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RATIONAL NUMBERS, , 1, , CHAPTER, , 1, , Rational Numbers, , 1.1 Introduction, In Mathematics, we frequently come across simple equations to be solved. For example,, the equation, 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, the collection of natural numbers and obtained the whole numbers. Even whole numbers, will not be sufficient to solve equations of type, 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, correspond to natural numbers. One may think that we have enough numbers to solve all, simple equations with the available list of integers. Now consider the equations, 2x = 3, , (4), , 5x + 7 = 0, , (5), , for which we cannot find a solution from the integers. (Check this), We need the numbers, , 3, −7, to solve equation (4) and, to solve, 2, 5, , equation (5). This leads us to the collection of rational numbers., We have already seen basic operations on rational, numbers. We now try to explore some properties of operations, on the different types of numbers seen so far., 2021–22
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2, , MATHEMATICS, , 1.2 Properties of Rational Numbers, 1.2.1 Closure, (i) Whole numbers, 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? under addition., 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 number., , Multiplication, , 0 × 3 = 0, a whole number, Whole numbers are closed, 3 × 7 = ... . Is it a whole number? under multiplication., In general, if a and b are any two, whole numbers, their product ab, is a whole number., , Division, , 5, , which is not a, 8, whole number., , 5÷8=, , Whole numbers are not closed, under subtraction., , Whole numbers are not closed, under division., , 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, , Addition, , – 6 + 5 = – 1, an integer, Is – 7 + (–5) an integer?, 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, Is 5 – 7 an integer?, – 6 – 8 = – 14, an integer, , 2021–22, , Remarks, , Integers are closed under, addition., , Integers are closed under, subtraction.
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RATIONAL NUMBERS, , – 6 – (– 8) = 2, an integer, Is 8 – (– 6) an integer?, In general, for any two integers, a and b, a – b is again an integer., Check if b – a is also an integer., Multiplication, , 5 × 8 = 40, an integer, Is – 5 × 8 an integer?, – 5 × (– 8) = 40, an integer, In general, for any two integers, a and b, a × b is also an integer., , Division, , 5÷8=, , Integers are closed under, multiplication., , 5, , which is not, 8, an integer., , Integers are not closed, under division., , You have seen that whole numbers are closed under addition and multiplication but, not under subtraction and division. However, integers are closed under addition, subtraction, and multiplication but not under division., (iii) Rational numbers, p, Recall that a number which can be written in the form q , where p and q are integers, 2 6 9, and q ≠ 0 is called a rational number. For example, − , ,, are all rational, 3 7 −5, , p, numbers. Since the numbers 0, –2, 4 can be written in the form q , they are also, rational numbers. (Check it!), , (a) You know how to add two rational numbers. Let us add a few pairs., 3 ( −5), 21 + ( − 40) − 19, +, =, =, (a rational number), 8, 7, 56, 56, − 3 ( − 4), − 15 + ( −32), +, = ..., =, Is it a rational number?, 8, 5, 40, 4 6, +, = ..., Is it a rational number?, 7 11, 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) Will the difference of two rational numbers be again a rational number?, We have,, −5 2 −5 × 3 – 2 × 7 −29, − =, =, 7, 3, 21, 21, 2021–22, , (a rational number), , 3
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4, , MATHEMATICS, , 5 4, 25 − 32, − =, = ..., 8 5, 40, , Is it a rational number?, , 3 − 8, − = ..., Is it a rational number?, 7 5 , Try this for some more pairs of rational numbers. We find that rational numbers, 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, −8 3 2 6, × =, ; × =, 3 5, 15 7 5 35, , (both the products are rational numbers), , 4 −6, ×, = ..., Is it a rational number?, 5 11, Take some more pairs of rational numbers and check that their product is again, a rational number., 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., −5 2 − 25, (d) We note that, (a rational number), ÷ =, 3 5, 6, 2 5, −3 − 2, ÷ = ... . Is it a rational number?, ÷, = ... . Is it a rational number?, 7 3, 8, 9, −, , 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, if we exclude zero then the collection of, all other rational numbers is, closed under division., , TRY THESE, Fill in the blanks in the following table., Numbers, , Closed under, , addition, , subtraction, , multiplication, , division, , Yes, , Yes, , ..., , No, , Integers, , ..., , Yes, , ..., , No, , Whole numbers, , ..., , ..., , Yes, , ..., , Natural numbers, , ..., , No, , ..., , ..., , Rational numbers, , 2021–22
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RATIONAL NUMBERS, , 1.2.2 Commutativity, (i) Whole numbers, Recall the commutativity of different operations for whole numbers by filling the, following table., Operation, , Addition, , Numbers, , 0+7=7+0=7, 2 + 3 = ... + ... = ...., For any two whole, numbers a and b,, a+b=b+a, , Remarks, , Addition is commutative., , Subtraction, , ........., , Subtraction is not commutative., , Multiplication, , ........., , Multiplication is commutative., , Division, , ........., , Division is not commutative., , Check whether the commutativity of the operations hold for natural numbers also., (ii) Integers, Fill in the following table and check the commutativity of different operations for, integers:, Operation, , Addition, Subtraction, , Numbers, , ........., Is 5 – (–3) = – 3 – 5?, , Remarks, , Addition is commutative., Subtraction is not commutative., , Multiplication, , ........., , Multiplication is commutative., , Division, , ........., , Division is not commutative., , (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 , + = ... and, 5 3, , Is, , − 6 −8 −8 −6 , + = + ?, 5 3 3 5, 2021–22, , 5
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6, , MATHEMATICS, , −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., , Is, , (b) Subtraction, 2 5 5 2, − = − ?, 3 4 4 3, 1 3 3 1, − = − ?, Is, 2 5 5 2, You will find that subtraction is not commutative for rational numbers., Note that subtraction is not commutative for integers and integers are also rational, numbers. So, subtraction will not be commutative for rational numbers too., , Is, , (c) Multiplication, We have,, , −7 6 − 42 6 −7 , × =, = × , 3 5 15 5 3 , , −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., So division is not commutative for rational numbers., , Is, , TRY THESE, Complete the following table:, Numbers, , Commutative for, , addition, , subtraction, , multiplication, , division, , Yes, , ..., , ..., , ..., , Integers, , ..., , No, , ..., , ..., , Whole numbers, , ..., , ..., , Yes, , ..., , Natural numbers, , ..., , ..., , ..., , No, , Rational numbers, , 2021–22
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RATIONAL NUMBERS, , 1.2.3 Associativity, (i) Whole numbers, Recall the associativity of the four operations for whole numbers through this table:, Operation, , Numbers, , Remarks, , Addition, , ........., , Addition is associative, , Subtraction, , ........., , Subtraction is not associative, , Multiplication, , Division, , Is 7 × (2 × 5) = (7 × 2) × 5?, Is 4 × (6 × 0) = (4 × 6) × 0?, For any three whole, numbers a, b and c, a × (b × c) = (a × b) × c, , Multiplication is associative, , Division is not associative, , ........., , Fill in this table and verify the remarks given in the last column., Check for yourself the associativity of different operations for natural numbers., (ii) Integers, Associativity of the four operations for integers can be seen from this table, Operation, , Numbers, , Remarks, , Addition, , Is (–2) + [3 + (– 4)], = [(–2) + 3)] + (– 4)?, Is (– 6) + [(– 4) + (–5)], = [(– 6) +(– 4)] + (–5)?, For any three integers a, b and c, a + (b + c) = (a + b) + c, , Addition is associative, , Subtraction, , Is 5 – (7 – 3) = (5 – 7) – 3?, , Subtraction is not associative, , Multiplication, , Is 5 × [(–7) × (– 8), = [5 × (–7)] × (– 8)?, Is (– 4) × [(– 8) × (–5)], = [(– 4) × (– 8)] × (–5)?, For any three integers a, b and c, a × (b × c) = (a × b) × c, , Multiplication is associative, , Division, , Is [(–10) ÷ 2] ÷ (–5), = (–10) ÷ [2 ÷ (– 5)]?, , Division is not associative, , 2021–22, , 7
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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, + + and +, 2 7 3 , 2, , 3 − 4, + . Are the two sums equal?, 7 3 , , 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, You already know that subtraction is not associative for integers, then what, about rational numbers., Is, , −2 − 4 1 2 − 4 1, −, −, = − − ?, 3 5 2 3 5 2, , Check for yourself., Subtraction is not associative for rational numbers., (c) Multiplication, Let us check the associativity for multiplication., −7 5 2 −7 10 −70 −35, × × = × =, =, 3 4 9 3 36 108 54, , −7 5 2, × × = ..., 3 4 9, We find that, , −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., 2021–22
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RATIONAL NUMBERS, , (d) Division, Recall that division is not associative for integers, then what about rational numbers?, Let us see if, , 1 −1 2 1 −1 2, ÷, ÷ = ÷ ÷, 2 3 5 2 3 5, , We have, LHS =, , 1 −1 2 1 −1 5 , ÷ ÷ = ÷ × , 2 3 5 2 3 2, , =, , RHS, , 2, 5, is ), 5, 2, , (reciprocal of, , 1 5, ÷ − = ..., 2 6, , 1 −1 2, = ÷ ÷, 2 3 5, , 1 −3 2, −3 2, ÷ = ..., = × ÷, =, 2 5, 2 1 5, Is LHS = RHS? Check for yourself. You will find that division is, not associative for rational numbers., , TRY THESE, Complete the following table:, Numbers, , Associative for, addition, , subtraction, , multiplication, , division, , Rational numbers, , ..., , ..., , ..., , No, , Integers, , ..., , ..., , Yes, , ..., , Whole numbers, , Yes, , ..., , ..., , ..., , Natural numbers, , ..., , No, , ..., , ..., , Example 1: Find, , Solution:, , 3 −6 −8 5 , + + + , 7 11 21 22 , , 3 −6 −8 5 , + + + , 7 11 21 22 , , =, , 198 −252 −176 105 , +, +, +, (Note that 462 is the LCM of, 462 462 462 462 , 7, 11, 21 and 22), , =, , 198 − 252 − 176 + 105 −125, =, 462, 462, 2021–22, , 9
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10, , MATHEMATICS, , We can also solve it as., 3 −6 −8 5, + + +, 7 11 21 22, , 3 −8 −6 5 , = + + +, , 7 21 11 22 , , 9 + ( −8) −12 + 5 , +, =, 21 22 , , (by using commutativity and associativity), , (LCM of 7 and 21 is 21; LCM of 11 and 22 is 22), , 1 −7 22 −147 −125, + =, =, 21 22 , 462, 462, Do you think the properties of commutativity and associativity made the calculations easier?, , =, , − 4 3 15 −14 , × ×, ×, , 5, 7 16 9 , Solution: We have, , Example 2: Find, , − 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 , × ×, ×, (Using commutativity and associativity), = , 5 16 7 9 , =, , −3 −2 1, × =, 4 3 2, , 1.2.4 The role of zero (0), Look at the following., 2+0=0+2=2, – 5 + 0 = ... + ... = – 5, −2 −2, −2, + ... = 0 + =, 7 7, 7, , 2021–22, , (Addition of 0 to a whole number), (Addition of 0 to an integer), (Addition of 0 to a rational number)
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RATIONAL NUMBERS, , You have done such additions earlier also. Do a few more such additions., What do you observe? You will find that when you add 0 to a whole number, the sum, is again that whole number. This happens for integers and rational numbers also., In general,, 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,, where c 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, , (Multiplication of 1 with a whole number), , −2, −2, × 1 = ... × ... =, 7, 7, 3, 3, 3, × ... = 1 × =, 8, 8, 8, , What do you find?, You will find that when you multiply any rational number with 1, you get back the same, rational number as the product. Check this for a few more rational numbers. You will find, that, a × 1 = 1 × a = a for any rational number a., We say that 1 is the multiplicative identity for rational numbers., Is 1 the multiplicative identity for integers? For whole numbers?, , THINK, DISCUSS AND WRITE, If a property holds for rational numbers, will it also hold for integers? For whole, numbers? Which will? Which will not?, 1.2.6 Negative of a number, While studying integers you have come across negatives of integers. What is the negative, 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., 2, For the rational number , we have,, 3, , 2 , + −, 3 , , 2, 2 + ( −2), =0, =, 3, 3, , 2021–22, , 11
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12, , MATHEMATICS, , 2 2, − + = 0, 3, 3, , Also,, , (How?), , −8, − 8, + ... = ... + = 0, 9 , 9, , Similarly,, , −11, −11, ... + , , , + ... = 0, =, 7 , 7 , In general, for a rational number, that −, , a , a, , we have, + −, b, b, , a , = −, b , , a a, + = 0 . We say, b b, , a, a, a, is the additive inverse of and, is the additive inverse of, b, b, b, , , −, , a, ., b, , 1.2.7 Reciprocal, By which rational number would you multiply, , 8, , to get the product 1? Obviously by, 21, , 21, 8 21, , since, ×, = 1., 8, 21 8, −5, 7, Similarly,, must be multiplied by, so as to get the product 1., 7, −5, , 7, 21, 8, −5, is the reciprocal of, and, is the reciprocal of, ., −5, 8, 21, 7, Can you say what is the reciprocal of 0 (zero)?, Is there a rational number which when multiplied by 0 gives 1? Thus, zero has no reciprocal., c, We say that a rational number is called the reciprocal or multiplicative inverse of, d, a a c, another non-zero rational number if × = 1., b b d, 1.2.8 Distributivity of multiplication over addition for rational, numbers, , We say that, , To understand this, consider the rational numbers, , −3 2, −5, ., , and, 4 3, 6, , −3 2 −5 −3 (4) + ( −5) , × + =, ×, , 4 3 6 , 4 , 6, , =, Also, , −3 −1, 3 1, × =, =, 24 8, 4 6, , −3 2, −3 × 2 − 6 −1, × =, =, =, 4 3, 4×3, 12, 2, , 2021–22
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RATIONAL NUMBERS, , 5, −3 −5, ×, =, 8, 4, 6, −1 5 1, −3 2 −3 −5 , + =, × +, × =, , 2 8 8, 4 3 4, 6, , And, Therefore, , −3 2 −5 −3, × + = , ×, 4 3 6 4, , Thus,, , Distributivity of Multiplication over Addition, and Subtraction., For all rational numbers a, b, and c,, a (b + c) = ab + ac, a (b – c) = ab – ac, , 2 −3 −5 , × , +, 3 4, 6, , TRY THESE, 7 −3 7 5 , Find using distributivity. (i) × + × , 5 12 5 12 , , 4 9 −3 , 9, (ii) × + × , 16 12 16 9 , , Example 3: Write the additive inverse of the following:, −7, (i), 19, , When you use distributivity, you, split a product as a sum or, difference of two products., , 21, (ii), 112, , Solution:, (i), , 7, −7, −7, 7, −7+7 0, =, is the additive inverse of, because, +, =, =0, 19, 19, 19 19, 19, 19, , (ii) The additive inverse of, , 21, − 21, is, 112, 112, , (Check!), , Example 4: Verify that – (– x) is the same as x for, (i) x =, , 13, 17, , (ii) x =, , Solution: (i) We have, x =, The additive inverse of x =, The same equality, , −21, 31, , 13, 17, , 13 −13 , 13, −13, +, = 0., is – x =, since, 17 17 , 17, 17, , 13 −13 , −13 13, +, = 0 , shows that the additive inverse of, is, , 17, 17, 17, 17, , 13, −13 , or − , =, , i.e., – (– x) = x., , 17 , 17, −21, 21, −21 21, is – x =, since, +, = 0., 31, 31, 31 31, −21 21, 21 −21, +, = 0 , shows that the additive inverse of, The same equality, is, ,, 31 31, 31, 31, i.e., – (– x) = x., , (ii) Additive inverse of x =, , 2021–22, , 13
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14, , MATHEMATICS, , Example 5: Find, Solution:, , 2 −3 1 3 3, ×, −, − ×, 5 7 14 7 5, , 2 −3 1 3 3, 2 −3 3 3 1, ×, −, − × = ×, − × −, (by commutativity), 5 7 14 7 5, 5 7 7 5 14, 2 −3 −3 3 1, + × −, = ×, 5 7 7 5 14, , −3 2 3 1, + −, (by distributivity), 7 5 5 14, −3, 1, − 6 − 1 −1, ×1−, =, =, =, 7, 14, 14, 2, =, , EXERCISE 1.1, 1. Using appropriate properties find., , 2 , 2 3 5 3 1, × −, × + − ×, (ii), 3 5 2 5 6, 5 , 2. Write the additive inverse of each of the following., (i) −, , 2, −5, (ii), 8, 9, 3. Verify that – (– x) = x for., , (i), , (iii), , 5., , 6., 7., 8., 9., , (iv), , 2, −9, , (v), , 19, −6, , 11, 13, (ii) x = −, 15, 17, Find the multiplicative inverse of the following., − 13, 1, − 5 −3, ×, (i) – 13, (ii), (iii), (iv), 19, 5, 8, 7, −2, (v) – 1 ×, (vi) – 1, 5, Name the property under multiplication used in each of the following., −4, −4, 4, 13 −2 −2 −13, ×1= 1×, =−, ×, =, ×, (i), (ii) −, 5, 5, 5, 17, 7, 7, 17, −19, 29, ×, =1, (iii), 29, −19, −7, 6, Multiply, by the reciprocal of, ., 16, 13, 1 , 4 1, 4, Tell what property allows you to compute × 6 × as × 6 × ., 3, 3 , 3 3, 8, 1, Is the multiplicative inverse of −1 ? Why or why not?, 9, 8, 1, Is 0.3 the multiplicative inverse of 3 ? Why or why not?, 3, , (i) x =, , 4., , −6, −5, , 3 1 3 1 2, ×, − × +, 7 6 2 14 5, , 2021–22
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RATIONAL NUMBERS, , 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 number that is equal to its negative., 11. Fill in the blanks., (i) Zero has ________ reciprocal., (ii) The numbers ________ and ________ are their own reciprocals, (iii) The reciprocal of – 5 is ________., 1, , where x ≠ 0 is ________., x, (v) The product of two rational numbers is always a _______., (vi) The reciprocal of a positive rational number is ________., , (iv) Reciprocal of, , 1.3 Representation of Rational Numbers on the, Number Line, You have learnt to represent natural numbers, whole numbers, integers, and rational numbers on a number line. Let us revise them., Natural numbers, (i), , 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., , Whole numbers, (ii), Integers, , The line extends, indefinitely on both sides., Do you see any numbers, between –1, 0; 0, 1 etc.?, , (iii), Rational numbers, (iv), , The line extends indefinitely, on both sides. But you can, now see numbers between, –1, 0; 0, 1 etc., , (v), , The point on the number line (iv) which is half way between 0 and 1 has been, labelled, , 1, . Also, the first of the equally spaced points that divides the distance between, 2, , 1, , as on number line (v). How would you, 3, label the second of these division points on number line (v)?, , 0 and 1 into three equal parts can be labelled, , 2021–22, , 15
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16, , MATHEMATICS, , The point to be labelled is twice as far from and to the right of 0 as the point, 1, 1, 2, labelled . So it is two times , i.e., . You can continue to label equally-spaced points on, 3, 3, 3, the number line in the same way. In this continuation, the next marking is 1. You can, see that 1 is the same as, Then comes, , 3, ., 3, , 7, 4 5 6, , , (or 2), and so on as shown on the number line (vi), 3, 3 3 3, , (vi), , Similarly, to represent, , 1, , the number line may be divided into eight equal parts as, 8, , shown:, 1, to name the first point of this division. The second point of, 8, 2, 3, division will be labelled , the third point , and so on as shown on number, 8, 8, , We use the number, , line (vii), (vii), Any rational number can be represented on the number line in this way. In a rational, number, the numeral below the bar, i.e., the denominator, tells the number of equal, 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, 4, such as means four of nine equal parts on the right of 0 (number line viii) and, 9, −7, 1, for, , we make 7 markings of distance each on the left of zero and starting, 4, 4, −7, from 0. The seventh marking is, [number line (ix)]., 4, (viii), , (ix), , 2021–22
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RATIONAL NUMBERS, , TRY THESE, Write the rational number for each point labelled with a letter., (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)., 3, 7, and ?, How many rational numbers are there between, 10, 10, 4 5, 6, and ., You may have thought that they are only ,, 10 10, 10, 3, 30, 7, 70, 31 32 33, ,, ,, But you can also write, as, and, as, . Now the numbers,, 10, 100, 10 100, 100 100 100, 3, 7, 68 69, , ..., ,, , are all between, and . The number of these rational numbers is 39., 10, 10, 100 100, 3, 3000, 7, 7000, Also, can be expressed as, and, as, . Now, we see that the, 10, 10000, 10, 10000, 7, 3001 3002, 6998 6999, 3, ,, ,...,, ,, are between, and, . These, 10, 10000 10000, 10000 10000, 10, are 3999 numbers in all., 3, In this way, we can go on inserting more and more rational numbers between, 10, 7, and . So unlike natural numbers and integers, the number of rational numbers between, 10, two rational numbers is not definite. Here is one more example., −1, 3, How many rational numbers are there between, and ?, 10, 10, 0 1 2, Obviously , ,, are rational numbers between the given numbers., 10 10 10, , rational numbers, , 2021–22, , 17
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18, , MATHEMATICS, , If we write, , −1, 3, 30000, −10000, as, and, as, , we get the rational numbers, 10, 10, 100000, 100000, , −1, 3, −9999 −9998, −29998 29999, ,, ,...,, ,, , between, and, ., 10, 10, 100000 100000, 100000 100000, 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., 0, −20, Solution: –2 can be written as, and 0 as, ., 10, 10, −19 −18 −17 −16 −15, −1, ,, ,, ,, ,, , ... ,, Thus we have, between –2 and 0., 10 10 10 10 10, 10, You can take any three of these., , Example 7: Find any ten rational numbers between, , 5, −5, and ., 8, 6, , −5, 5, and to rational numbers with the same denominators., 6, 8, −5 × 4 −20, 5 × 3 15, =, =, and, 6×4, 24, 8 × 3 24, −20, 15, −19 −18 −17, 14, Thus we have, as the rational numbers between, and, ., ,, ,, ,...,, 24, 24, 24 24 24, 24, You can take any ten of these., , Solution: We first convert, , Another Method, 3, 1, Let us find rational numbers between 1 and 2. One of them is 1.5 or 1 or . This is the, 2, 2, mean of 1 and 2. You have studied mean in Class VII., 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, rational numbers., 1, 1, and ., 4, 2, Solution: We find the mean of the given rational numbers., , Example 8: Find a rational number between, , 3 1 3, 1 1, 1+ 2 , + ÷ 2 = , ÷ 2 = × =, 4 2, 4, 4 2 8, 3, 1, 1, lies between and ., 8, 4, 2, This can be seen on the number line also., 2021–22
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RATIONAL NUMBERS, , 3, 1 1, We find the mid point of AB which is C, represented by + ÷ 2 = ., 4 2, 8, 1 3 1, We find that < < ., 4 8 2, a+b, If a and b are two rational numbers, then, is a rational number between a and, 2, a+b, b such that a <, < b., 2, This again shows that there are countless number of rational numbers between any, two given rational numbers., , 1, 1, and ., 2, 4, Solution: We find the mean of the given rational numbers., , Example 9: Find three rational numbers between, , As given in the above example, the mean is, , 3, 1 3 1, and < < ., 8, 4 8 2, , We now find another rational number between, of, , 1, 3, and . That is,, 4, 8, , 1, 3, and . For this, we again find the mean, 4, 8, , 1 3, 5 1, 5, + ÷ 2 = × =, 4 8, 8 2 16, 1, 5, 3 1, <, < <, 4 16 8 2, , Now find the mean of, Thus we get, , Thus,, , 3 1, 3, 1, 7, 7 1, and . We have, + ÷ 2 = × =, 8 2, 8, 2, 8 2 16, , 1, 5, 3 7 1, <, < <, < ., 4 16 8 16 2, , 5 3 7, 1, 1, , ,, are the three rational numbers between and ., 16 8 16, 4, 2, , This can clearly be shown on the number line as follows:, , In the same way we can obtain as many rational numbers as we want between two, given rational numbers . You have noticed that there are countless rational numbers between, any two given rational numbers., 2021–22, , 19
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20, , MATHEMATICS, , EXERCISE 1.2, 1. Represent these numbers on the number line. (i), , 7, 4, , (ii), , −5, 6, , −2 −5 −9, , ,, on the number line., 11 11 11, 3. Write five rational numbers which are smaller than 2., , 2. Represent, , 4. Find ten rational numbers between, , −2, 1, and ., 5, 2, , 5. Find five rational numbers between., 2, 4, −3, 5, and, (ii), and, 3, 5, 2, 3, 6. Write five rational numbers greater than –2., , (i), , 7. Find ten rational numbers between, , (iii), , 1, 1, and, 4, 2, , 3, 3, and ., 4, 5, , WHAT HAVE WE DISCUSSED?, 1. Rational numbers are closed under the operations of addition, subtraction and multiplication., 2. The operations addition and multiplication are, (i) commutative for rational numbers., (ii) associative for rational numbers., 3. The rational number 0 is the additive identity for rational numbers., 4. The rational number 1 is the multiplicative identity for rational numbers., 5. The additive inverse of the rational number, , a, a, is − and vice-versa., b, b, , 6. The reciprocal or multiplicative inverse of the rational number, , c, a, a c, is, if × = 1 ., b d, 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. Between any two given rational numbers there are countless rational numbers. The idea of mean, helps us to find rational numbers between two rational numbers., , 2021–22
