RATIONAL NUMBERS Rational Numbers1

1 RATIONAL NUMBERS IntroductionIn Mathematics, we frequently come across simple equations to be solved. For example,the equationx + 2 =13(1)is solved when x = 11, because this value of x satisfies the given equation. The solution11 is a natural number . On the other hand, for the equationx + 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 tothe collection of natural NUMBERS and obtained the whole NUMBERS . Even whole numberswill not be sufficient to solve equations of typex + 18 =5(3)Do you see why ? We require the number 13 which is not a whole number . Thisled us to think of integers, (positive and negative). Note that the positive integerscorrespond to natural NUMBERS . One may think that we have enough NUMBERS to solve allsimple equations with the available list of integers. Now consider the equations2x =3(4)5x + 7 =0(5)for which we cannot find a solution from the integers.

2 (Check this)We need the NUMBERS 32 to solve equation (4) and 75 to solveequation (5). This leads us to the collection of RATIONAL have already seen basic operations on rationalnumbers. We now try to explore some properties of operationson the different types of NUMBERS seen so NumbersCHAPTER12022-232 Properties of RATIONAL Closure(i) whole numbersLet us revisit the closure property for all the operations on whole NUMBERS in + 5 = 5, a whole numberWhole NUMBERS are closed4 + 7 = .. Is it a whole number ?under general, a + b is a wholenumber for any two wholenumbers a and 7 = 2, which is not aWhole NUMBERS are not closedwhole 3 = 0, a whole numberWhole NUMBERS are closed3 7 = .. Is it a whole number ?under general, if a and b are any twowhole NUMBERS , their product abis a whole 8 = 58, which is not awhole for closure property under all the four operations for natural NUMBERS .

3 (ii)IntegersLet us now recall the operations under which integers are 6 + 5 = 1, an integerIntegers are closed underIs 7 + ( 5) an integer? 8 + 5 an integer?In general, a + b is an integerfor any two integers a and 5 = 2, an integerIntegers are closed underIs 5 7 an integer?subtraction. 6 8 = 14, an integerWhole NUMBERS are not closedunder NUMBERS 3 6 ( 8) = 2, an integerIs 8 ( 6) an integer?In general, for any two integersa and b, a b is again an if b a is also an 8 = 40, an integerIntegers are closed underIs 5 8 an integer?multiplication. 5 ( 8) = 40, an integerIn general, for any two integersa and b, a b is also an 8 = 58, which is notIntegers are not closedan have seen that whole NUMBERS are closed under addition and multiplication butnot under subtraction and division. However, integers are closed under addition, subtractionand multiplication but not under division.

4 (iii) RATIONAL numbersRecall that a number which can be written in the form pq, where p and q are integersand q 0 is called a RATIONAL number . For example, 23 , 67, 95 are all rationalnumbers. Since the NUMBERS 0, 2, 4 can be written in the form pq, they are alsorational NUMBERS . (Check it!)(a)You know how to add two RATIONAL NUMBERS . Let us add a few ( 5)87 +=21 ( 40)195656+ =(a RATIONAL number )3( 4)85 + =15 ( 32)..40 + =Is it a RATIONAL number ?46711+ =..Is it a RATIONAL number ?We find that sum of two RATIONAL NUMBERS is again a RATIONAL number . Check itfor a few more pairs of RATIONAL say that RATIONAL NUMBERS are closed under addition. That is, for anytwo 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,5273 =5 3 27292121 =(a RATIONAL number )2022-234 MATHEMATICSTRY THESE5485 = 25 3240 =.

5 Is it a RATIONAL number ?3785 =..Is it a RATIONAL number ?Try this for some more pairs of RATIONAL NUMBERS . We find that RATIONAL numbersare closed under subtraction. That is, for any two RATIONAL NUMBERS a andb, a b is also a RATIONAL number .(c)Let us now see the product of two RATIONAL =8 326;15 7535 =(both the products are RATIONAL NUMBERS )46511 =..Is it a RATIONAL number ?Take some more pairs of RATIONAL NUMBERS and check that their product is againa RATIONAL say that RATIONAL NUMBERS are closed under multiplication. Thatis, for any two RATIONAL NUMBERS a and b, a b is also a rationalnumber.(d)We note that 5225356 =(a RATIONAL number ) = . Is it a RATIONAL number ? =. 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 RATIONAL NUMBERS are not closed under , if we exclude zero then the collection of, all other RATIONAL NUMBERS isclosed under in the blanks in the following underadditionsubtractionmultiplicationdi visionRational NUMBERS Commutativity(i) whole numbersRecall the commutativity of different operations for whole NUMBERS by filling thefollowing + 7 = 7 + 0 = 7 Addition is + 3 =.

6 + .. = ..For any two wholenumbers a and b,a + b = b + aSubtraction ..Subtraction is not ..Multiplication is ..Division is not whether the commutativity of the operations hold for natural NUMBERS also.(ii)IntegersFill in the following table and check the commutativity of different operations forintegers:OperationNumbersRemarksAddit ion ..Addition is 5 ( 3) = 3 5?Subtraction is not ..Multiplication is ..Division is not commutative.(iii) RATIONAL NUMBERS (a)AdditionYou know how to add two RATIONAL NUMBERS . Let us add a few pairs 51521and37217321 + =+= So,2 5 5237 73 + = + Also, + 6583 = .. and Is + = + 65836583?2022-236 MATHEMATICSTRY THESEIs3 1 1387 78 + = + ?You find that two RATIONAL NUMBERS can be added in any order. We say thataddition is commutative for RATIONAL NUMBERS .

7 That is, for any two rationalnumbers a and b, a + b = b + a.(b)SubtractionIs2 5 5 23 4 4 3 = ?Is1 3 3 12 5 5 2 = ?You will find that subtraction is not commutative for RATIONAL that subtraction is not commutative for integers and integers are also rationalnumbers. So, subtraction will not be commutative for RATIONAL NUMBERS too.(c)MultiplicationWe have, = = 736542156573Is = 89474789?Check for some more such will find that multiplication is commutative for RATIONAL general, a b = b a for any two RATIONAL NUMBERS a and b.(d)DivisionIs5 3 35?47 74 = You will find that expressions on both sides are not division is not commutative for RATIONAL the following table:NumbersCommutative foradditionsubtractionmultiplicationdivi sionRational NUMBERS Associativity(i) whole numbersRecall the associativity of the four operations for whole NUMBERS through this is is not associativeMultiplicationIs 7 (2 5) = (7 2) 5?

8 Multiplication is associativeIs 4 (6 0) = (4 6) 0?For any three wholenumbers a, b and ca (b c) = (a b) cDivision ..Division is not associativeFill in this table and verify the remarks given in the last for yourself the associativity of different operations for natural NUMBERS .(ii)IntegersAssociativity of the four operations for integers can be seen from this tableOperationNumbersRemarksAdditionIs ( 2) + [3 + ( 4)]Addition is associative= [( 2) + 3)] + ( 4)?Is ( 6) + [( 4) + ( 5)]= [( 6) +( 4)] + ( 5)?For any three integers a, b and ca + (b + c) = (a + b) + cSubtractionIs 5 (7 3) = (5 7) 3?Subtraction is not associativeMultiplicationIs 5 [( 7) ( 8)Multiplication is associative= [5 ( 7)] ( 8)?Is ( 4) [( 8) ( 5)]= [( 4) ( 8)] ( 5)?For any three integers a, b and ca (b c) = (a b) cDivisionIs [( 10) 2] ( 5)Division is not associative= ( 10) [2 ( 5)]?]

9 2022-238 MATHEMATICS(iii) RATIONAL NUMBERS (a)AdditionWe have 235272793563303010 ++=+== 235152793561563010 ++=+== So,2352 35356356 ++=++ Find ++ + + 123743123743and. Are the two sums equal?Take some more RATIONAL NUMBERS , add them as above and see if the two sumsare equal. We find that addition is associative for RATIONAL NUMBERS . Thatis, for any three RATIONAL NUMBERS a, b and c, a + (b + c) = (a + b) + c.(b)SubtractionYou already know that subtraction is not associative for integers, then whatabout RATIONAL = 234512234512?Check for is not associative for RATIONAL NUMBERS .(c)MultiplicationLet us check the associativity for 27 10703534 9336 10854 = == = find that75 27 5234 9349 = Is26 4264?375375 = Take some more RATIONAL NUMBERS and check for observe that multiplication is associative for RATIONAL NUMBERS .

10 That isfor any three RATIONAL NUMBERS a, b and c, a (b c) = (a b) NUMBERS 9 TRY THESE(d)DivisionRecall that division is not associative for integers, then what about RATIONAL NUMBERS ?Let us see if 11 2112235235 = We have, LHS = 11 2235 = 121352 (reciprocal of 25 is 52)= 1526 = .. RHS=112235 = 132215 = 3 225 = ..Is LHS = RHS? Check for yourself. You will find that division isnot associative for RATIONAL the following table:NumbersAssociative foradditionsubtractionmultiplicationdivi sionRational 1: Find 36857112122 +++ Solution: 36857112122 +++ =198462252462176462105462+ + + (Note that 462 is the LCM of7, 11, 21 and 22)=198252 176 105462 + = 125462 2022-2310 MATHEMATICSWe can also solve it +++ = 38657211122 +++ (by using commutativity and associativity)= 9 82112 522+ + + ( ) (LCM of 7 and 21 is 21;LCM of 11 and 22 is 22)= 172122 + = 22 147125462462 =Do you think the properties of commutativity and associativity made the calculations easier?