P.1 Real Numbers. Real Numbers. What you should learn

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1 7_P.qp /7/6 9:8 AM Chpter P Pge Prerequisites P. Rel Numbers Wht you should lern Rel Numbers Rel numbers re used in everydy life to describe quntities such s ge, miles per gllon, nd popultion. Rel numbers re represented by symbols such s 5, 9,,, , 8., 冪,, nd 䊏 䊏. 冪 Here re some importnt subsets (ech member of subset B is lso member of set A) of the set of rel numbers. 再,,,,...冎 䊏 䊏 䊏 Represent nd clssify rel numbers. Order rel numbers nd use inequlities. Find the bsolute vlues of rel numbers nd the distnce between two rel numbers. Evlute lgebric epressions. Use the bsic rules nd properties of lgebr. Set of nturl numbers Why you should lern it 再,,,,,...冎 Set of whole numbers 再...,,,,,,,,...冎 Set of integers A rel number is rtionl if it cn be written s the rtio p兾q of two integers, where q. For instnce, the numbers Rel numbers re used in every spect of our lives, such s finding the surplus or deficit in the federl budget. See Eercises 8 88 on pge ,.5, nd re rtionl. The deciml representtion of rtionl number either repets 共s in.5 兲 or termintes 共s in.5兲. A rel number tht cnnot be written s the rtio of two integers is clled irrtionl. Irrtionl numbers hve infinite nonrepeting deciml representtions. For instnce, the numbers 7 55 冪 nd re irrtionl. (The symbol mens is pproimtely equl to. ) Figure P. shows subsets of rel numbers nd their reltionships to ech other. Rel numbers re represented grphiclly by rel number line. The point on the rel number line is the origin. Numbers to the right of re positive nd numbers to the left of re negtive, s shown in Figure P.. The term nonnegtive describes number tht is either positive or zero. Aln Schein Photogrphy/Corbis Rel numbers Origin Negtive direction Figure P. Positive direction. 5 Every point on the rel number line corresponds to ectly one rel number. Figure P. Rtionl numbers The Rel Number Line There is one-to-one correspondence between rel numbers nd points on the rel number line. Tht is, every point on the rel number line corresponds to ectly one rel number, clled its coordinte, nd every rel number corresponds to ectly one point on the rel number line, s shown in Figure P.. Irrtionl numbers One-to-One Correspondence π.75 Integers Negtive integers Noninteger frctions (positive nd negtive) Whole numbers Every rel number corresponds to ectly one point on the rel number line. Nturl numbers Figure P. Zero Subsets of Rel Numbers

2 7_P.qp /7/6 9:8 AM Pge Section P. Rel Numbers Ordering Rel Numbers One importnt property of rel numbers is tht they re ordered. Definition of Order on the Rel Number Line If nd b re rel numbers, is less thn b if b is positive. This order is denoted by the inequlity < b. This reltionship cn lso be described by sying tht b is greter thn nd writing b >. The inequlity b mens tht is less thn or equl to b, nd the inequlity b mens tht b is greter thn or equl to. The symbols <, >,, nd, re inequlity symbols. Geometriclly, this definition implies tht < b if nd only if lies to the left of b on the rel number line, s shown in Figure P.. b Figure P. < b if nd only if lies to the left of b. Emple Interpreting Inequlities Describe the subset of rel numbers represented by ech inequlity.. b. > c. < Figure P.5 Solution. The inequlity denotes ll rel numbers less thn or equl to, s shown in Figure P.5. b. The inequlity > denotes ll rel numbers greter thn, s shown in Figure P.6. c. The inequlity < mens tht nd <. The double inequlity denotes ll rel numbers between nd, including but not including, s shown in Figure P.7. Now try Eercise (). > Figure P.6 < Figure P.7 Inequlities cn be used to describe subsets of rel numbers clled intervls. In the bounded intervls below, the rel numbers nd b re the endpoints of ech intervl. Bounded Intervls on the Rel Number Line Nottion Intervl Type Inequlity Grph, b Closed b b, b Open < < b b, b < b b, b < b b STUDY TIP The endpoints of closed intervl re included in the intervl. The endpoints of n open intervl re not included in the intervl.

3 7_P.qp /7/6 9:8 AM Pge Chpter P Prerequisites The symbols, positive infinity, nd, negtive infinity, do not represent rel numbers. They re simply convenient symbols used to describe the unboundedness of n intervl such s, or,. Unbounded Intervls on the Rel Number Line Nottion Intervl Type Inequlity Grph,, Open, b, b Open, Entire rel line > b b < b b < < STUDY TIP An intervl is unbounded when it continues indefinitely in one or both directions. Emple Using Inequlities to Represent Intervls Use inequlity nottion to describe ech of the following.. c is t most. b. All in the intervl, 5 Solution. The sttement c is t most cn be represented by c. b. All in the intervl, 5 cn be represented by < 5. Now try Eercise 5. Emple Interpreting Intervls Give verbl description of ech intervl.., b., c., Solution. This intervl consists of ll rel numbers tht re greter thn nd less thn. b. This intervl consists of ll rel numbers tht re greter thn or equl to. c. This intervl consists of ll negtive rel numbers. Now try Eercise 7. Additionl Emples Use inequlity nottion to describe ech of the following.. m is t lest. b. All in the intervl [, ] c. w is t lest nd t most 5. d. All q in the intervl, Solution. m b. c. w 5 d. < q < The Lw of Trichotomy sttes tht for ny two rel numbers nd b, precisely one of three reltionships is possible: b, < b, or > b. Lw of Trichotomy

4 7_P.qp /7/6 9:8 AM Pge 5 Section P. Rel Numbers 5 Absolute Vlue nd Distnce The bsolute vlue of rel number is its mgnitude, or the distnce between the origin nd the point representing the rel number on the rel number line. Definition of Absolute Vlue If is rel number, the bsolute vlue of is if, if <. Eplortion Absolute vlue epressions cn be evluted on grphing utility. When evluting n epression such s 8, prentheses should surround the epression s shown below. Notice from this definition tht the bsolute vlue of rel number is never negtive. For instnce, if 5, then The bsolute vlue of rel number is either positive or zero. Moreover, is the only rel number whose bsolute vlue is. So,. Emple Evluting the Absolute Vlue of Number Evlute for () > nd (b) <. Solution. If >, then nd b. If <, then nd. Now try Eercise 5.. Evlute ech epression. Wht cn you conclude?. 6 b. c. 5 d. 5 Properties of Absolute Vlue.... b b b b, b Absolute vlue cn be used to define the distnce between two points on the rel number line. For instnce, the distnce between nd is 7 7 s shown in Figure P.8. Figure P.8 7 The distnce between nd is 7. Distnce Between Two Points on the Rel Number Line Let nd b be rel numbers. The distnce between nd b is d, b b b.

5 7_P.qp /7/6 9:8 AM Pge 6 6 Chpter P Prerequisites Algebric Epressions One chrcteristic of lgebr is the use of letters to represent numbers. The letters re vribles, nd combintions of letters nd numbers re lgebric epressions. Here re few emples of lgebric epressions. 5,,, 7 y Definition of n Algebric Epression An lgebric epression is combintion of letters (vribles) nd rel numbers (constnts) combined using the opertions of ddition, subtrction, multipliction, division, nd eponentition. The terms of n lgebric epression re those prts tht re seprted by ddition. For emple, hs three terms: nd 5 re the vrible terms nd 8 is the constnt term. The numericl fctor of vrible term is the coefficient of the vrible term. For instnce, the coefficient of 5 is 5, nd the coefficient of is. To evlute n lgebric epression, substitute numericl vlues for ech of the vribles in the epression. Here re two emples. Vlue of Vlue of Epression Vrible Substitute Epression When n lgebric epression is evluted, the Substitution Principle is used. It sttes, If b, then cn be replced by b in ny epression involving. In the first evlution shown bove, for instnce, is substituted for in the epression 5. Common Error A common error is to use the wrong order of opertions when evluting epressions. Remind students tht the order of opertions is s follows.. First do opertions tht occur within symbols of grouping.. Then evlute eponentil epressions.. Then do multiplictions nd divisions from left to right.. Finlly do dditions nd subtrctions from left to right. Bsic Rules of Algebr There re four rithmetic opertions with rel numbers: ddition, multipliction, subtrction, nd division, denoted by the symbols, or,, nd or. Of these, ddition nd multipliction re the two primry opertions. Subtrction nd division re the inverse opertions of ddition nd multipliction, respectively. Subtrction: Add the opposite of b. Division: Multiply by the reciprocl of b. b b If b, then b b b. In these definitions, b is the dditive inverse (or opposite) of b, nd b is the multiplictive inverse (or reciprocl) of b. In the frctionl form b, is the numertor of the frction nd b is the denomintor.

6 7_P.qp /7/6 9:8 AM Pge 7 Becuse the properties of rel numbers below re true for vribles nd lgebric epressions, s well s for rel numbers, they re often clled the Bsic Rules of Algebr. Try to formulte verbl description of ech property. For instnce, the Commuttive Property of Addition sttes tht the order in which two rel numbers re dded does not ffect their sum. Section P. Rel Numbers 7 Bsic Rules of Algebr Let, b, nd c be rel numbers, vribles, or lgebric epressions. Property Emple Commuttive Property of Addition: Commuttive Property of Multipliction: Associtive Property of Addition: Associtive Property of Multipliction: Distributive Properties: Additive Identity Property: Multiplictive Identity Property: Additive Inverse Property: b b b b b c b c bc bc b c b c bc c bc 5y 5y 6 6 Multiplictive Inverse Property:, 5 5 y8 y y 8y y y 8 y Becuse subtrction is defined s dding the opposite, the Distributive Properties re lso true for subtrction. For instnce, the subtrction form of b c b c is b c b c. Properties of Negtion nd Equlity Let, b, nd c be rel numbers, vribles, or lgebric epressions. Property Emple b b b b b b b If then c b c If b, then c bc. 8. If c b c, then b If c bc nd c, then b. 9 STUDY TIP Be sure you see the difference between the opposite of number nd negtive number. If is lredy negtive, then its opposite,, is positive. For instnce, if, then.

7 7_P.qp /7/6 9:9 AM Pge 8 8 Chpter P Prerequisites Properties of Zero Let nd b be rel numbers, vribles, or lgebric epressions.. nd..,. is undefined. 5. Zero-Fctor Property: If b, then or b. Properties nd Opertions of Frctions Let, b, c, nd d be rel numbers, vribles, or lgebric epressions such tht b nd d.. Equivlent Frctions: if nd only if d bc. b c d. Rules of Signs: nd b b b b b c. Generte Equivlent Frctions: c b bc,. Add or Subtrct with Like Denomintors: b ± c b ± c b 5. Add or Subtrct with Unlike Denomintors: b ± c d ± bc d bd 6. Multiply Frctions: b c c d bd 7. Divide Frctions: c b c d b d d c bc, Emple 5 Properties nd Opertions of Frctions 5. Add frctions with unlike denomintors b. Divide frctions. 7 STUDY TIP The or in the Zero-Fctor Property includes the possibility tht either or both fctors my be zero. This is n inclusive or, nd it is the wy the word or is generlly used in mthemtics. Activities. Use inequlity nottion to describe the set of nonnegtive numbers. Answer:. Use intervl nottion to describe the inequlity 6 <. Answer: 6,. Evlute:. Answer:. Find the distnce between nd 6. Answer: 57 STUDY TIP In Property of frctions, the phrse if nd only if implies two sttements. One sttement is: If b cd, then d bc. The other sttement is: If d bc, where b nd d, then b cd. Now try Eercise. If, b, nd c re integers such tht b c, then nd b re fctors or divisors of c. A prime number is n integer tht hs ectly two positive fctors: itself nd. For emple,,, 5, 7, nd re prime numbers. The numbers, 6, 8, 9, nd re composite becuse they cn be written s the product of two or more prime numbers. The number is neither prime nor composite. The Fundmentl Theorem of Arithmetic sttes tht every positive integer greter thn cn be written s the product of prime numbers. For instnce, the prime fctoriztion of is. Point out to students tht to dd or subtrct frctions with unlike denomintors, they cn either use Property 5 of frctions s in Emple 5(), or they cn rewrite the frctions with like denomintors using the lest common denomintor (LCD) of the frctions.

8 7_P.qp /7/6 9:9 AM Pge 9 Section P. Rel Numbers 9 P. Eercises See for worked-out solutions to odd-numbered eercises. Vocbulry Check Fill in the blnks. p. A rel number is if it cn be written s the rtio of two integers, where q. q. numbers hve infinite nonrepeting deciml representtions.. The distnce between point on the rel number line nd the origin is the of the rel number.. Numbers tht cn be written s the product of two or more prime numbers re clled numbers. 5. Integers tht hve ectly two positive fctors, the integer itself nd, re clled numbers. 6. An lgebric epression is combintion of letters clled nd rel numbers clled. 7. The of n lgebric epression re those prts seprted by ddition. 8. The numericl fctor of vrible term is the of the vrible term. 9. The sttes: If b, then or b. In Eercises 6, determine which numbers re () nturl numbers, (b) whole numbers, (c) integers, (d) rtionl numbers, nd (e) irrtionl numbers.. 9, 7, 5,,,,,,. 5, 7, 7,,., 5,, 8,.., ,,....,,,.....,., 5.,, 6,, 7.5,, 6. 5, 7, 5, 9,.,, 6, In Eercises 7, use clcultor to find the deciml form of the rtionl number. If it is nonterminting deciml, write the repeting pttern In Eercises 6, use clcultor to rewrite the rtionl number s the rtio of two integers In Eercises 7 nd 8, pproimte the numbers nd plce the correct inequlity symbol (< or >) between them ,.6,,,, 7, In Eercises 9, plot the two rel numbers on the rel number line. Then plce the correct inequlity symbol (< or >) between them..5, 9., 8.., 7., ,. 8 7, 7 In Eercises 5, () verblly describe the subset of rel numbers represented by the inequlity, (b) sketch the subset on the rel number line, nd (c) stte whether the intervl is bounded or unbounded > 7. < < <. 5. <. < 6 In Eercises 8, use inequlity nd intervl nottion to describe the set.. is negtive.. z is t lest. 5. y is nonnegtive. 6. y is no more thn p is less thn 9 but no less thn. 8. The nnul rte of infltion r is epected to be t lest.5%, but no more thn 5%. In Eercises 9, use intervl nottion to describe the grph

9 7_P.qp /7/6 9:9 AM Pge Chpter P Prerequisites.... In Eercises 5 8, give verbl description of the intervl. 5. 6, 6., 7., 8., In Eercises 9 5, evlute the epression In Eercises 55 6, evlute the epression for the given vlues of nd y. Then use grphing utility to verify your result. y c for nd y 56. y for nd y 57. y for nd y 58. for nd y y 59. for nd y 6. for nd y y y y In Eercises 6 66, plce the correct symbol <, >, or between the pir of rel numbers () 6 In Eercises 67 7, find the distnce between nd b , b , b , b 7. b, , b , b 5.65 In Eercises 7 76, use bsolute vlue nottion to describe the sitution. 7. The distnce between nd 5 is no more thn. 7. The distnce between nd is t lest c y is t lest si units from. 76. y is t most two units from. 77. While trveling on the Pennsylvni Turnpike, you pss milepost 57 ner Pittsburgh, then milepost 6 ner Gettysburg. How mny miles do you trvel between these two mileposts? 78. The temperture in Bismrck, North Dkot ws 6F t noon, then F t midnight. Wht ws the chnge in temperture over the -hour period? Budget Vrince In Eercises 79 8, the ccounting deprtment of compny is checking to determine whether the ctul epenses of deprtment differ from the budgeted epenses by more thn $5 or by more thn 5%. Fill in the missing prts of the tble, nd determine whether the ctul epense psses the budget vrince test. Budgeted Actul Epense, b Epense, b.5b 79. Wges $,7 $,56 8. Utilities $9 $ Tes $7,6 $7,5 8. Insurnce $575 $6 Federl Deficit In Eercises 8 88, use the br grph, which shows the receipts of the federl government (in billions of dollrs) for selected yers from 995 through 5. In ech eercise you re given the ependitures of the federl government. Find the mgnitude of the surplus or deficit for the yer. (Source: U.S. Office of Mngement nd Budget) Receipts (in billions of dollrs) Yer Receipts Ependitures Receipts Ependitures $55.8 billion $6. billion $7.9 billion 86. $86.9 billion 87. $57.6 billion $7. billion

10 7_P.qp /7/6 9:9 AM Pge Section P. Rel Numbers In Eercises 89 9, identify the terms. Then identify the coefficients of the vrible terms of the epression In Eercises 95 98, evlute the epression for ech vlue of. (If not possible, stte the reson.) In Eercises 99 6, identify the rule(s) of lgebr illustrted by the sttement z z In Eercises 7 6, perform the opertion(s). (Write frctionl nswers in simplest form.) In Eercises 7, use clcultor to evlute the epression. (Round your nswer to two deciml plces.) h 6, h 6 y y h 6 5 Epression Vlues () (b) 96. () (b) () (b) 98. () (b) () Use clcultor to complete the tble. (b) Use the result from prt () to mke conjecture bout the vlue of 5n s n pproches.. () Use clcultor to complete the tble. (b) Use the result from prt () to mke conjecture bout the vlue of 5n s n increses without bound. Synthesis n n n,, 5n True or Flse? In Eercises 5 nd 6, determine whether the sttement is true or flse. Justify your nswer. 5. Let > b, then where nd b. > b, b c 6. Becuse then b c c c c b c, b. In Eercises 7 nd 8, use the rel numbers A, B, nd C shown on the number line. Determine the sign of ech epression. C B A 7. () A 8. () C (b) B A (b) A C u v u v. 9. Eplortion Consider nd () Are the vlues of the epressions lwys equl? If not, under wht conditions re they not equl? (b) If the two epressions re not equl for certin vlues of u nd v, is one of the epressions lwys greter thn the other? Eplin.. Think About It Is there difference between sying tht rel number is positive nd sying tht rel number is nonnegtive? Eplin.. Writing Describe the differences mong the sets of whole numbers, nturl numbers, integers, rtionl numbers, nd irrtionl numbers.. Writing Cn it ever be true tht for ny rel number? Eplin.