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 4 + 7 = ... . Is it a whole number? In general, a + b is a whole number for any two whole numbers a and b. Whole numbers are closed under addition. Subtraction 5 – 7 = – 2, which is not a whole number. Whole numbers are not closed under subtraction. Multiplication 0 × 3 = 0, a whole number 3 × 7 = ... . Is it a whole number? In general, if a and b are any two whole numbers, their product ab is a whole number. Whole numbers are closed under multiplication. Division 5 ÷ 8 = 5 8 , which is not a whole number. 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 Remarks 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. Integers are closed under addition. Subtraction 7 – 5 = 2, an integer Is 5 – 7 an integer? – 6 – 8 = – 14, an integer Integers are closed under subtraction. RATIONAL NUMBERS 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 pRecall that a number which can be written in the form , where p and q are integersq 26and q ≠ 0 is called a rational number. For example, − , are all rational37pnumbers. Since the numbers 0, –2, 4 can be written in the form , they are alsoq rational numbers. (Check it!) (a) You know how to add two rational numbers. Let us add a few pairs. 3(5) 21+− −19− ( 40) + = = (a rational number)87 5656 −3(−4) −+− 15 (32) + = =... Is it a rational number?85 40 46+ = ... Is it a rational number?711We 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, −52 −× ×753–2 −29 − = = (a rational number)7 321 21 54 25 −32− = = ... Is it a rational number?8540 3 −8⎛⎞ −⎜⎟= ... Is it a rational number?⎝⎠75 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. −24 −832 6× =; ×= (both the products are rational numbers)3 5 157535 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. (d) We note that 52 3 5 −÷= 25 6 − (a rational number) 25 ...73÷=. 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, if we exclude zero then the collection of, all other rational numbers is closed under division. Fill in the blanks in the following table. Numbers Closed under addition subtraction multiplication division Rational numbers Integers Whole numbers Natural numbers Yes ... ... ... Yes Yes ... No ... ... Yes ... No No ... ... RATIONAL NUMBERS 1.2.2 Commutativity (i) Whole numbers Recall the commutativity of different operations for whole numbers by filling the following table. Operation Numbers Remarks Addition 0 + 7 = 7 + 0 = 7 2 + 3 = ... + ... = .... For any two whole numbers a and b, a + b = b + a 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 Numbers Remarks Addition Subtraction Multiplication Division ......... Is 5 – (–3) = – 3 – 5? ......... ......... Addition is commutative. Subtraction is not commutative. Multiplication is commutative. Division is not commutative. (iii) Rational numbers (a) Addition You know how to add two rational numbers. Let us add a few pairs here. −251 5 −21⎛⎞ += and +=⎜⎟3721 7 321 ⎝⎠ −255 −2⎛⎞ So, +=+⎜ ⎟377 3⎝⎠ −⎛⎞ ⎛⎞ 6 −8 –8 −6 Also, +⎜ ⎟= ... and +⎜ ⎟=...⎝⎠⎝⎠ 53 35 −6 −8 −8 −6⎛ ⎞⎛ ⎞⎛ ⎞ +=+Is ⎜ ⎟⎜ ⎟⎜ ⎟?⎝ ⎠⎝ ⎠⎝ ⎠53 3 5 −311 −3⎛⎞ +=+Is ⎜⎟?877 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 2552Is −=−?3443 1331Is −=−?2552 You will find that subtraction is not commutative for rational numbers. (c) Multiplication −76 −42 6 −7⎛⎞ We have, ×= =×⎜⎟3 5 15 5 ⎝⎠ 3 8 −4 −4 −8−⎛⎞ ⎛⎞ × =×Is ⎜⎟ ⎜⎟ ?⎝⎠ ⎝⎠97 79 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 −533 −5⎛⎞ ÷=÷?Is ⎜⎟477 4⎝⎠ You will find that expressions on both sides are not equal. So division is not commutative for rational numbers. RATIONAL NUMBERS 1.2.3 Associativity (i) Whole numbers Recall the associativity of the four operations for whole numbers through this table: 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 (iii) Rational numbers (a) Addition We have −2 ⎡3 ⎛−5 ⎞⎤ −2 ⎛−7 ⎞−27 −9++ =+==⎢⎜⎟⎥ ⎜⎟3 ⎣5 ⎝ 6 ⎠⎦ 3 ⎝ 30 ⎠ 30 10 ⎡−23⎤ ⎛−5 ⎞−1 ⎛−5 ⎞−27 −9++ =+ ==⎢ ⎥⎜⎟ ⎜⎟⎣ 35⎦⎝ 6 ⎠ 15 ⎝ 6 ⎠ 30 10 −2 ⎡3 ⎛−5 ⎞⎤ ⎡−23⎤ ⎛−5 ⎞ So, +⎢+⎜ ⎟⎥=⎢ +⎥+⎜ ⎟3 ⎣5 ⎝ 6 ⎠⎦ ⎣ 35⎦⎝ 6 ⎠ −1 ⎡3 ⎛− 4⎞⎤ ⎡−13⎤ ⎛− 4⎞Find +⎢+⎜ ⎟⎥ and ⎢ +⎥+⎜ ⎟ . Are the two sums equal? 2 ⎣7 ⎝ 3 ⎠⎦ ⎣ 27⎦⎝ 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 −2 ⎡− 41⎤⎡ 2 ⎛− 4⎞⎤ 1− −=− − ?Is ⎢ ⎥⎢⎜⎟⎥3 ⎣ 52⎦⎣ 3 ⎝ 5 ⎠⎦ 2 Check for yourself. Subtraction is not associative for rational numbers. (c) Multiplication Let us check the associativity for multiplication. −7 ⎛ 52 ⎞−7 10 −70 −35××=×= =⎜⎟3 ⎝ 49 ⎠ 3 36 108 54 ⎛−75⎞ 2× ×=...⎜⎟⎝ 34⎠ 9 −7 ⎛ 52 ⎞ ⎛−75 ⎞ 2× ×= ××We find that ⎜⎟⎜ ⎟3 ⎝ 49 ⎠⎝ 34 ⎠ 9 2 ⎛−64 ⎞⎛ 2 −6 ⎞ 4× ×=× × ?Is ⎜ ⎟⎜ ⎟3 ⎝ 75 ⎠⎝ 37 ⎠ 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. RATIONAL NUMBERS (d) Division 1 12 ⎡1 −⎤ 2− 1⎡ ⎤ ⎛⎞ Let us see if ÷⎢ ÷⎥=⎢÷⎜ ⎟⎥÷ 23522 5⎣ ⎦⎣ ⎝⎠⎦ 1 ⎛−12 ⎞ 1 ⎛−15⎞ 25We have, LHS = ÷⎜ ÷⎟= ÷⎜ ×⎟⎠ (reciprocal of is )2 ⎝35 ⎠2 ⎝32 52 1 ⎛5÷−⎞ = ⎜⎟= ...2 ⎝ 6 ⎠⎡1 ⎛⎞ 1−⎤2 RHS = ⎢÷⎜ ⎟⎥÷ 23 5⎣ ⎝⎠⎦ ⎛1 −3 ⎞ 2 −32×÷= ⎜⎟ = ÷ = ...⎝21 ⎠5 25 Is LHS = RHS? Check for yourself. You will find that division is not associative for rational numbers. Complete the following table: 3 −6 −85⎛ ⎞⎛ ⎞⎛ ⎞ +++Example 1: Find ⎜ ⎟⎜ ⎟⎜ ⎟7 11 21 22 ⎝ ⎠⎝ ⎠⎝ ⎠ 3 −6 −85⎛ ⎞⎛ ⎞⎛ ⎞ +++Solution: ⎜ ⎟⎜ ⎟⎜ ⎟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 We can also solve it as. 3 −6 −85⎛ ⎞⎛ ⎞ +++⎜ ⎟⎜ ⎟7 11 21 22 ⎝ ⎠⎝ ⎠ ⎡3 −⎤ −8 65⎛⎞⎡ ⎤ = ⎢+⎜⎟⎥+⎢+ ⎥ (by using commutativity and associativity)7 21 ⎦ 11 22 ⎦⎣ ⎝⎠⎣ ⎡9(8) +−⎤⎡ −+ 12 5⎤ = + (LCM of 7 and 21 is 21; LCM of 11 and 22 is 22) ⎢ ⎥⎢ ⎥⎣21 ⎦⎣ 22 ⎦1 −7⎛⎞ 22 −147 −125 = +⎜ ⎟= = 21 22 462⎝⎠462 Do you think the properties of commutativity and associativity made the calculations easier? −4 315 ⎛−14⎞Example 2: Find ×××⎜ ⎟5 716 ⎝9 ⎠ Solution: We have −4 315 ⎛−14⎞×× ×⎜⎟5 716 ⎝9 ⎠ ⎛43 15 ×− ⎞×⎞⎛ ( 14) = −×⎜⎟⎜ ⎟⎝57×⎠⎝ 16 ×9 ⎠ −12 ⎛−35 −×12 (35) ⎞− 1×= = = ⎜⎟35 ⎝24 ⎠ 35×24 2 We can also do it as. −4 315 ⎛−14⎞×× ×⎜⎟5 716 ⎝9 ⎠ ⎛−415 ⎞⎡3 ⎛− ⎤ 14 ⎞× ×× = ⎜ ⎟⎢⎜ ⎟⎥(Using commutativity and associativity)⎝5 16 ⎠⎣7 ⎝9 ⎠⎦−3 −2⎛⎞ 1×= ⎜⎟= 43 2⎝⎠1.2.4 The role of zero (0) Look at the following. 2 + 0 = 0 + 2 = 2 (Addition of 0 to a whole number) – 5 + 0 = ... + ... = – 5 (Addition of 0 to an integer) −2 −2 −2⎛⎞ + ... = 0 + ⎜⎟= (Addition of 0 to a rational number)⎝⎠ 7 77 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 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 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. 2For the rational number , we have,3 2 2 2(2) ⎛+−+−⎞ =0⎜ ⎟⎠= 3 ⎝ 33 ⎛−2⎞ 2 Also, ⎜ ⎟+ = 0 (How?)⎝⎠33 −8 −8⎛⎞ Similarly, +... = ... +⎜ ⎟=0⎝⎠ 99 ⎛−11⎞ ⎛−11⎞... +⎜ ⎟= ⎜ ⎟+=... 0⎝ ⎠⎝ ⎠77 aa ⎛ a⎞⎛ a⎞ a=− =0In general, for a rational number , we have, +−⎜ ⎟⎜ ⎟+ . We say bb ⎝ b⎠⎝ b⎠ b a aa ⎛−a⎞that − is the additive inverse of and is the additive inverse of ⎜⎟.⎝⎠bbb b 1.2.7 Reciprocal 8 By which rational number would you multiply , to get the product 1? Obviously by21 21 821 ,since ×=1. 8 21 8 −57 Similarly, must be multiplied by so as to get the product 1.7 −521 87 −5We say that is the reciprocal of and − is the reciprocal of .8215 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. cWe say that a rational number is called the reciprocal or multiplicative inverse ofda acanother rational number if ×=1.bb d 1.2.8 Distributivity of multiplication over addition for rational numbers −32 −5To understand this, consider the rational numbers , and . 43 6 −3 ⎧2 ⎛⎞ 5 −3 ⎧(4) +− −⎫ ( 5) ⎫×⎨+⎜ ⎟ ⎬= ×⎨ ⎬⎝⎠ 4 ⎩36 ⎭4 ⎩ 6 ⎭ 3 −1 31− ⎛⎞ = ×⎜ ⎟= = ⎝⎠4 6 248 −32 −× −632 −1Also × = == 43 4 ×312 2 2 −31 33 Example 5: Find × −−× 5 7 14 75 2 −3133 2 −333 1 Solution: × −−× = × −×− (by commutativity)5 7 14 75 5 7 7514 2 −3 −33⎛⎞ 1 = × +⎜ ⎟×− ⎝⎠ 57 7514 32 3⎞−⎛ 1 = ⎜ +⎟− (by distributivity)⎝⎠755 14 −31 −−61 −1×−= 1 = = 7 14 14 2 EXERCISE 1.1 1. Using appropriate properties find. 23531 2 ⎛3 1312−×+−× ×−⎞−×+ ×(i) (ii) ⎜⎟⎝⎠35256 5 7 62145 2. Write the additive inverse of each of the following. 2 −5 −62 19 (i) (ii) (iii) (iv) (v)8 9 −5 −9−6 3. Verify that – (– x) = x for. 11 13 (i) x = (ii) x =−15 17 4. Find the multiplicative inverse of the following. −13 1 −5 −3 (i) – 13 (ii) (iii) (iv) ×195 87 −2 (v) – 1 × (vi) – 1 5 5. Name the property under multiplication used in each of the following. −4 −44 13 −2 −2 −13 ×=× − −=(i) 11 = (ii) ××5 5 5 177717 −19 29 (iii) ×=129 −19 −76. Multiply 136 by the reciprocal of 16 . 1 ⎛ 4⎞⎛ 1 ⎞ 47. Tell what property allows you to compute ×6 × as ×6 × .⎜ ⎟⎜ ⎟3 ⎝ 3⎠⎝ 3 ⎠ 3 8. Is 98 the multiplicative inverse of −181 ? Why or why not? 9. Is 0.3 the multiplicative inverse of 3 13 ? Why or why not? 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 (iv) Reciprocal of , where x ≠ 0 is ________.(v) The product of two rational numbers is always a _______. (vi) The reciprocal of a positive rational number is ________. x 1.3 Representation of Rational Numbers on the 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 0 and 1 into three equal parts can be labelled , as on number line (v). How would you3 label the second of these division points on number line (v)? The point to be labelled is twice as far from and to the right of 0 as the point 1 12 labelled . So it is two times , i.e., . You can continue to label equally-spaced points on 3 33 the number line in the same way. The next marking is 1. You can see that 1 is the same as 3. 3 Then comes 456 ,,333 (or 2), 7 3 and so on as shown on the number line (vi) (vi) 1 Similarly, to represent , the number line may be divided into eight equal parts as8 1We use the number to name the first point of this division. The second point of 8 division will be labelled 2 8 , the third point 3 8 , and so on as shown on number line (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 from 0. The seventh marking is [number line (ix)]. 9 for −7 4 , we make 7 markings of distance 1 4 each on the left of zero and starting −7 4 If we write 1 10 − as 10000 100000 − and 3 10 as 30000 100000 , we get the rational numbers 9999− 9998− 29998− 29999 1− 3 , ,..., , , between and100000 100000 100000 100000 10 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. −20 0 Solution: –2 can be written as and 0 as10 10. −19 −18 −17 −16 −15 −1Thus we have , , , , ,..., between –2 and 0.1010101010 10 You can take any three of these. Example 7: Find any ten rational numbers between −5 and 5. 68 −55 Solution:We first convert and to rational numbers with the same denominators.685420 53× 15 −× − ==and64 24 × 24 × 83 −20 15−19 −18 −17 14Thus we have , , ,..., as the rational numbers between and 242424 24 24 24. You can take any ten of these. Another Method 13Let us find rational numbers between 1 and 2. One of them is 1.5 or 1 or . This is the 22 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. 11Example 8: Find a rational number between and .42 Solution:We find the mean of the given rational numbers. 11 12 313⎛ ⎞ ⎛+⎞⎜ +⎟÷2 = ⎜ ⎟÷= ×2 = ⎝⎠⎝⎠42 4428 3 11lies between and 2.84 This can be seen on the number line also. RATIONAL NUMBERS ⎛11⎞ 3⎜+ ÷2We find the mid point of AB which is C, represented by ⎟ =⎝42⎠8. 131 We find that <<.482 abIf aand bare two rational numbers, then +is a rational number between aand2ab+bsuch that a< < b.2 This again shows that there are countless number of rational numbers between any two given rational numbers. 11Example 9: Find three rational numbers between and . 42 Solution: We find the mean of the given rational numbers. 3 131 As given in the above example, the mean is and <<.8482 13 We now find another rational number between 4 and 8 . For this, we again find the mean 13 ⎛13⎞ 51 5 of and . That is, ⎜+⎟÷2 = ×=48 ⎝48⎠82 16 1 5 31< << 416 8 2 31 ⎛31⎞ 71 7 Now find the mean of and . We have, ⎜+ ÷2 =⎟× = 82 ⎝82⎠8 2 16 1 5 371 Thus we get < <<<.416 816 2 537 11 Thus, , , are the three rational numbers between and 2.168164 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. EXERCISE 1.2 7 −5 Represent these numbers on the number line. (i) (ii)46 25−−−9 , , on the number line.11 11 11 Write five rational numbers which are smaller than 2. −214. Find ten rational numbers between and .52 5. Find five rational numbers between. 24 −35 11 (i) and (ii) and (iii) and35 2342 6. Write five rational numbers greater than –2. 7. Find ten rational numbers between 3 and 3 . 54 WHAT HAVE WE DISCUSSED?