 2 25 3 5 6 − − ÷ = (arationalnumber) 2 5 ... 7 3 ÷ = . Is it a rational number? 3 2 ... 8 9 − − ÷ = . Is it a rational number? Can you say that rational numbers are closed under division? We find that for any rational number a, a ÷ 0 is not defined. So rational numbers are not closed under division. However,ifweexcludezerothenthecollectionof,allotherrationalnumbersis closedunderdivision. Fillintheblanksinthefollowingtable. Numbers Closed under addition subtraction multiplication division Rationalnumbers Yes Yes ... No Integers ... Yes ... No Wholenumbers ... ... Yes ... Naturalnumbers ... No ... ...
- 5. RATIONAL NUMBERS 5 1.2.2 Commutativity (i) Wholenumbers Recall the commutativity of different operations for whole numbers by filling the followingtable. Operation Numbers Remarks Addition 0 + 7 = 7 + 0 = 7 Additioniscommutative. 2 + 3 = ... + ... = .... For any two whole numbers a and b, a + b = b + a Subtraction ......... Subtractionisnotcommutative. Multiplication ......... Multiplicationiscommutative. Division ......... Divisionisnotcommutative. Checkwhetherthecommutativityoftheoperationsholdfornaturalnumbersalso. (ii) Integers Fill in the following table and check the commutativity of different operations for integers: Operation Numbers Remarks Addition ......... Additioniscommutative. Subtraction Is 5 – (–3) = – 3 – 5? Subtractionisnotcommutative. Multiplication ......... Multiplicationiscommutative. Division ......... Divisionisnotcommutative. (iii) Rational numbers (a) Addition You know how to add two rational numbers. Let us add a few pairs here. 2 5 1 5 2 1 and 3 7 21 7 3 21 − − + = + = So, 2 5 5 2 3 7 7 3 − − + = + Also, 6 8 5 3 − − + = ... and – 8 6 ... 3 5 − + = Is 6 8 8 6 5 3 3 5 − − − − + = + ?
- 6. 6 MATHEMATICS TRY THESE Is 3 1 1 3 8 7 7 8 − − + = + ? You find that two rational numbers can be added in any order. We say that addition is commutative for rational numbers. That is, for any two rational numbers a and b, a + b = b + a. (b) Subtraction Is 2 5 5 2 3 4 4 3 − = − ? Is 1 3 3 1 2 5 5 2 − = − ? Youwillfindthatsubtractionisnotcommutativeforrationalnumbers. (c) Multiplication We have, 7 6 42 6 7 3 5 15 5 3 − − − × = = × Is 8 4 4 8 9 7 7 9 − − − − × = × ? Check for some more such products. You will find that multiplication is commutative for rational numbers. In general, a × b = b × a for any two rational numbers a and b. (d) Division Is 5 3 3 5 ? 4 7 7 4 − − ÷ = ÷ You will find that expressions on both sides are not equal. Sodivisionisnotcommutativeforrationalnumbers. Completethefollowingtable: Numbers Commutative for addition subtraction multiplication division Rationalnumbers Yes ... ... ... Integers ... No ... ... Wholenumbers ... ... Yes ... Naturalnumbers ... ... ... No
- 7. RATIONAL NUMBERS 7 1.2.3 Associativity (i) Wholenumbers Recalltheassociativityofthefouroperationsforwholenumbersthroughthistable: Operation Numbers Remarks Addition ......... Additionisassociative Subtraction ......... Subtractionisnotassociative Multiplication Is 7 × (2 × 5) = (7 × 2) × 5? Multiplicationisassociative Is 4 × (6 × 0) = (4 × 6) × 0? For any three whole numbers a, b and c a × (b × c) = (a × b) × c Division ......... Divisionisnotassociative Fillinthistableandverifytheremarksgiveninthelastcolumn. Checkforyourselftheassociativityofdifferentoperationsfornaturalnumbers. (ii) Integers Associativityofthefouroperationsforintegerscanbeseenfromthistable Operation Numbers Remarks Addition Is (–2) + [3 + (– 4)] Additionisassociative = [(–2) + 3)] + (– 4)? Is (– 6) + [(– 4) + (–5)] = [(– 6) +(– 4)] + (–5)? For any three integers a, b and c a + (b + c) = (a + b) + c Subtraction Is 5 – (7 – 3) = (5 – 7) – 3? Subtractionisnotassociative Multiplication Is 5 × [(–7) × (– 8) Multiplicationisassociative = [5 × (–7)] × (– 8)? Is (– 4) × [(– 8) × (–5)] = [(– 4) × (– 8)] × (–5)? For any three integers a, b and c a × (b × c) = (a × b) × c Division Is [(–10) ÷ 2] ÷ (–5) Divisionisnotassociative = (–10) ÷ [2 ÷ (– 5)]?
- 8. 8 MATHEMATICS (iii) Rational numbers (a) Addition We have 2 3 5 2 7 27 9 3 5 6 3 30 30 10 − − − − − − + + = + = = 2 3 5 1 5 27 9 3 5 6 15 6 30 10 − − − − − − + + = + = = So, 2 3 5 2 3 5 3 5 6 3 5 6 − − − − + + = + + Find 1 3 4 1 3 4 and 2 7 3 2 7 3 − − − − + + + + .Arethetwosumsequal? Take some more rational numbers, add them as above and see if the two sums are equal. We find that addition is associative for rational numbers. That is, for any three rational numbers a, b and c, a + (b + c) = (a + b) + c. (b) Subtraction Is 2 4 1 2 4 1 ? 3 5 2 3 5 2 − − − − − = − − Checkforyourself. Subtraction is not associative for rational numbers. (c) Multiplication Letuschecktheassociativityformultiplication. 7 5 2 7 10 70 35 3 4 9 3 36 108 54 − − − − × × = × = = 7 5 2 ... 3 4 9 − × × = Wefindthat 7 5 2 7 5 2 3 4 9 3 4 9 − − × × = × × Is 2 6 4 2 6 4 ? 3 7 5 3 7 5 − − × × = × × Take some more rational numbers and check for yourself. We observe that multiplication is associative for rational numbers. That is for any three rational numbers a, b and c, a × (b × c) = (a × b) × c.
- 9. RATIONAL NUMBERS 9 TRY THESE (d) Division Let us see if 1 1 2 1 1 2 2 3 5 2 2 5 − − ÷ ÷ = ÷ ÷ We have, LHS = 1 1 2 2 3 5 − ÷ ÷ = 1 1 5 2 3 2 − ÷ × (reciprocal of 2 5 is 5 2 ) = 1 5 2 6 ÷ − = ... RHS = 1 1 2 2 3 5 − ÷ ÷ = 1 3 2 2 1 5 − × ÷ = 3 2 2 5 − ÷ = ... Is LHS = RHS? Check for yourself. You will find that division is not associative for rational numbers. Completethefollowingtable: Numbers Associative for addition subtraction multiplication division Rationalnumbers ... ... ... No Integers ... ... Yes ... Wholenumbers Yes ... ... ... Naturalnumbers ... Yes ... ... Example 1: Find 3 6 8 5 7 11 21 22 − − + + + Solution: 3 6 8 5 7 11 21 22 − − + + + = 198 252 176 105 462 462 462 462 − − + + + (Note that 462 is the LCM of 7, 11, 21 and 22) = 198 252 176 105 462 − − + = 125 462 −
- 10. 10 MATHEMATICS We can also solve it as. 3 6 8 5 7 11 21 22 − − + + + = 3 8 6 5 7 21 11 22 − − + + + (byusingcommutativityandassociativity) = 9 ( 8) 12 5 21 22 + − − + + (LCM of 7 and 21 is 21; LCM of 11 and 22 is 22) = 1 7 21 22 − + = 22 147 125 462 462 − − = Doyouthinkthepropertiesofcommutativityandassociativitymadethecalculationseasier? Example 2: Find 4 3 15 14 5 7 16 9 − − × × × Solution: We have 4 3 15 14 5 7 16 9 − − × × × = 4 3 15 ( 14) 5 7 16 9 × × − − × × × = 12 35 12 ( 35) 1 35 24 35 24 2 − − − × − × = = × We can also do it as. 4 3 15 14 5 7 16 9 − − × × × = 4 15 3 14 5 16 7 9 − − × × × (Usingcommutativityandassociativity) = 3 2 4 3 − − × = 1 2 1.2.4 The role of zero (0) Lookatthefollowing. 2 + 0 = 0 + 2 = 2 (Addition of 0 to a whole number) – 5 + 0 = ... + ... = – 5 (Addition of 0 to an integer) 2 7 − + ... = 0 + 2 7 − = 2 7 − (Addition of 0 to a rational number)
- 11. RATIONAL NUMBERS 11 THINK, DISCUSS AND WRITE You have done such additions earlier also. Do a few more such additions. Whatdoyouobserve?Youwillfindthatwhenyouadd0toawholenumber,thesum isagainthatwholenumber.Thishappensforintegersandrationalnumbersalso. Ingeneral, a + 0 = 0 + a = a, where a is a whole number b + 0 = 0 + b = b, where b is an integer c + 0 = 0 + c = c, wherec is a rational number Zero is called the identity for the addition of rational numbers. It is the additive identity for integers and whole numbers as well. 1.2.5 The role of 1 We have, 5 × 1 = 5 = 1 × 5 (Multiplicationof1withawholenumber) 2 7 − × 1 = ... × ... = 2 7 − 3 8 × ... = 1 × 3 8 = 3 8 What do you find? You will find that when you multiply any rational number with 1, you get back that rationalnumberastheproduct.Checkthisforafewmorerationalnumbers.Youwillfind that, a × 1 = 1 × a = a for any rational number a. We say that 1 is the multiplicative identity for rational numbers. Is1themultiplicativeidentityforintegers?Forwholenumbers? If a property holds for rational numbers, will it also hold for integers? For whole numbers?Whichwill?Whichwillnot? 1.2.6 Negative of a number Whilestudyingintegersyouhavecomeacrossnegativesofintegers.Whatisthenegative of 1? It is – 1 because 1 + (– 1) = (–1) + 1 = 0 So, what will be the negative of (–1)? It will be 1. Also, 2 + (–2) = (–2) + 2 = 0, so we say 2 is the negative or additive inverse of –2 and vice-versa. In general, for an integer a, we have, a + (– a) = (– a) + a = 0; so, a is the negative of – a and – a is the negative of a. For the rational number 2 3 , we have, 2 2 3 3 + − = 2 ( 2) 0 3 + − =
- 12. 12 MATHEMATICS Also, 2 2 3 3 − + = 0 (How?) Similarly, 8 ... 9 − + = 8 ... 0 9 − + = 11 ... 7 − + = 11 ... 0 7 − + = Ingeneral,forarationalnumber a b ,wehave, 0 a a a a b b b b + − = − + = .Wesay that a b − is the additive inverse of a b and a b is the additive inverse of a b − . 1.2.7 Reciprocal By which rational number would you multiply 8 21 , to get the product 1? Obviously by 21 8 21 , since 1 8 21 8 × = . Similarly, 5 7 − must be multiplied by 7 5− so as to get the product 1. We say that 21 8 is the reciprocal of 8 21 and 7 5− is the reciprocal of 5 7 − . Can you say what is the reciprocal of 0 (zero)? Istherearationalnumberwhichwhenmultipliedby0gives1? Thus,zerohasnoreciprocal. We say that a rational number c d is called the reciprocal or multiplicative inverse of anotherrationalnumber a b if 1 a c b d × = . 1.2.8 Distributivity of multiplication over addition for rational numbers To understand this, consider the rational numbers 3 2 , 4 3 − and 5 6 − . 3 2 5 4 3 6 − − × + = 3 (4) ( 5) 4 6 − + − × = 3 1 4 6 − − × = 3 1 24 8 = Also 3 2 4 3 − × = 3 2 6 1 4 3 12 2 − × − − = = ×
- 13. RATIONAL NUMBERS 13 TRY THESE And 3 5 4 6 − − × = 5 8 Therefore 3 2 3 5 4 3 4 6 − − − × + × = 1 5 1 2 8 8 − + = Thus, 3 2 5 4 3 6 − − × + = 3 2 3 5 4 3 4 6 − − − × + × Find using distributivity. (i) 7 3 7 5 5 12 5 12 − × + × (ii) 9 4 9 3 16 12 16 9 − × + × Example 3: Writetheadditiveinverseofthefollowing: (i) 7 19 − (ii) 21 112 Solution: (i) 7 19 is the additive inverse of 7 19 − because 7 19 − + 7 19 = 7 7 0 19 19 − + = = 0 (ii) The additive inverse of 21 112 is 21 112 − (Check!) Example 4: Verify that – (– x) is the same as x for (i) x = 13 17 (ii) 21 31 x − = Solution: (i) We have, x = 13 17 The additive inverse of x = 13 17 is – x = 13 17 − since 13 13 0 17 17 − + = . The same equality 13 13 0 17 17 − + = , shows that the additive inverse of 13 17 − is 13 17 or 13 17 − − = 13 17 , i.e., – (– x) = x. (ii) Additive inverse of 21 31 x − = is – x = 21 31 since 21 21 0 31 31 − + = . The same equality 21 21 0 31 31 − + = , shows that the additive inverse of 21 31 is 21 31 − , i.e., – (– x) = x. Distributivity of Multi- plication over Addition and Subtraction. For all rational numbers a, b and c, a (b + c) = ab + ac a (b – c) = ab – ac When you use distributivity, you split a product as a sum or difference of two products.
- 14. 14 MATHEMATICS Example 5: Find 2 3 1 3 3 5 7 14 7 5 − × − − × Solution: 2 3 1 3 3 5 7 14 7 5 − × − − × = 2 3 3 3 1 5 7 7 5 14 − × − × − (bycommutativity) = 2 3 3 3 1 5 7 7 5 14 − − × + × − = 3 2 3 1 7 5 5 14 − + − (bydistributivity) = 3 1 1 7 14 − × − = 6 1 1 14 2 − − − = EXERCISE 1.1 1. Usingappropriatepropertiesfind. (i) 2 3 5 3 1 3 5 2 5 6 − × + − × (ii) 2 3 1 3 1 2 5 7 6 2 14 5 × − − × + × 2. Writetheadditiveinverseofeachofthefollowing. (i) 2 8 (ii) 5 9 − (iii) 6 5 − − (iv) 2 9− (v) 19 6− 3. Verify that – (– x) = x for. (i) x = 11 15 (ii) 13 17 x = − 4. Findthemultiplicativeinverseofthefollowing. (i) – 13 (ii) 13 19 − (iii) 1 5 (iv) 5 3 8 7 − − × (v) – 1 2 5 − × (vi) – 1 5. Namethepropertyundermultiplicationusedineachofthefollowing. (i) 4 4 4 1 1 5 5 5 − − × = × = − (ii) 13 2 2 13 17 7 7 17 − − − − × = × (iii) 19 29 1 29 19 − × = − 6. Multiply 6 13 by the reciprocal of 7 16 − . 7. Tell what property allows you to compute 1 4 1 4 6 as 6 3 3 3 3 × × × × . 8. Is 8 9 themultiplicativeinverseof 1 1 8 − ?Whyorwhynot? 9. Is0.3themultiplicativeinverseof 1 3 3 ? Why or why not?
- 15. RATIONAL NUMBERS 15 10. Write. (i) The rational number that does not have a reciprocal. (ii) The rational numbers that are equal to their reciprocals. (iii) Therationalnumberthatisequaltoitsnegative. 11. Fillintheblanks. (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 ≠ 0 is ________. (v) The product of two rational numbers is always a _______. (vi) The reciprocal of a positive rational number is ________. 1.3 Representation of Rational Numbers on the Number Line Youhavelearnttorepresentnaturalnumbers,wholenumbers,integers andrationalnumbersonanumberline.Letusrevisethem. Naturalnumbers (i) Wholenumbers (ii) Integers (iii) Rational numbers (iv) (v) The point on the number line (iv) which is half way between 0 and 1 has been labelled 1 2 .Also, the first of the equally spaced points that divides the distance between 0 and 1 into three equal parts can be labelled 1 3 , as on number line (v). How would you labelthesecondofthesedivisionpointsonnumberline(v)? The line extends indefinitely only to the right side of 1. The line extends indefinitely to the right, but from 0. There are no numbers to the left of 0. The line extends indefinitely on both sides. Do you see any numbers between –1, 0; 0, 1 etc.? The line extends indefinitely on both sides. But you can now see numbers between –1, 0; 0, 1 etc.
- 16. 16 MATHEMATICS The point to be labelled is twice as far from and to the right of 0 as the point labelled 1 3 .Soitistwotimes 1 3 ,i.e., 2 3 .Youcancontinuetolabelequally-spacedpointson thenumberlineinthesameway.Thenextmarkingis1.Youcanseethat1isthesameas 3 3 . Then comes 4 5 6 , , 3 3 3 (or 2), 7 3 and so on as shown on the number line (vi) (vi) Similarly, to represent 1 8 , the number line may be divided into eight equal parts as shown: We use the number 1 8 to name the first point of this division.The second point of division will be labelled 2 8 , the third point 3 8 , and so on as shown on number line(vii) (vii) Anyrationalnumbercanberepresentedonthenumberlineinthisway.Inarational number,thenumeralbelowthebar,i.e.,thedenominator,tellsthenumberofequal parts into which the first unit has been divided. The numeral above the bar i.e., the numerator, tells ‘how many’of these parts are considered. So, a rational number such as 4 9 means four of nine equal parts on the right of 0 (number line viii) and for 7 4 − , we make 7 markings of distance 1 4 each on the left of zero and starting from0.Theseventhmarkingis 7 4 − [numberline(ix)]. (viii) (ix)
- 17. RATIONAL NUMBERS 17 TRY THESE Writetherationalnumberforeachpointlabelledwithaletter. (i) (ii) 1.4 Rational Numbers between Two Rational Numbers Can you tell the natural numbers between 1 and 5? They are 2, 3 and 4. How many natural numbers are there between 7 and 9? There is one and it is 8. How many natural numbers are there between 10 and 11? Obviously none. List the integers that lie between –5 and 4. They are – 4, – 3, –2, –1, 0, 1, 2, 3. How many integers are there between –1 and 1? How many integers are there between –9 and –10? You will find a definite number of natural numbers (integers) between two natural numbers(integers). How many rational numbers are there between 3 10 and 7 10 ? You may have thought that they are only 4 5 , 10 10 and 6 10 . But you can also write 3 10 as 30 100 and 7 10 as 70 100 . Now the numbers, 31 32 33 , , 100 100 100 68 69 , ... , 100 100 , are all between 3 10 and 7 10 . The number of these rational numbers is 39. Also 3 10 can be expressed as 3000 10000 and 7 10 as 7000 10000 . Now, we see that the rational numbers 3001 3002 6998 6999 , ,..., , 10000 10000 10000 10000 are between 3 10 and 7 10 . These are 3999 numbers in all. In this way, we can go on inserting more and more rational numbers between 3 10 and 7 10 .Sounlikenaturalnumbersandintegers,thenumberofrationalnumbersbetween tworationalnumbersisnotdefinite.Hereisonemoreexample. How many rational numbers are there between 1 10 − and 3 10 ? Obviously 0 1 2 , , 10 10 10 arerationalnumbersbetweenthegivennumbers.
- 18. 18 MATHEMATICS If we write 1 10 − as 10000 100000 − and 3 10 as 30000 100000 , we get the rational numbers 9999 9998 , ,..., 100000 100000 − − 29998 100000 − , 29999 100000 , between 1 10 − and 3 10 . You will find that you get countless rational numbers between any two given rational numbers. Example 6: Write any 3 rational numbers between –2 and 0. Solution: –2 can be written as 20 10 − and 0 as 0 10 . Thus we have 19 18 17 16 15 1 , , , , , ..., 10 10 10 10 10 10 − − − − − − between –2 and 0. You can take any three of these. Example 7: Find any ten rational numbers between 5 6 − and 5 8 . Solution:Wefirstconvert 5 6 − and 5 8 torationalnumberswiththesamedenominators. 5 4 20 6 4 24 − × − = × and 5 3 15 8 3 24 × = × Thus we have 19 18 17 14 , , ,..., 24 24 24 24 − − − as the rational numbers between 20 24 − and 15 24 . You can take any ten of these. Another Method Letusfindrationalnumbersbetween1and2.Oneofthemis 1.5or 1 1 2 or 3 2 .Thisisthe mean of 1 and 2.You have studied mean in ClassVII. We find that between any two given numbers, we need not necessarily get an integer but there will always lie a rational number. We can use the idea of mean also to find rational numbers between any two given rationalnumbers. Example 8: Find a rational number between 1 4 and 1 2 . Solution:We find the mean of the given rational numbers. 1 1 2 4 2 + ÷ = 1 2 3 1 3 2 4 4 2 8 + ÷ = × = 3 8 lies between 1 4 and 1 2 . This can be seen on the number line also.
- 19. RATIONAL NUMBERS 19 We find the mid point ofAB which is C, represented by 1 1 2 4 2 + ÷ = 3 8 . We find that 1 3 1 4 8 2 < < . If a and b are two rational numbers, then 2 a b+ is a rational number between a and b such that a < 2 a b+ < b. This again shows that there are countless number of rational numbers between any twogivenrationalnumbers. Example 9: Find three rational numbers between 1 4 and 1 2 . Solution: We find the mean of the given rational numbers. As given in the above example, the mean is 3 8 and 1 3 1 4 8 2 < < . Wenowfindanotherrationalnumberbetween 1 3 and 4 8 .Forthis,weagainfindthemean of 1 3 and 4 8 . That is, 1 3 2 4 8 + ÷ = 5 1 5 8 2 16 × = 1 5 3 1 4 16 8 2 < < < Now find the mean of 3 1 and 8 2 . We have, 3 1 2 8 2 + ÷ = 7 1 8 2 × = 7 16 Thus we get 1 5 3 7 1 4 16 8 16 2 < < < < . Thus, 5 3 7 , , 16 8 16 are the three rational numbers between 1 1 and 4 2 . Thiscanclearlybeshownonthenumberlineasfollows: In the same way we can obtain as many rational numbers as we want between two givenrationalnumbers.Youhavenoticedthattherearecountlessrationalnumbersbetween anytwogivenrationalnumbers.
- 20. 20 MATHEMATICS EXERCISE 1.2 1. Represent these numbers on the number line. (i) 7 4 (ii) 5 6 − 2. Represent 2 5 9 , , 11 11 11 − − − on the number line. 3. Writefiverationalnumberswhicharesmallerthan2. 4. Find ten rational numbers between 2 1 and 5 2 − . 5. Findfiverationalnumbersbetween. (i) 2 3 and 4 5 (ii) 3 2 − and 5 3 (iii) 1 4 and 1 2 6. Writefiverationalnumbersgreaterthan–2. 7. Find ten rational numbers between 3 5 and 3 4 . WHAT HAVE WE DISCUSSED? 1. Rationalnumbersareclosedundertheoperationsofaddition,subtractionandmultiplication. 2. Theoperationsadditionandmultiplicationare (i) commutativeforrationalnumbers. (ii) associative for rational numbers. 3. The rational number 0 is theadditive identity for rational numbers. 4. Therationalnumber1isthemultiplicativeidentityforrationalnumbers. 5. The additive inverse of the rational number a b is a b − and vice-versa. 6. The reciprocal or multiplicative inverse of the rational number a b is c d if 1 a c b d × = . 7. Distributivity of rational numbers: For all rational numbers a, b and c, a(b + c) = ab + ac and a(b – c) = ab – ac 8. Rational numbers can be represented on a number line. 9. Betweenanytwogivenrationalnumberstherearecountlessrationalnumbers.Theideaofmean helpsustofindrationalnumbersbetweentworationalnumbers.