Rational Numbers and Decimal Representation

Transcription

1 0 CHAPTER The Rel Numbers nd Their Representtions. Rtionl Numbers nd Deciml Representtion Properties nd Opertions The set of rel numbers is composed of two importnt mutully exclusive subsets: the rtionl numbers nd the irrtionl numbers. (Two sets re mutully exclusive if they contin no elements in common.) Recll from Section. tht quotients of integers re clled rtionl numbers. Think of the rtionl numbers s being mde up of ll the frctions (quotients of integers with denomintor not equl to zero) nd ll the integers. Any integer cn be written s the quotient of two integers. For exmple, the integer 9 cn be written s the quotient 9, or 8, or, nd so on. Also, cn be expressed s quotient of integers s or 0, nd so on. (How cn the integer 0 be written s quotient of integers?) Since both frctions nd integers cn be written s quotients of integers, the set of rtionl numbers is defined s follows. Rtionl Numbers Rtionl numbers x x is quotient of two integers, with denomintor not 0 A rtionl number is sid to be in lowest terms if the gretest common fctor of the numertor (top number) nd the denomintor (bottom number) is. Rtionl numbers re written in lowest terms by using the fundmentl property of rtionl numbers. Fundmentl Property of Rtionl Numbers If, b, nd k re integers with b 0 nd k 0, then k b k b. EXAMPLE Write 4 in lowest terms. Since the gretest common fctor of nd 4 is 8, The clcultor reduces 4 to lowest terms, s illustrted in Exmple. In the bove exmple, 4. If we multiply the numertor of the frction on the left by the denomintor of the frction on the right, we obtin 08. If we multiply the denomintor of the frction on the left by the numertor of the frction on the right, we obtin The result is the sme in both cses. One wy of determining whether two frctions re equl is to perform this test. If the product of the extremes ( nd in this cse) equls the product of the mens (4 nd ), the frctions re equl. This test for equlity of rtionl numbers is clled the cross-product test.

2 . Rtionl Numbers nd Deciml Representtion Cross-Product Test for Equlity of Rtionl Numbers For rtionl numbers b nd cd, b 0, d 0, b c d if nd only if d b c. Benjmin Bnneker ( 80) spent the first hlf of his life tending frm in Mrylnd. He gined reputtion loclly for his mechnicl skills nd bilities in mthemticl problem solving. In he cquired stronomy books from neighbor nd devoted himself to lerning stronomy, observing the skies, nd mking clcultions. In 89 Bnneker joined the tem tht surveyed wht is now the District of Columbi. Bnneker published lmncs yerly from 9 to 80. He sent copy of his first lmnc to Thoms Jefferson long with n impssioned letter ginst slvery. Jefferson subsequently chmpioned the cuse of this erly Africn- Americn mthemticin. The opertion of ddition of rtionl numbers cn be illustrted by the sketches in Figure 0. The rectngle t the top left is divided into three equl portions, with one of the portions in color. The rectngle t the top right is divided into five equl prts, with two of them in color. The totl of the res in color is represented by the sum To evlute this sum, the res in color must be redrwn in terms of common unit. Since the lest common multiple of nd is, redrw both rectngles with prts. See Figure. In the figure, of the smll rectngles re in color, so In generl, the sum. my be found by writing b nd cd with the common denomintor bd, retining this denomintor in the sum, nd dding the numertors: b c d. b c d d bc bd bd d bc. bd _ + + = FIGURE 0 FIGURE

3 CHAPTER The Rel Numbers nd Their Representtions A similr cse cn be mde for the difference between rtionl numbers. A forml definition of ddition nd subtrction of rtionl numbers follows. Adding nd Subtrcting Rtionl Numbers If b nd cd re rtionl numbers, then b c d d bc bd nd b c d d bc. bd This forml definition is seldom used in prctice. In prcticl problems involving ddition nd subtrction of rtionl numbers, we usully rewrite the frctions with the lest common multiple of their denomintors, clled the lest common denomintor. The results of Exmple re illustrted in this screen. EXAMPLE () Add:. 0 The lest common multiple of nd 0 is 0. Now write nd 0 with denomintors of 0, nd then dd the numertors. Proceed s follows: Since 0, nd since 0 0, Thus, 4 0, (b) Subtrct: The lest common multiple of 80 nd 00 is The product of two rtionl numbers is defined s follows. Multiplying Rtionl Numbers If b nd cd re rtionl numbers, then b c d c bd.

4 . Rtionl Numbers nd Deciml Representtion EXAMPLE Find ech of the following products. To illustrte the results of Exmple, we use prentheses round the frction fctors. () (b) In prctice, multipliction problem such s this is often solved by using slsh mrks to indicte tht common fctors hve been divided out of the numertor nd denomintor. 8 0 is divided out of the terms nd 8; is divided out of nd 0. FOR FURTHER THOUGHT The Influence of Spnish Coinge on Stock Prices Until August 8, 000, when decimliztion of the U.S. stock mrket begn, mrket prices were reported with frctions hving denomintors with powers of, such s nd. Did 4 8 you ever wonder why this ws done? During the erly yers of the United Sttes, prior to the minting of its own coinge, the Spnish eight-reles coin, lso known s the Spnish milled dollr, circulted freely in the sttes. Its frctionl prts, the four reles, two reles, nd one rel, were known s pieces of eight, nd described s such in pirte nd tresure lore. When the New York Stock Exchnge ws founded in 9, it chose to use the Spnish milled dollr s its price bsis, rther thn the deciml bse s proposed by Thoms Jefferson tht sme yer. In the September 99 issue of COINge, Tom Delorey s rticle The End of Pieces of Eight gives the following ccount: As the Spnish dollr nd its frctions continued to be legl tender in Americ longside the deciml coins until 8, there ws no urgency to chnge the system nd by the time the Spnish-Americn money ws withdrwn in 8, pricing stocks in eighths of dollr nd no less ws trdition crved in stone. Being somewht conservtive orgniztion, the NYSE sw no need to fix wht ws not broken. All prices on the U.S. stock mrkets re now reported in decimls. (Source: Stock price tbles go to deciml listings, The Times Picyune, June, 000.) For Group Discussion Consider this: Hve you ever herd this old cheer? Two bits, four bits, six bits, dollr. All for the (home tem), stnd up nd holler. The term two bits refers to cents. Discuss how this cheer is bsed on the Spnish eight-reles coin.

5 4 CHAPTER The Rel Numbers nd Their Representtions In frction, the frction br indictes the opertion of division. Recll tht, in the previous section, we defined the multiplictive inverse, or reciprocl, of the nonzero number b. The multiplictive inverse of b is b. We cn now define division using multiplictive inverses. Definition of Division If nd b re rel numbers, b 0, then b b. Erly U.S. cents nd hlf cents used frctions to denote their denomintions. The hlf cent used nd the cent used. (See Exercise 8 for photo of n interesting error coin.) The coins shown here were prt of the collection of Louis E. Elisberg, Sr., tht ws uctioned by Bowers nd Meren, Inc., severl yers go. Louis Elisberg ws the only person ever to ssemble complete collection of United Sttes coins. The hlf cent pictured sold for $0,000 nd the cent sold for $,00. The cent shown in Exercise 8 went for mere $90. You hve probbly herd the rule, To divide frctions, invert the divisor nd multiply. But hve you ever wondered why this rule works? To illustrte it, suppose tht you hve 8 of gllon of milk nd you wish to find how mny qurts you hve. Since qurt is 4 of gllon, you must sk yourself, How mny 4s re there in 8? This would be interpreted s 8 or The fundmentl property of rtionl numbers discussed erlier cn be extended to rtionl number vlues of, b, nd k. With 8, b 4, nd k 4 (the reciprocl of b 4), 8 4 b k b k Now notice tht we begn with the division problem 8 4 which, through series of equivlent expressions, led to the multipliction problem 8 4. So dividing by 4 is equivlent to multiplying by its reciprocl, 4. By the definition of multipliction of frctions, , nd thus there re or qurts in 8 gllon.* We now stte the rule for dividing b by cd Dividing Rtionl Numbers If b nd cd re rtionl numbers, where cd 0, then b c d b d c d bc. * is mixed number. Mixed numbers re covered in the exercises for this section.

6 . Rtionl Numbers nd Deciml Representtion EXAMPLE 4 Find ech of the following quotients. This screen supports the results in Exmple 4(b) nd (c). () (b) (c) There is no integer between two consecutive integers, such s nd 4. However, rtionl number cn lwys be found between ny two distinct rtionl numbers. For this reson, the set of rtionl numbers is sid to be dense. Density Property of the Rtionl Numbers If r nd t re distinct rtionl numbers, with r t, then there exists rtionl number s such tht r s t. To find the rithmetic men, or verge, of n numbers, we dd the numbers nd then divide the sum by n. For two numbers, the number tht lies hlfwy between them is their verge. Yer Number (in thousnds) 99,0 99,9 99,0 998, 999,4 000,8 Source: U.S. Bureu of Lbor Sttistics. EXAMPLE () Find the rtionl number hlfwy between nd. First, find their sum. Now divide by The number hlfwy between nd is 4. (b) The tble in the mrgin shows the number of lbor union or employee ssocition members, in thousnds, for the yers Wht is the verge number, in thousnds, for this six-yer period? To find this verge, divide the sum by. The computtion in Exmple (b) is shown here.,0,9,0,,4,8 The verge to the nerest whole number of thousnds is,8. 9,8,80.8

7 CHAPTER The Rel Numbers nd Their Representtions Repeted ppliction of the density property implies tht between two given rtionl numbers re infinitely mny rtionl numbers. It is lso true tht between ny two rel numbers there is nother rel number. Thus, we sy tht the set of rel numbers is dense. Simon Stevin (48 0) worked s bookkeeper in Belgium nd becme n engineer in the Netherlnds rmy. He is usully given credit for the development of decimls While hs repeting deciml representtion., the clcultor rounds off in the finl deciml plce displyed. Deciml Form of Rtionl Numbers Up to now in this section, we hve discussed rtionl numbers in the form of quotients of integers. Rtionl numbers cn lso be expressed s decimls. Deciml numerls hve plce vlues tht re powers of 0. For exmple, the deciml numerl is red four hundred eighty-three nd thirty-nine thousnd, four hundred seventy-five millionths. The plce vlues re s shown here.... Hundreds Tens Given rtionl number in the form b, it cn be expressed s deciml most esily by entering it into clcultor. For exmple, to write 8 s deciml, enter, then enter the opertion of division, then enter 8. Press the equls key to find the following equivlence. Of course, this sme result my be obtined by long division, s shown in the mrgin. By this result, the rtionl number 8 is the sme s the deciml.. A deciml such s., which stops, is clled terminting deciml. Other exmples of terminting decimls re 4 Ones., Not ll rtionl numbers cn be represented by terminting decimls. For exmple, convert 4 into deciml by dividing into 4 using clcultor. The disply shows., or perhps.4. However, we see tht the long division process, shown in the mrgin, indictes tht we will ctully get...., with the digits repeting over nd over indefinitely. To indicte this, we write br (clled vinculum) over the block of digits tht repets. Therefore, we cn write 4.. A deciml such s., which continues indefinitely, is clled repeting deciml. Other exmples of repeting decimls re.4, Deciml point., 0 8 Tenths.., nd Hundredths nd Thousndths Ten-thousndths... Hundred-thousndths Millionths

8 . Rtionl Numbers nd Deciml Representtion Although only ten deciml digits re shown, ll three frctions hve decimls tht repet endlessly. Becuse of the limittions of the disply of clcultor, nd becuse some rtionl numbers hve repeting decimls, it is importnt to be ble to interpret clcultor results ccordingly when obtining repeting decimls. While we shll distinguish between terminting nd repeting decimls in this book, some mthemticins prefer to consider ll rtionl numbers s repeting decimls. This cn be justified by thinking this wy: if the division process leds to reminder of 0, then zeros repet without end in the deciml form. For exmple, we cn consider the deciml form of 4 s follows. 4.0 By considering the possible reminders tht my be obtined when converting quotient of integers to deciml, we cn drw n importnt conclusion bout the deciml form of rtionl numbers. If the reminder is never zero, the division will produce repeting deciml. This hppens becuse ech step of the division process must produce reminder tht is less thn the divisor. Since the number of different possible reminders is less thn the divisor, the reminders must eventully begin to repet. This mkes the digits of the quotient repet, producing repeting deciml. To find bsebll plyer s btting verge, we divide the number of hits by the number of t-bts. A surprising prdox exists concerning verges; it is possible for Plyer A to hve higher btting verge thn Plyer B in ech of two successive yers, yet for the two-yer period, Plyer B cn hve higher totl btting verge. Look t the chrt. Yer Plyer A Plyer B Two yertotl In both individul yers, Plyer A hd higher verge, but for the two-yer period, Plyer B hd the higher verge. This is n exmple of Simpson s prdox from sttistics. Deciml Representtion of Rtionl Numbers Any rtionl number cn be expressed s either terminting deciml or repeting deciml. To determine whether the deciml form of quotient of integers will terminte or repet, we use the following rule. Criteri for Terminting nd Repeting Decimls A rtionl number b in lowest terms results in terminting deciml if the only prime fctor of the denomintor is or (or both). A rtionl number b in lowest terms results in repeting deciml if prime other thn or ppers in the prime fctoriztion of the denomintor. The justifiction of this rule is bsed on the fct tht the prime fctors of 0 re nd, nd the deciml system uses ten s its bse. EXAMPLE Without ctully dividing, determine whether the deciml form of the given rtionl number termintes or repets. () 8 Since 8 fctors s, the deciml form will terminte. No primes other thn or divide the denomintor.

9 8 CHAPTER The Rel Numbers nd Their Representtions (b) 0 0. Since ppers s prime fctor of the denomintor, the deciml form will repet. (c) First write the rtionl number in lowest terms. Since, the deciml form will terminte. We hve seen tht rtionl number will be represented by either terminting or repeting deciml. Wht bout the reverse process? Tht is, must terminting deciml or repeting deciml represent rtionl number? The nswer is yes. For exmple, the terminting deciml. represents rtionl number. The results of Exmple re supported in this screen.... = Terminting or Repeting? One of the most bffling truths of elementry mthemtics is the following: Most people believe tht.9 hs to be less tht, but this is not the cse. The following rgument shows otherwise. Let x Then 9x 9 x x x Subtrct. Therefore, Similrly, it cn be shown tht ny terminting deciml cn be represented s repeting deciml with n endless string of 9s. For exmple, nd This is wy of justifying tht ny rtionl number my be represented s repeting deciml. See Exercises 9 nd 9 for more on EXAMPLE Write ech terminting deciml s quotient of integers. (). (b) Repeting decimls cnnot be converted into quotients of integers quite so quickly. The steps for mking this conversion re given in the next exmple. (This exmple uses bsic lgebr.) EXAMPLE 8 Find quotient of two integers equl to.8. Step : Let x.8, so x Step : Multiply both sides of the eqution x by 00. (Use 00 since there re two digits in the prt tht repets, nd 00 0.) x x x Step : Subtrct the expressions in Step from the finl expressions in Step. 00x x x 8 (Recll tht x x nd Step 4: Solve the eqution 99x 8 by dividing both sides by x 8 99x x x.8 00x x 99x.

10 . Rtionl Numbers nd Deciml Representtion 9 This result my be checked with clcultor. Remember, however, tht the clcultor will only show finite number of deciml plces, nd my round off in the finl deciml plce shown.