Introduction to Real Numbers and Algebraic Expressions Real-World Application 1.1 Introduction to Algebra 1.2 The Real Numbers = — > de , Surface temperatures on Mars vary from —128°C 1.5 Addition of Real Numbers during polar night to 27°C at the equator during midday at the closest point in orbit to the sun. Find the difference between the highest value and the lowest value in this temperature range. 1.4 Subtraction of Real Numbers 1.5 Multiplication of Real Numbers 1.5 Division of Real Numbers .7 Properties of Real Numbers 1.5 Simplifying Expressions, Order of Operations Source: Mars Institute This problem appears as Example 13 in Section 1.4. 1 т L 3 ' sm" o A E a й Eee Objectives Evaluate algebraic expressions by substitution. Translate phrases to algebraic expressions. Mu, is a EEE 4 > LT PEL Ae x a 1, Translate this problem to an equation. Use the graph below. Mountain Peaks. There are 92 mountain peaks in the United States higher than 14,000 ft. The bar graph below shows data for six of these. How much higher is Mt. Fairweather than Mt. Rainer? | Mountain Peaks in the United States E | 20 | 3 19 + MENE ep ue Een AL EAA AA 17-3 ses \ 16-- 15: if. (feet in thousands) Source: US. Department of the Interior, Geological Survey — Answer on page A-1 2 EE ES e A T TR USE F1 TAN DA pas Sr | ня TE 7 Ji Y r ай : ib UN ua <A = = PE - i ; omy aD RAC n= ol iy HAS я Те = x : =! в ГИГ Ë 1 LL CO En cea ssgaERad IER a En En ee AE free a U ey ra ' у y A : e E E ï À ñ 1 ” x + 4 E т я a E 5 - 7 CHAPTER 1: Introduction to Real Numbers and Algebraic Expressions SECTEUR TO ALGEBRA = EE E ES WEES IE Sr de.) ia elena pr a = The study of algebra involves the use of equations to solve problems. Equations are constructed from algebraic expressions. The purpose of this | | section is to introduce you to the types of expressions encountered in SEDA SR Evaluating Algebraic Expressions In arithmetic, you have worked with expressions such as 5 49 + 75, 8 x 6.07, 29 — 14, and —. 6 Tv In algebra, we use certain letters for numbers and work with algebraic expres- | dic: sions such as я | 8 Xx y, 29 — tf, and —. | + 79, Xx + 5 D Sometimes a letter can represent various numbers. In that case, we call the letter a variable. Let a = your age. Then a is a variable since a changes from year to year. Sometimes a letter can stand for just one number. In that case, we call the letter a constant. Let b = your date of birth. Then b is a constant. i Where do algebraic expressions occur? Most often we encounter : them when we are solving applied problems. For example, consider the bar graph shown at left, one that we might find in a book or maga- | M4 zine. Suppose we want to know how much higher Mt. McKinley is than Mt. Evans. Using arithmetic, we might simply subtract. But lets; see how we can find this out using algebra. We translate the problem: into a statement of equality, an equation. It could be done as follows: Height of How much Height of Mt. Evans plus more is Mt McKinley 14,264 + = 20,320. Note that we have an algebraic expression, 14,264 + x, on the left of ef To equals sign. To find the number x, we can subtract 14,264 on both sides oí: the equation: 14,264 + x = 20,320 © whe 14,264 + x — 14,264 = 20,320 — 14,264 5 x = 6056. i EX! This value of x gives the answer, 6056 ft. We call 14,264 + x an algebraic expression and 14,264 + x = 20,3208 an algebraic equation. Note that there is no equals sign, =, in an algebraic’ expression. : In arithmetic, you probably would do this subtraction without ever con- sidering an equation. In algebra, more complex problems are difficult to solve without first writing an equation. ES EX] Do Exercise 1. Do! A Ч ЛЕЙ ig: ~ An algebraic expression consists of variables, constants, numerals, and Fi ration signs. When we replace a variable with a number, we say that we are substituting for the variable. This process is called evaluating the expression. a PS | EXA PLE 1 Evaluate x + y when x = 37 and y = 29. We substitute 37 for x and 29 for y and carry out the addition: roblems. se of this E 1algebra. 3 a X+y=37 + 29 = 66. € number 66 is called the value of the expression. io. AE E y E Algebraic expressions involving multiplication can be written in several Ways. For example, “8 times a” can be written as 8Xa 8a, (8a), orsimply 8a. e Two letters written together without an operation symbol, such as ab, also in- ic expres- | dicate a multiplication. E ÿ EXAI PLE 2 Evaluate 3y when y = 14. or O 3y= 314) = 42 Ч case, we A Do Exercises 2- 4, >sincea just one . our date | a В Е AMPLE 3 Area of a Rectangle. The area A of a rectangle of length / and * dth wis given by the formula A = Jw. Find the area when lis 24.5 in. and he We Бенина 24.5 in. for 7 and 16 in. for w and carry out the pe counter . -onsider | т maga- У er is is. = Jw = (24.5 in) (16 in.) ut let's | me te Nes roblem = (24.5) (16) (in.) (in.) и follows: = 392 in”, or 392 square inches. E Algebraic expressions involving division can also be written in several of the | E Ways. For example, “8 divided by t” can be written as ides of E 8 1 | 8 +1, + 8/1, or 8 ‘where the fraction bar is a division symbol. Pa E e EXAMPLE 4 Evaluate —_ when a = 63 and b = 9. 20.320 E | 3 We substitute 63 for a and 9 for b and carry out the division: . ÿ чо D 39 rcon- ; 0 solve Do Exercises 6 and 7. 2. Evaluate a + b when a = 38 and b = 26, 3. Evaluate x — y when x = 57 and у = 29, 4. Evaluate.4t when t = 15. 5. Find the area of a rectangle when [is 24 ft and w is 8 ft. 6. Evaluate a/b when a = 200 and Ь = 8. 7. Evaluate 10p/q when p = 40 and q = 25. Answers on page A-1 3 1.1 Introduction to Algebra 8. Motorcycle Travel. Find the MN EXAMPLE 6 Motorcycle Travel. pas it takes to travel 660 mi if | Ed takes a trip on his motorcycle. He the speed is 55 mph. | wants to travel 660 mi on a particular | day. The time 7, in hours, that it takes to travel 660 mi is given by 660 [= — r ¥ where r is the speed of Ed's motorcycle. | Find the time of travel if the speed ris {60 mph. We substitute 60 for r and carry out the division: 660 660 { i — = — = llhr. | r 60 hr — — | Do Exercise 8. Answer on page A-1 am CALCULATOR CORNER a, | Doll Ed a e A ai wlll am o o E ой ne ie: nn Ё и “A Гы B Evaluating Algebraic Expressions To the student and the instructor: This book contains a series of optional | discussions on using a calculator. A calculator is not a requirement for this textbook. There are many kinds of | calculators and different instructions for their usage. We have included instructions here for the scientific keys on a graphing calculator such as a T|-84 Plus. Be sure to consult your users manual as well. Also, check with your instructor about whether you are allowed to use a calculator in the course. : Note that there are options above the keys as well as on them. To access the option written on a key, simply | press the key. The options written in blue above the keys are accessed by first pressing the blue E key and then pressing the key corresponding to the desired option. The green options are accessed by first pressing the green CD key. To turn the calculator on, press the @ key at the bottom left-hand corner of the keypad. You should see a blinking rectangle, or cursor, on the screen. If you do not see the cursor, try adjusting the display contrast. To do this, first press @ID and then press and hold (A to increase the contrast or (© to decrease the contrast. To turn the calculator off, press ED Сет). (OFF is the second operation associated with the @ key.) The calculator will turn itself off automatically after about five minutes of no activity. We can evaluate algebraic expressions on a calculator by making the appropriate substitutions, keeping in mind the rules for order of operations, and then carrying out the resulting calculations. To evaluate 12m/n when m = 8 and n= 16, as in Example 5, we enter 12 - 8/16 by pressing OOOO ED - The result is 6. 128/16 Exercises: Evaluate. 12m 1. — when m = 42 and n = 9 4. 27xy, when x = 12.7 and y = 100.4 2. a + b, when a = 8.2 and b = 3.7 5 3a + 2b, when a = 2.9 and b = 5.7 3. b — a, when a = 7.6 and b = 9.4 6. 24 + 3b, when a = 7.3 and b = 5.1 4 CHAPTER 1: Introduction to Real Numbers and Algebraic Expressions 9) Translating to Algebraic Expressions “algebra, we translate problems to equations. The different parts of an equa- Е ion are translations of word phrases to algebraic expressions. It is easier to translate if we know that certain words often translate to certain operation symbols. KEYWORDS, PHRASES, AND CONCEPTS subtract multiply divide subtracted from multiplied by divided by difference product quotient minus times Jus less than of more than decreased by increased by take away В À 4 | EXAMPLE 7 "Translate to an algebraic expression: | Twice (or two times) some number. ma ru AE i. { Think of some number, say, 8. We can write 2 times 8 as 2 x 8, or 2 - 8. We multiplied by 2. Do the same thing using a variable. We can use any vari- ‘able we wish, such as x, y, m, or n. Let's use y to stand for some number. If we та I multiply by 2, we get an expression god, 2X% 2-y, or 2y. | In algebra, 2y is the expression generally used. | > AMPLE 8 Translate to an algebraic expression: i: Thirty-eight percent of some number. * Let n = the number. The word “of” translates to a multiplication symbol, 50 we get the following expressions as a translation: pr 38% - n, his, 0.38 x n, or 0.38n. nd iE , EXAMPLE 9 Translate to an algebraic expression: Seven less than some number. i PTE Ë - We let x represent the number. | w if the number were 23, then 7 less than 23 is 16, that is, (23 — 7), not 1-2 If we knew the number to be 345, then the translation would be le that 7 — x is not a correct translation of the expression in re 9. he expression 7—Xxisa translation of “seven minus some number” o To the student; In the preface, at the front of the text, you will find a Student Organizer card. This pullout card will help you keep track of important dates and useful contact information. You can also use it to plan time for class, study, work, and relaxation. By managing your time wisely, you will provide yourself the best possible opportunity to be successful in this course. 5 1.1 Introduction to Algebra =—— TIT Ea lid e a A | — Translate to an algebraic expression. E EXAMPLE 10 Translate to an algebraic expression: 9. Eight less than some number Eighteen more than a number. We let t = the number. Now if the number were 26, then the translation would be 26 + 18, or 10. Eight more than some number 18 + 26. If we knew the number to be 174, then the translation would be - ar 174 + 18, or 18 + 174. If the number is 7, then the translation is = t+18, or 18 +t. : a соо 3 tak 11. Bout lesan some number EXAMPLE 11 Translate to an algebraic expression: A number divided by 5. : h We let Ë : 3. m = the number. | an 12. Half of a number ‚Now if the number were 76, then the translation would be 76 = 5, or 76/5, or b: | % If the number were 213, then the translation would be 213 = 5, or 213/5, § ; or 23 If the number is m, then the translation is | m ; m=5 m/5 or —. y о | 13. Six more than eight times some Е is Ш EXAMPLE 12 Translate each phrase to an algebraic expression. | ALOEBRAIC EXPRESSION | Five more than some number n+50r5+n | 14. The difference of two numbers | Е aly | Half of a number —t,—,ort/2 5 2 E Five more than three times some number Зр + 5, ог5 + 3p € The difference of two numbers X— Six less than the product of two numbers mn — 6 15. Fifty-nine percent of some Seventy-six percent of some number 76%z, or 0.762 number Four less than twice some number 2% = 4 а \ я = Do Exercises 9-17. 16. Two hundred less than the product of two numbers 17. The sum of two numbers 1 Answers on page A-1 6 CHAPTER 1: Introduction to Real Numbers | and Algebraic Expressions TA o LN AAN TA EE 8, OF id be 5, Or 4 3/5, fl ) EXERCISE SET MathXL ~~ MyMathlab — InterAct — Math Tutor Digital Video Student's For Extra Help ven Center” tor D Y Solutions e | ао: & 4 es o == r Tame Ma Sa a a en oy = Substitute to find values of the expressions in each of the following applied problems. EL Commuting Time. It takes Erin 24 min less time to commute to work than it does George. Suppose that the variable x stands for the time it takes George to get to work. Then x — 24 stands for the time it takes Erin to get to work. How long does it take Erin to get to work if it takes George 56 min? 93 min? 105 min? 3 An ofa Титан The area À of a triangle with base b and height h is given by A = 3 bh. Find the area when b = 45 m (meters) and h = 86 m. RE cs [SES 5, Distance Traveled. A driver who drives at a constant “speed of r mph for t hr will travel a distance d mi given by d = rt mi. How far will a driver travel at a speed of 65 mph for 4 hr? a 7. Hockey Goal. The front of a regulation hockey goal is a rectangle that is 6 ft wide and 4 ft high. Find its area. * Source: National Hockey League 2. Enrolfment Costs. At Emmett Community College, it costs $600 to enroll in the 8 A.M. section of Elementary Algebra. Suppose that the variable n stands for the number of students who enroll. Then 600n stands for the total amount of money collected for this course. How much is collected if 34 students enroll? 78 students? 250 students? 4, Area of a Parallelogram. The area A of a parallelogram with base b and height h is given by A = bh. Find the area of the parallelogram when the height is 15.4 cm (centimeters) and the base is 6.5 cm. I | | ¡Ah | I 4 нений. Y b 6. Simple Interest, The simple interest 7 on a principal of P dollars at interest rate r for time ft, in years, is given by [ = Prt. Find the simple interest on a principal of $4800 at 9% for 2 yr. (Hint: 9% = 0.09.) 8. Zoology A great white shark has triangular teeth. Each tooth measures about 5 cm across the base and has a height of 6 cm. Find the surface area of one side of one tooth. (See Exercise 3.) 1 Exercise Set 1.1 «Le rE Evaluate. À | ; 9. 8x, when x = 7 10. 6y, when y = 7 E e с р + 1-7 whent=24anda=3 TEME PRA 4 Li 3 3 13. when =2andg=6 14. 2, when y = 15 and z = 25 E + + 4 15. 2% when x = 10 and y = 20 16. F7", when p = 2 and q = 16 в «J 3 8 17. &—*, when x = 20 and y = 4 18. =", when m = 16 and n = 6 b Translate each phrase to an algebraic expression. Use any letter for the variable unless directed otherwise. e, 19. Seven more than some number 20. Nine more than some number Ë | | E | | 21. Twelve less than some number 22. Fourteen less than some number ! 23. Some number increased by four 24. Some number increased by thirteen | a | 25. bmore than a 26. ¢ more than d i 27. x divided by y 28. c divided by h | y 29. x plus w 30. s added to ! 31. m subtracted from n 32. p subtracted from q 33. The sum of two numbers | 34. The sum of nine and some number 35. Twice some number 36. Three times some number 37. Three multiplied by some number 38. The product of eight and some number 8 CHAPTER 1: Introduction to Real Numbers and Algebraic Expressions a CITA NL ET EE - = д er A 39. Six more than four times some number 40. Two more than six times some number | 41. Eight less than the product of two numbers 42. The product of two numbers minus seven E В В 43. Five less than twice some number 44, Six less than seven times some number E 8 у ~ 45, Three times some number plus eleven 46. Some number times 8 plus 5 i Е 47. The sum of four times a number plus three times 48. Five times a number minus eight times another number another number =I В. E с „49. The product of 89% and your salary 50. 67% of the women attending == м sl. Your salary after a 5% salary increase if your salary 52. The price of a blouse after a 30% reduction if the price A before the increase was s before the reduction was P Le В 53, Danielle drove at a speed of 65 mph for £ hours. How far 54. Juan has d dollars before spending $29.95 on a DVD of did Danielle travel? the movie Chicago. How much did Juan have after the ыы purchase? A 55. Lisa had $50 before spending x dollars on pizza. How 56. Dino drove his pickup truck at 55 mph for r hours. How “much money remains? far did he travel? a. | a 1 ÿ the student and the instructor: The Discussion and Writing exercises are meant to be answered with one or more sentences. ~ They can be discussed and answered collaboratively by the entire class or by small groups. Because of their open-ended nature, the answers to these exercises do not appear at the back of the book. They are denoted by the symbol Dy. 7. Dw If the length of a rectangle is doubled, does the 58. Day If the height and the base of a triangle are doubled, area double? Why or why not? what happens to the area? Explain. La г F3. = AN Мун pi " SYNTHESIS | J ÈS (a Ti bie student and the instructor: The Synthesis exercises found at the end of most exercise sets challenge students to combine co ncepts or skills studied in that section or in preceding parts of the text. A aluate. TA y EN a—-?b+ X 5 2 9 Fe when a = 20, b = 10,andc =5 60. — — — + —, when x = 30 and y = 6 : i! e = ди! — 61. ‚when b = land ¢ = 12 62. * —% hen w= 5, y=6,andz=1 9 | A Exercise Set 1.1 THE REAL NUMBERS . Objectives State the integer that corresponds to a real-world situation. E Graph rational numbers on a number line. na Convert from fraction notation to decimal notation for a rational number. A set is a collection of objects. For our purposes, we will most often be considering sets of numbers. One way to name a set uses what is called roster notation. For example, roster notation for the set containing the num- bers 0, 2, and 5 is {0, 2, 5}. Sets that are part of other sets are called subsets. In this section, we be- come acquainted with the set of real numbers and its various subsets. Two important subsets of the real numbers are listed below using roster notation. Determine which of two real numbers is greater and indicate which, using < or >; given an inequality like a > b, write another inequality with the same meaning. Determine whether an inequality like — 3 = 5 is true or false. 2 Find the absolute value of a real number. NATURAL NUMBERS The set of natural numbers = (1,2,3,...). These are the numbers used for counting. The set of whole numbers = {0, 1,2, 3,...}. This is the set of natural numbers with 0 included. Study Tips We can represent these sets on a number line. The natural numbers a” numbered exercises. Tutoring is provided free to students who have bought a new text- book with a special access card bound with the book. Tutoring is available by toll- free telephone, toll-free fax, e-mail, and the Internet. White-board technology allows tutors and students to actually see problems worked while they “talk” live in real time during the tutoring sessions. If you purchased a book without this card, you can purchase an access code through your bookstore using ISBN 0-201-72170-8. (This is also discussed in the Preface.) 10 CHAPTER 1: “Introduction to Real Numbers and Algebraic Expressions THE AW MATH TUTOR those to the right of zero. The whole numbers are the natural numbe CENTER | and zero. w-bc.com/tutorcenter | The AW Math Tutor Center is | Natural numbers _ staffed by highly qualified | F | 7 mathematics instructors who | rr ES provide students with tutoring | De EN on text examples and odd- | Whkslé numtiés | EEE Zn A O оао: Г Вос! aa We create a new set, called the integers, by starting with the whole num- bers, 0, 1, 2, 3, and so on. For each natural number 1, 2, 3, and so on, we ob- tain a new number to the left of zero on the number line: For the number 1, there will be an opposite number | (negative 1). For the number 2, there will be an opposite number 2 (negative 2). For the number 3, there will be an opposite number 3 (negative 3), and so on. The integers consist of the whole numbers and these new numbers. The set of integers = {...,—5,—4,-3,-2,-1,0,1,2,3,4,5,...}. We picture the integers on a number line as follows. Integers 0, neither positive nor negative Negative integers | Positive integers той A ften 4 В led 5432101723405 A111 - be- 3 e Opposites ster 2у En call these new numbers to the left of 0 negative integers. The natural E ие numbers are also called positive integers. Zero is neither positive nor nega- iv Ay call —1 and 1 opposites of each other. Similarly, —2 and 2 are oppo- sites, —3 and 3 are opposites, —100 and 100 are opposites, and 0 is its own of Ee Pairs of opposite numbers like —3 and 3 are the same distance from x 0. ‚The integers extend infinitely on the number line to the left and right of zero. a. ik a “EN Integers and the Real World | 3 Integers correspond to many real-world problems and situations. The follow- ing examples will help you get ready to translate problem situations that in- q 3 “volve integers to mathematical language. ых | ] EXAMPLE 1 Tellwhich integer corresponds to this situation: The tempera- | tured is 4 degrees below zero. are com Temperature: +11” | Low: —4° | Humidity: 60% 7 38 fo b- 8 | e The integer —4 corresponds to the situation. The temperature is —4°. _ = № le | EX IPLE 2 “jeopardy” Tell which integer corresponds to this situation: mum A GA ES | ae ‘A contestant missed a $600 question on the television game show “Jeopardy.” PET TYR Missing a $600 question means — 600. À Ш ое a $600 question causes a $600 loss on the score—that is, the con- stant earns —600 dollars. 11 1.2 The Real Numbers a je EA ce UPA = del; Ji EN pedo y E илов слоеный Pr a State the integers that correspond to the given situation. 1. The halfback gained 8 yd on the first down. The quarterback was “sacked for a 5-yd loss on the second down. 2, Temperature High and Low. The highest recorded temperature in Nevada is 125*F on June 29, 1994, in Laughlin. The lowest recorded temperature in Nevada is 50°F below zero on June 8, 1937, in San Jacinto. Sources: National Climatic Data Center, Asheville, NC, and Storm Phillips. STORMFAX, INC. m \ 1 с 3. Stock Decrease. The price of E Wendy's stock decreased from " $41 per share to $38 per share T over a recent time period. Si Source: The New York Stock Exchange IT p! о! m is wi be 4. At 10 sec before liftoff, ignition occurs. At 156 sec after liftoff, the first stage is detached from the rocket. 5. A submarine dove 120 ft, rose 50 ft, and then dove 80 ft. US EXAMPLE 3 «инте НН Не Hlepation, Tell which integer corresponds to this situation: The shores of California's largest lake, the Salton Sea, are 227 ft below sea level. Source: National Geographic, February 2005, p. 88. Salton Sea, by Joel K. Bourne, Jr, Senior Writer. | | | | | | | | | 1 The integer —227 corresponds to the situation. The elevation is —227% КК p Chongpe, situation: The price of Pearson Education stock decreased from $27 per shar to $11 per share over a recent time period. The price of Safeway stock in creased from $20 per share to $22 per share over à recent time period. Source: The New York Stock Exchange | The integer —16 corresponds to the decrease in the stock value. The in ger 2 represents the increase in stock value. & EXAMPLE 4 > 7 Tell which integers correspond to | | | | | i | = ется Do Exercises 1 —5. Db! The Rational Numbers We created the set of Integers by obtaining a negative number for each nati ral number and also including 0. To create a larger number system, called set of rational numbers, we consider quotients of integers with nonzero di sors. The following are some examples of rational numbers: 2 2 7 23 1 TB ET 1 4, — 0, o 2.4, —0.17, 10-—: 3 3 1 > 2 The number —3 (read “negative two-thirds”) can also be named = or : that is, бей. b b —b The number 2.4 can be named 1; or 2 and —0.17 can be named — 405 - We el describe the set of rational numbers as follows. RATIONAL NUMBERS a The set of rational numbers = the set of numbers 5 ot equal to 0 (b # 0). where a and b are integers and bisn ituation: ea level r Writer, Note that this new set of numbers, the rational numbers, contains the wh hole numbers, the integers, the arithmetic numbers (also called the non- Tegative rational numbers), and the negative rational numbers. We picture the rational numbers on a number line as follows. Negative rational 0 Positive rational numbers numbers dl — pt ОЕ ИВ Яна! ай —— To graph a number means to find and mark its point on the number line. Si Some rational numbers are graphed in the preceding figure. Exa PLE 5 Graph: 3 ES The number 2 can be named 23, or 2.5. Its graph is halfway between 2 and 3. Y PLE 6 Graph: —3.2. р | The graph of —3.2 is 5 of the way from —3 to —4. | EXAMPLE 7 Graph: 3. : The number # can be named 1 5 Or 1.625. The graph is 2 of the way from lito 2. Be... 8. Notation for Rational Numbers Each rational number can be named using fraction or decimal notation. PLE 8 Convert to decimal notation: — 2. о 0.6 2 5 Graph on a number line, 7 6. — > = == Kan Wr A CALCULATOR Ca CORNER Negative Numbers on a Calculator; Converting to Decimal Notation We use the opposite key D to enter negative numbers on a graphing calculator. Note that this is different from the C=) key, which is used for the operation of subtraction. To convert —3 to decimal notation, as in Example 8, we press (=) (3) + 3) ED. The result is —0.625. — 5/8 = 625 Exercises: Convert each of the following negative numbers to decimal notation. 3 | 9 | A 4 20 1 | 9 a, жабы A ce 8 5 57 11 > 20 9 Е 7 1 g 2 25 13 1.2 The Real Numbers Convert to decimal notation. A CALCULATOR EN CORNER Approximating Square Roots and 77 Square roots are found by pressing ED CG. (V is the second operation associated with the ED key To find an approximation for V48, we press ED 7) (5) 6) OP. The approxima- tion 6.92820323 is displayed. To find 8 - V13, we press LW OOO TWD. The approximation 28.8444102 is displayed. The number 7 is used widely enough to have its own key. [is the second operation associated with the E) key.) To approximate ar, we press ED 7) CUE. The approximation 3.141592654 is displayed. Exercises: Approximate. 1. V76 2. V317 3. 15 - V20 4. 29 + \/42 5 7 6. 29-7 7. т. 13° В. 5-п +: М237 14 peating part—in this case, 0.63. Thus, 1; E EXAMPLE 9 Convert to decimal notation: Fi PE VE SR EN DEE EE - CHAPTER 1: Introduction to Real Numbers and Algebraic Expressions Decimal notation for — 2 is —0.625. We consider —0.625 to be a termina ing decimal. Decimal notation for some numbers repeats. | 11 0.6 3 6 3.. 1)7.0000 6 6 Dividing 1 1 | | \ e a foals hh 1 © an © Zo id со © E 5 i We can abbreviate repeating decimal notation by writing a bar over the re = 0.63. [9 EA fa AI I Each rational number can be expressed in either terminating or repeating decimal notation. The following are other examples to show how each rational number can be named using fraction or decimal notation: 27 3 13 = = 097 —g = —8.75, ma == 0. 87 8 5 —2.16. 100 0 0=—, 8 Do Exercises 9-11. [o The Real Numbers and Order Every rational number has a point on the number line. However, there art some points on the line for which there is no rational number. These poin correspond to what are called irrational numbers. What kinds of numbers are irrational? One example is the number 7 which is used in finding the area and the circumference of a circle: A = m” and C = Zar. Another example of an irrational number is the square L root of 2, named V2. It is the length of the diagonal of a square with sides of length 1. It is also the number that when multiplied by itself gives 2— that is, V2 - VI «22 There is no rational number that can be multiplied by itself to get 2. But the following are rational approximations: : 1.4 is an approximation of V2 because (1.4)* = 1.96; = 1.9881; 1.4142 is an even better approximation because (1.4142)° = 1.99996164. 1.41 is a better approximation because (1.41)? We can find rational approximations for square roots using a calculator. Dm p—r on pp pm pa = о ee e de Чон erminat- TA 2 § De nal notation for rational numbers either terminates or repeats. E Decimal notation for irrational numbers neither terminates nor “Some other examples of irrational numbers are V3, —V8, VII, and 0 121221222122221.... Whenever we take the square root of a number that is not a perfect square, we will get an irrational number. E The rational numbers and the irrational numbers together correspond to E all the points on a number line and make up what is called the real-number “system. Irrational numbers -2.5 “= | | 0 E | - -4 -3 > —1 || | Rational numbers le +-—> Real numbers 3 — ®t — @ | no the re- REAL NUMBERS The set of real numbers = The set of all numbers corresponding to points on the number line. 4 $: The real numbers consist of the rational numbers and the irrational num- 1 bers. The following figure shows the relationships among various kinds of = numbers. A. т E еее + i Positive e | | DZ $ 2r can Ш — Integers Negative integers: — Rational numbers -— —L Anos IA | Rational numbers that are not 2 = > are mE 3 7 ei + ; A 0.47,... unts eu = | E 3 a: y Irrational numbers: V2, 7, —V3, V13, 5.363663666.. . , . .. A ORDER q ] | №! numbers are named in order on the number line, with larger numbers 1 58 named farther to the right. For any two numbers on the line, the one to the left a pr is less than the one to the right. a | В = EE | | } | 1 | 1 | | | I 1 | | | 1 | | = IN | | 1 | | 1 1 | 1 Ï 1 | | fi 1 ЕР В -9 4 7-6-5-4 3-2-1 0 1 2 3 4 5 6 7 8 9 i Lo conor] 64. à o We use the symbol < to mean “is less than.” The sentence —8 < 6 means 5 io is less than 6.” The symbol > means “is greater than.” The sentence a 3 > —7 means “—3 is greater than — 7.” The sentences —8 < 6 and —3 > —7 are inequalities. Use either < or > for | | to writea true sentence. 12. —3 L] 7 13. —8 [| —5 14. 7 [_] —10 15. 3.1 [_] —9.5 16, — = [] -1 3 520-3 19. —4.78 [] -5.01 Answers on page A-1 15 1.2 The Real Numbers

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