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270 CHAPTER 6 6.3 The Real Numbers and Their Representations Rational Numbers and Decimal Representation Properties and Operations The set of real numbers is composed of two important mutually exclusive subsets: the rational numbers and the irrational numbers. (Two sets are mutually exclusive if they contain no elements in common.) Recall from Section 6.1 that quotients of integers are called rational numbers. Think of the rational numbers as being made up of all the fractions (quotients of integers with denominator not equal to zero) and all the integers. Any integer can be written as the quotient of two integers. For example, the integer 9 can be written as the quotient 91, or 182, or 273, and so on. Also, 5 can be expressed as a quotient of integers as 51 or 102, and so on. (How can the integer 0 be written as a quotient of integers?) Since both fractions and integers can be written as quotients of integers, the set of rational numbers is defined as follows. Rational Numbers Rational numbers x x is a quotient of two integers, with denominator not 0 A rational number is said to be in lowest terms if the greatest common factor of the numerator (top number) and the denominator (bottom number) is 1. Rational numbers are written in lowest terms by using the fundamental property of rational numbers. Fundamental Property of Rational Numbers If a, b, and k are integers with b 0 and k 0, then ak a . bk b EXAMPLE 1 Write 3654 in lowest terms. Since the greatest common factor of 36 and 54 is 18, 36 2 18 2 . 54 3 18 3 The calculator reduces 3654 to lowest terms, as illustrated in Example 1. 36 2 In the above example, 54 3. If we multiply the numerator of the fraction on the left by the denominator of the fraction on the right, we obtain 36 3 108. If we multiply the denominator of the fraction on the left by the numerator of the fraction on the right, we obtain 54 2 108. The result is the same in both cases. One way of determining whether two fractions are equal is to perform this test. If the product of the “extremes” (36 and 3 in this case) equals the product of the “means” (54 and 2), the fractions are equal. This test for equality of rational numbers is called the cross-product test. An Addison-Wesley product. Copyright © 2004 Pearson Education, Inc. 6.3 Rational Numbers and Decimal Representation 271 Cross-Product Test for Equality of Rational Numbers For rational numbers ab and cd, a c b d Benjamin Banneker (1731 – 1806) spent the first half of his life tending a farm in Maryland. He gained a reputation locally for his mechanical skills and abilities in mathematical problem solving. In 1772 he acquired astronomy books from a neighbor and devoted himself to learning astronomy, observing the skies, and making calculations. In 1789 Banneker joined the team that surveyed what is now the District of Columbia. Banneker published almanacs yearly from 1792 to 1802. He sent a copy of his first almanac to Thomas Jefferson along with an impassioned letter against slavery. Jefferson subsequently championed the cause of this early AfricanAmerican mathematician. b 0, d 0, if and only if a d b c. The operation of addition of rational numbers can be illustrated by the sketches in Figure 10. The rectangle at the top left is divided into three equal portions, with one of the portions in color. The rectangle at the top right is divided into five equal parts, with two of them in color. The total of the areas in color is represented by the sum 2 1 . 3 5 To evaluate this sum, the areas in color must be redrawn in terms of a common unit. Since the least common multiple of 3 and 5 is 15, redraw both rectangles with 15 parts. See Figure 11. In the figure, 11 of the small rectangles are in color, so 1 2 5 6 11 . 3 5 15 15 15 In general, the sum a c b d may be found by writing ab and cd with the common denominator bd, retaining this denominator in the sum, and adding the numerators: c ad bc ad bc a . b d bd bd bd 1_ 3 2_ 5 1_ + 2_ 3 5 FIGURE 10 5 __ 15 6 __ 15 5 + __ 6 = 11 __ __ 15 15 15 FIGURE 11 An Addison-Wesley product. Copyright © 2004 Pearson Education, Inc. 272 CHAPTER 6 The Real Numbers and Their Representations A similar case can be made for the difference between rational numbers. A formal definition of addition and subtraction of rational numbers follows. Adding and Subtracting Rational Numbers If ab and cd are rational numbers, then a c ad bc and b d bd a c ad bc . b d bd This formal definition is seldom used in practice. In practical problems involving addition and subtraction of rational numbers, we usually rewrite the fractions with the least common multiple of their denominators, called the least common denominator. EXAMPLE 2 (a) Add: 2 1 . 15 10 The least common multiple of 15 and 10 is 30. Now write 215 and 110 with denominators of 30, and then add the numerators. Proceed as follows: The results of Example 2 are illustrated in this screen. Since 30 15 2, 2 22 4 , 15 15 2 30 and since 30 10 3, 1 13 3 . 10 10 3 30 2 1 4 3 7 . 15 10 30 30 30 Thus, (b) Subtract: 69 173 . 180 1200 The least common multiple of 180 and 1200 is 3600. 69 3460 207 3460 207 3253 173 180 1200 3600 3600 3600 3600 The product of two rational numbers is defined as follows. Multiplying Rational Numbers If ab and cd are rational numbers, then a c ac . b d bd An Addison-Wesley product. Copyright © 2004 Pearson Education, Inc. 6.3 EXAMPLE To illustrate the results of Example 3, we use parentheses around the fraction factors. 3 Rational Numbers and Decimal Representation 273 Find each of the following products. (a) 37 21 3 7 4 10 4 10 40 (b) 5 3 53 15 1 15 1 18 10 18 10 180 12 15 12 In practice, a multiplication problem such as this is often solved by using slash marks to indicate that common factors have been divided out of the numerator and denominator. 1 1 5 3 1 1 18 10 6 2 6 3 is divided out of the terms 3 and 18; 5 is divided out of 5 and 10. 2 1 12 FOR FURTHER THOUGHT The Influence of Spanish Coinage on Stock Prices Until August 28, 2000, when decimalization of the U.S. stock market began, market prices were reported with fractions having denominators 5 3 with powers of 2, such as 17 and 112 . Did 4 8 you ever wonder why this was done? During the early years of the United States, prior to the minting of its own coinage, the Spanish eight-reales coin, also known as the Spanish milled dollar, circulated freely in the states. Its fractional parts, the four reales, two reales, and one real, were known as pieces of eight, and described as such in pirate and treasure lore. When the New York Stock Exchange was founded in 1792, it chose to use the Spanish milled dollar as its price basis, rather than the decimal base as proposed by Thomas Jefferson that same year. In the September 1997 issue of COINage, Tom Delorey’s article “The End of ‘Pieces of Eight’” gives the following account: by the time the Spanish-American money was withdrawn in 1857, pricing stocks in eighths of a dollar—and no less—was a tradition carved in stone. Being somewhat a conservative organization, the NYSE saw no need to fix what was not broken. All prices on the U.S. stock markets are now reported in decimals. (Source: “Stock price tables go to decimal listings,” The Times Picayune, June 27, 2000.) For Group Discussion Consider this: Have you ever heard this old cheer? “Two bits, four bits, six bits, a dollar. All for the (home team), stand up and holler.” The term two bits refers to 25 cents. Discuss how this cheer is based on the Spanish eight-reales coin. As the Spanish dollar and its fractions continued to be legal tender in America alongside the decimal coins until 1857, there was no urgency to change the system—and An Addison-Wesley product. Copyright © 2004 Pearson Education, Inc. 274 CHAPTER 6 The Real Numbers and Their Representations In a fraction, the fraction bar indicates the operation of division. Recall that, in the previous section, we defined the multiplicative inverse, or reciprocal, of the nonzero number b. The multiplicative inverse of b is 1b. We can now define division using multiplicative inverses. Definition of Division If a and b are real numbers, b 0, then a 1 a . b b Early U.S. cents and half cents used fractions to denote their denominations. The half cent used 1 1 and the cent used . (See 200 100 Exercise 18 for a photo of an interesting error coin.) The coins shown here were part of the collection of Louis E. Eliasberg, Sr., that was auctioned by Bowers and Merena, Inc., several years ago. Louis Eliasberg was the only person ever to assemble a complete collection of United States coins. The half cent pictured sold for $506,000 and the cent sold for $27,500. The cent shown in Exercise 18 went for a mere $2970. You have probably heard the rule, “To divide fractions, invert the divisor and multiply.” But have you ever wondered why this rule works? To illustrate it, suppose that you have 78 of a gallon of milk and you wish to find how many quarts you have. Since a quart is 14 of a gallon, you must ask yourself, “How many 14s are there in 78?” This would be interpreted as 7 1 7 8 or . 8 4 1 4 The fundamental property of rational numbers discussed earlier can be extended to rational number values of a, b, and k. With a 78, b 14, and k 4 (the reciprocal of b 14), 7 7 4 4 a ak 8 8 7 4 . b bk 1 1 8 1 4 4 Now notice that we began with the division problem 78 14 which, through a series of equivalent expressions, led to the multiplication problem 78 41. So dividing by 14 is equivalent to multiplying by its reciprocal, 41. By the definition of multiplication of fractions, 7 4 28 7 , 8 1 8 2 and thus there are 72 or 3 12 quarts in 78 gallon.* We now state the rule for dividing ab by cd. Dividing Rational Numbers If ab and cd are rational numbers, where cd 0, then a c a d ad . b d b c bc 1 *3 2 is a mixed number. Mixed numbers are covered in the exercises for this section. An Addison-Wesley product. Copyright © 2004 Pearson Education, Inc. 6.3 EXAMPLE This screen supports the results in Example 4(b) and (c). 4 Rational Numbers and Decimal Representation 275 Find each of the following quotients. (a) 7 3 15 45 9 5 9 3 5 15 5 7 35 7 5 7 (b) 4 3 4 14 56 8 7 8 8 7 14 7 3 21 37 3 3 1 2 1 2 4 2 1 2 1 (c) 4 9 9 1 9 4 9 4 18 2 There is no integer between two consecutive integers, such as 3 and 4. However, a rational number can always be found between any two distinct rational numbers. For this reason, the set of rational numbers is said to be dense. Density Property of the Rational Numbers If r and t are distinct rational numbers, with r t, then there exists a rational number s such that r s t. To find the arithmetic mean, or average, of n numbers, we add the numbers and then divide the sum by n. For two numbers, the number that lies halfway between them is their average. Year Number (in thousands) 1995 16,360 1996 16,269 1997 16,110 First, find their sum. 1998 16,211 Now divide by 2. 1999 16,477 2000 16,258 Source: U.S. Bureau of Labor Statistics. EXAMPLE 5 (a) Find the rational number halfway between 23 and 56. 2 5 4 5 9 3 3 6 6 6 6 2 3 1 3 3 2 2 2 2 4 The number halfway between 23 and 56 is 34. (b) The table in the margin shows the number of labor union or employee association members, in thousands, for the years 1995–2000. What is the average number, in thousands, for this six-year period? To find this average, divide the sum by 6. 16,360 16,269 16,110 16,211 16,477 16,258 97,685 6 6 The computation in Example 5(b) is shown here. 16,280.8 The average to the nearest whole number of thousands is 16,281. An Addison-Wesley product. Copyright © 2004 Pearson Education, Inc. 276 CHAPTER 6 The Real Numbers and Their Representations Repeated application of the density property implies that between two given rational numbers are infinitely many rational numbers. It is also true that between any two real numbers there is another real number. Thus, we say that the set of real numbers is dense. Ten s On es De cim al p oin t Ten ths Hu ndr edt hs Tho usa ndt hs Ten -th ous and ths Hu ndr edtho usa Mil ndt lion hs ths ... Hu ndr eds Simon Stevin (1548 – 1620) worked as a bookkeeper in Belgium and became an engineer in the Netherlands army. He is usually given credit for the development of decimals. Decimal Form of Rational Numbers Up to now in this section, we have discussed rational numbers in the form of quotients of integers. Rational numbers can also be expressed as decimals. Decimal numerals have place values that are powers of 10. For example, the decimal numeral 483.039475 is read “four hundred eighty-three and thirty-nine thousand, four hundred seventy-five millionths.” The place values are as shown here. 4 8 3 . 3 0 ... 9 4 7 5 Given a rational number in the form ab, it can be expressed as a decimal most easily by entering it into a calculator. For example, to write 38 as a decimal, enter 3, then enter the operation of division, then enter 8. Press the equals key to find the following equivalence. .375 83.000 24 60 56 40 40 0 .3636 . . . 114.00000 . . . 33 70 66 40 33 Of course, this same result may be obtained by long division, as shown in the margin. By this result, the rational number 38 is the same as the decimal .375. A decimal such as .375, which stops, is called a terminating decimal. Other examples of terminating decimals are 1 .25 , 4 70 66 40 . 3 .375 8 . . 7 .7 , and 10 89 .089 . 1000 Not all rational numbers can be represented by terminating decimals. For example, convert 411 into a decimal by dividing 11 into 4 using a calculator. The display shows .3636363636, or perhaps .363636364. However, we see that the long division process, shown in the margin, indicates that we will actually get .3636 . . . , with the digits 36 repeating over and over indefinitely. To indicate this, we write a bar (called a vinculum) over the “block” of digits that repeats. Therefore, we can write While 23 has a repeating decimal representation 23 .6 , the calculator rounds off in the final decimal place displayed. 4 .36 . 11 A decimal such as .36, which continues indefinitely, is called a repeating decimal. Other examples of repeating decimals are 5 .45, 11 1 .3, and 3 5 .83. 6 An Addison-Wesley product. Copyright © 2004 Pearson Education, Inc. 6.3 Although only ten decimal digits are shown, all three fractions have decimals that repeat endlessly. Rational Numbers and Decimal Representation 277 Because of the limitations of the display of a calculator, and because some rational numbers have repeating decimals, it is important to be able to interpret calculator results accordingly when obtaining repeating decimals. While we shall distinguish between terminating and repeating decimals in this book, some mathematicians prefer to consider all rational numbers as repeating decimals. This can be justified by thinking this way: if the division process leads to a remainder of 0, then zeros repeat without end in the decimal form. For example, we can consider the decimal form of 34 as follows. 3 .750 4 By considering the possible remainders that may be obtained when converting a quotient of integers to a decimal, we can draw an important conclusion about the decimal form of rational numbers. If the remainder is never zero, the division will produce a repeating decimal. This happens because each step of the division process must produce a remainder that is less than the divisor. Since the number of different possible remainders is less than the divisor, the remainders must eventually begin to repeat. This makes the digits of the quotient repeat, producing a repeating decimal. To find a baseball player’s batting average, we divide the number of hits by the number of at-bats. A surprising paradox exists concerning averages; it is possible for Player A to have a higher batting average than Player B in each of two successive years, yet for the two-year period, Player B can have a higher total batting average. Look at the chart. Decimal Representation of Rational Numbers Any rational number can be expressed as either a terminating decimal or a repeating decimal. To determine whether the decimal form of a quotient of integers will terminate or repeat, we use the following rule. Criteria for Terminating and Repeating Decimals Year Player A Player B 20 .500 1998 40 60 .300 1999 200 90 .450 200 10 .250 40 A rational number ab in lowest terms results in a terminating decimal if the only prime factor of the denominator is 2 or 5 (or both). A rational number ab in lowest terms results in a repeating decimal if a prime other than 2 or 5 appears in the prime factorization of the denominator. Two 80 100 year .333 .417 240 total 240 In both individual years, Player A had a higher average, but for the two-year period, Player B had the higher average. This is an example of Simpson’s paradox from statistics. The justification of this rule is based on the fact that the prime factors of 10 are 2 and 5, and the decimal system uses ten as its base. EXAMPLE 6 Without actually dividing, determine whether the decimal form of the given rational number terminates or repeats. (a) 7 8 Since 8 factors as 2 3, the decimal form will terminate. No primes other than 2 or 5 divide the denominator. An Addison-Wesley product. Copyright © 2004 Pearson Education, Inc. 278 CHAPTER 6 The Real Numbers and Their Representations (b) (c) 1= .9 9 9 9 9 99 99 Terminating or Repeating? One of the most baffling truths of elementary mathematics is the following: 1 .9999 . . . . Most people believe that .9 has to be less that 1, but this is not the case. The following argument shows otherwise. Let x .9999 . . . . Then 10x 9.9999 . . . x .9999. . . 9x 9 x 1. 6 75 First write the rational number in lowest terms. 6 2 75 25 Since 25 52, the decimal form will terminate. We have seen that a rational number will be represented by either a terminating or a repeating decimal. What about the reverse process? That is, must a terminating decimal or a repeating decimal represent a rational number? The answer is yes. For example, the terminating decimal .6 represents a rational number. 3 6 .6 10 5 The results of Example 7 are supported in this screen. 9 99 13 150 150 2 3 52. Since 3 appears as a prime factor of the denominator, the decimal form will repeat. 9 . .. EXAMPLE (a) .437 7 437 1000 Write each terminating decimal as a quotient of integers. (b) 8.2 8 82 41 2 10 10 5 Repeating decimals cannot be converted into quotients of integers quite so quickly. The steps for making this conversion are given in the next example. (This example uses basic algebra.) EXAMPLE 8 Find a quotient of two integers equal to .85. Step 1: Let x .85, so x .858585 . . . . Step 2: Multiply both sides of the equation x .858585 . . . by 100. (Use 100 since there are two digits in the part that repeats, and 100 10 2.) x .858585 . . . 100x 100.858585 . . . 100x 85.858585 . . . Step 3: Subtract the expressions in Step 1 from the final expressions in Step 2. Subtract. Therefore, 1 .9999 . . . . Similarly, it can be shown that any terminating decimal can be represented as a repeating decimal with an endless string of 9s. For example, .5 .49999 . . . and 2.6 2.59999 . . . . This is a way of justifying that any rational number may be represented as a repeating decimal. See Exercises 95 and 96 for more on .999 . . . 1. 100x 85.858585 . . . x .858585 . . . 99x 85 (Recall that x 1x and 100x x 99x. Step 4: Solve the equation 99x 85 by dividing both sides by 99. 99x 85 99x 85 99 99 85 x 99 85 .85 99 x .85 An Addison-Wesley product. Copyright © 2004 Pearson Education, Inc. 6.3 Rational Numbers and Decimal Representation 279 This result may be checked with a calculator. Remember, however, that the calculator will only show a finite number of decimal places, and may round off in the final decimal place shown. An Addison-Wesley product. Copyright © 2004 Pearson Education, Inc. 1 8 4 .50 .55 .6 20 30 .249 25 100 .666… 5 9 1 4 .5 9 5 1 10 400 4 5 16 48 21 28 15 35 8 48 3 8 9 10 5 7 7 12