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ben35651_ch06_301-402.indd Page 302 8/13/14 9:21 AM f-w-166 302 Math Activity /201/MH02188/ben35651_disk1of1/0078035651/ben35651_pagefiles 6.1 MATH ACTIVITY 6.1 Decimal Place Value with Base-Ten Pieces and Decimal Squares Virtual Manipulatives Purpose: Explore decimal place value concepts using Decimal Squares and base-ten pieces. Materials: Base-Ten Pieces and Decimal Squares in the Manipulative Kit or Virtual Manipulatives. www.mhhe.com/bbne *1. When the largest base-ten piece in your kit represents the unit, the other base-ten pieces take on the values shown here. Notice that the hundredths piece is divided into 10 equal parts to represent thousandths, and 1 part is shaded to represent 1 thousandth. 1 .1 .01 .001 a. Look at the four different base-ten pieces and describe five mathematical relationships between pairs of pieces. b. Form the collection of 1 unit piece, 4 tenths pieces, and 12 hundredths pieces. By using only your base-ten pieces and exchanging (trading) the pieces, it is possible to represent this collection in many different ways. Record some of these in a place value table like the one shown here. Units Tenths Hundredths 1 4 12 Thousandths 2. In the Decimal Squares model the unit square is divided into 10, 100, and 1000 equal parts to represent tenths, hundredths, and thousandths (respectively). Sort your deck of Decimal Squares into three piles according to color. a. Determine the smallest and largest decimal represented in each pile. b. How do the shaded amounts of each type of Decimal Square increase? c. List some relationships between the three types of Decimal Squares. .6 3. The two Decimal Squares shown at the left illustrate .6 5 .60 because both squares have the same amount of shading. In the deck of Decimal Squares there are three squares whose decimals equal .6. Sort your deck of Decimal Squares into piles so squares with the same shaded amount are in the same pile. a. Find all the decimals from the Decimal Squares that equal the following: .5, .35, .9, and .10 and write each corresponding equality statement. b. The two-place decimal .65 is not equal to a one-place decimal, such as .6 or .7. List all the other two-place decimals from your deck that are not equal to a one-place decimal. .60 c. The three-place decimal .375 is not equal to a two-place decimal. List all the other three-place decimals from your deck that are not equal to a two-place decimal. ben35651_ch06_301-402.indd Page 303 8/13/14 9:21 AM f-w-166 /201/MH02188/ben35651_disk1of1/0078035651/ben35651_pagefiles 303 Section 6.1 Decimals and Rational Numbers SECTION 6.1 DECIMALS AND RATIONAL NUMBERS Circular patterns of atoms in an iridium crystal, magnified more than 1 million times by a field ion microscope. PROBLEM OPENER A carpenter agrees that during a specified 30-hour period he be paid $15.50 every hour he works and that he pay $16.60 every hour he does not work. At the end of 30 hours, he finds he has earned $47.70. How many hours did he work?* Each dot in the remarkable photograph above is an atom in an iridium crystal. The circular patterns show the order and symmetry governing atomic structures. The diameters of atoms, and even the diameters of electrons contained in atoms, can be measured by decimals. Each atom in this picture has a diameter of .000000027 centimeter, and the diameter of an electron is .00000000000056354 centimeter. The use of decimals is not restricted to describing small objects. The U.S. gross national product (GNP) and the U.S. national income (NI) for selected 5-year periods are expressed to the nearest tenth of a billion dollars in Figure 6.1.† Figure 6.1 1985 1990 1995 2000 2005 2010 GNP (billions) $4244.0 $5835.0 $7444.3 $9989.2 $12,735.5 $14,848.7 NI (billions) $3696.3 $5059.5 $6522.3 $8938.9 $11,273.8 $12,828.2 In our daily lives we encounter decimals in representations of dollar amounts: $17.35, $12.09, $24.00, etc. In elementary school, pennies, dimes, and dollars are commonly used for teaching decimals. DECIMAL TERMINOLOGY AND NOTATION The word decimal comes from the Latin decem, meaning ten. Technically, any number written in base-ten positional numeration can be called a decimal. However, decimal most often refers only to numbers such as 17.38 and .45, which are expressed with *“Problems of the Month,” Mathematics Teacher. †Adapted from Statistical Abstract of the United States, 131st ed. (Washington, DC: Bureau of the Census, 2012). ben35651_ch06_301-402.indd Page 304 9/6/14 1:45 PM f-w-166 304 Chapter 6 /201/MH02188/ben35651_disk1of1/0078035651/ben35651_pagefiles Decimals: Rational and Irrational Numbers decimal points. A combination of a whole number and a decimal, such as 17.38, is also called a mixed decimal. There are currently many variations in decimal notation. In England, the decimal point is placed higher above the line than in the United States. In other European countries, a comma is used in place of a decimal point. A comma and a raised numeral denote a decimal in Scandinavian countries. United States 82.17 England 82?17 Europe 82,17 Scandinavian countries 82,17 The number of digits to the right of the decimal point is called the number of decimal places. There are two decimal places in 7.08 and one decimal place in 104.5. The positions of the digits to the left of the decimal point represent place values that are increasing powers of 10 (1, 10, 102, 103, . . . ). The positions to the right of the decimal point represent place values that are decreasing powers of 10 (1021, 1022, 1023, . . . ), or reciprocals of powers of 10 ( 101 , 101 2 , 101 3, . . . ) . In the decimal 5473.286 (Figure 6.2), the 2 represents 102 , the 8 6 8 represents 100 , and the 6 represents 1000 . Notice the similarity in pairs of names to the right and left of the units digit, for example, tens and tenths, hundreds and hundredths, etc. The convention in the following Historical Highlight of placing a small zero under the units digit helped to focus attention on these pairs of names. 5t ho u 4 h sand un s dre 7t en ds s 3u nit s 2t en ths 8h un dre 6t ho dths us an dth s 5 4 7 3 .2 8 6 Figure 6.2 HISTORICAL HIGHLIGHT The person most responsible for our use of decimals is Simon Stevin, a Dutchman. In 1585 Stevin wrote La Disme, the first book on the use of decimals. He not only stated the rules for computing with decimals but also pointed out their practical applications. Stevin showed that business calculations with decimals can be performed as easily as those involving only whole numbers. He recommended that the government adopt the decimal system and enforce its use. As decimals gained acceptance in the sixteenth and seventeenth centuries, a variety of notations were used. Many writers used a vertical bar in place of a decimal point. Here are eight examples of how 27.847 was written during this period. 27 u 847 27847 . . . E X A M P LE A 3 s 27(847) 27,8i4ii7iii 27 u 847 27847 o 27 847 27 s 0 8 s 1 4s 2 7 3 s Express the value of the digit marked by the arrow as a fraction whose denominator is a power of 10. 1. 47.35 Solution 1. 2. 6.089 5 100 2. 3. 14.07 9 1000 3. 0 10 ben35651_ch06_301-402.indd Page 305 8/13/14 9:21 AM f-w-166 /201/MH02188/ben35651_disk1of1/0078035651/ben35651_pagefiles Section 6.1 Decimals and Rational Numbers 305 Like whole numbers, decimals can be written in expanded form to show the powers of 10 (Figure 6.3). 473.2865 Figure 6.3 Content Standards 5.NBT.3 Read and write decimals to thousandths using base-ten numerals, number names, and expanded form. 1 1 1 1 4(102) 7(10) 3(1) 2 10 8 100 6 1000 5 10,000 Reading and Writing Decimals The digits to the left of the decimal point are read as a whole number, and the decimal point is read and. The digits to the right of the point are also read as a whole number, after which we say the name of the place value of the last digit. For example, 1208.0925 is read “one thousand two hundred eight and nine hundred twenty-five ten-thousandths.” Common Core State Standards Mathematics ⏐ ⏐ ⏐ ⏐ ↓ One thousand two hundred eight E XA M P LE B ⎫ ⎬ ⎭ ⎫ ⎬ ⎭ 1208 . 0925 ⏐ ⏐ ⏐ ⏐ ↓ and ⏐ ⏐ ⏐ ⏐ ↓ Nine hundred twenty-five ten-thousandths Write the name of each decimal. 1. 3.472 2. 16.14 3. .3775 Solution 1. Three and four hundred seventy-two thousandths. 2. Sixteen and fourteen hundredths. 3. Three thousand seven hundred seventy-five ten-thousandths. TRY IT! 6.1.1 1. Write the decimal .2865 in expanded form. 2. Add the fractions in the expanded form of .2865. 3. Write the name of the final fraction form of .2865 in words. One place where you may see the names of numbers is on bank checks. When writing an amount of money, some people write the decimal part of a dollar in words. Notice that on the bank check in Figure 6.4 on the next page it is unnecessary to write dollars or cents. The amount is in terms of dollars, and this unit is printed at the end of the line on which the amount of money is written. Some people write the decimal part of a dollar as a fraction. For example, the amount of this check might have been written “one 24 hundred seventy-seven and 100 .” ben35651_ch06_301-402.indd Page 306 8/13/14 9:21 AM f-w-166 306 Chapter 6 /201/MH02188/ben35651_disk1of1/0078035651/ben35651_pagefiles Decimals: Rational and Irrational Numbers 54-7001/2114 20 $ PAY TO THE ORDER OF DOLLARS FEDERAL SAVINGS BANK MEMO Figure 6.4 MODELS FOR DECIMALS Models are important for providing conceptual understanding and insight into the use of decimals. Often, there is a rush to begin decimal computation, before devoting time to examine decimals using concrete models that can help children understand decimals both for their practical applications, and as their part of the rational number system. There are several models for decimals but we will focus on the Decimal Square model because of its easy connection to fractions and percent. Decimal Squares The Decimal Squares model illustrates the part-to-whole concept of decimals and place value. Unit squares are divided into 10, 100, and 1000 equal parts (Figure 6.5), and the decimal tells what part of the square is shaded.* Figure 6.5 Tenths square Hundredths square Thousandths square .3 .35 .375 Each decimal in Figure 6.5 can be obtained by beginning with the fraction for the shaded amount of the square and obtaining the expanded form of the decimal. For example, the 375 . fraction for the square representing 375 parts out of 1000 is 1000 375 300 70 5 3 7 5 5 1 1 5 1 1 5 .375 1000 1000 1000 1000 10 100 1000 Place value table Similarly, the decimal for Tenths Hundredths Thousandths 3 7 5 3 35 is .3, and the decimal for is .35. 10 100 *Decimal Squares is a registered trademark of American Education Products, LLC. ben35651_ch06_301-402.indd Page 307 9/5/14 11:33 AM f-w-166 /201/MH02188/ben35651_disk1of1/0078035651/ben35651_pagefiles 307 Section 6.1 Decimals and Rational Numbers You can use models to repr esent decimal s. 1.0 one 0.1 one tenth 0.01 one hundredt h Try It There are 0 w hole grids shaded. Hundredths Ones Tenths Complete th e place-value chart that re the fraction presents of the grid th at is shaded at the right. There are squares shad ed out of a to tal of squares. In words, this is forty-two hu ndredths. This is the sa me as 4 tenths and 2 hundre dths. So, write 4 te nths and 2 hu nd re dt place-value ch hs in the art. 4 tenths 2 hundredths 4 tenths and 2 is 42 hundredt hundredths hs. Talk About It 2 Use Number Sen Marc has 6 pe se Paulo has nnies. How m 6 dimes. any times grea of 6 dimes th ter is the valu an 6 pennies? e Explain. 632Chap From My Math,ter 10 Fractions and Deci mals Grade 4, by M cGraw-Hill Ed of McGraw-H ucation. Copyrig ill Education. ht ©2013 by M cGraw-Hill 631_634_C10_ L01_116195.in dd 632 Copyright © The McGraw-Hill Com panies, Inc. 1. Education. Re printed by perm ission 10/14/11 10: 53 AM ben35651_ch06_301-402.indd Page 308 9/19/14 7:08 PM f-w-166 308 Chapter 6 E X A M P LE C /201/MH02188/ben35651_disk1of1/0078035651/ben35651_pagefiles Decimals: Rational and Irrational Numbers Describe the Decimal Square that would represent each fraction, and write the decimal for each fraction. 1. 4728 10,000 2. 6 100 Solution 1. A square with 4728 parts shaded out of 10,000 4000 700 20 8 4728 5 1 1 1 10,000 10,000 10,000 10,000 10,000 4 7 2 8 5 1 1 1 10 100 1000 10,000 5 .4728 2. A square with 6 parts shaded out of 100 6 0 6 5 1 5 .06 100 10 100 TRY IT! 6.1.2 Write the decimal numeral that matches each description. 1. A Decimal Square with 123 parts shaded out of 1000. 2. A Decimal Square with 101 parts shaded out of 10,000. Calculator Connection Example C shows that it is easy to obtain the decimal for a fraction whose denominator is a power of 10; it is just a matter of locating the decimal point. Try the example on your calculator by dividing 4728 by 10,000 and 6 by 100. It is also instructive to enter 4728 into a calculator and then repeatedly divide by 10. Each time the decimal point moves one digit to the left. Keystrokes 4728 View Screen ÷ 10 = 472.8 ÷ 10 = 47.28 ÷ 10 = 4.728 ÷ 10 = .4728 In general, to divide an integer by a power of 10, begin with the units digit and, for each factor of 10, count off a digit to the left to locate the decimal point. Number Line The number line is a common model for illustrating decimals. One method of marking off a unit from 0 to 1 is to use the edge of a Decimal Square, as shown in Figure 6.6 on the next page. This approach shows the relationship between a region model for a unit (the Decimal Square) and a linear model for a unit (the edge of a square). The Decimal Square can be used repeatedly to mark off tenths on the number line from 0 to 1, 1 to 2, etc. ben35651_ch06_301-402.indd Page 309 8/13/14 9:21 AM f-w-166 /201/MH02188/ben35651_disk1of1/0078035651/ben35651_pagefiles 309 Section 6.1 Decimals and Rational Numbers Figure 6.6 0 .3 1 2 Consider locating the point for .372 on a number line. One approach is to use the expanded form of the decimal .372 5 Content Standards 4.NF.6 Use decimal notation for fractions with denominators 10 or 100. For example, rewrite 0.62 as 62y100; describe a length as 0.62 meters; locate 0.62 on a number line diagram. 3 7 2 1 1 10 100 1000 and locate the point in several steps, as shown in Figure 6.7. First, the point for 103 (.3) is located at the end of the third interval, as in Figure 6.6. Second, the expanded form 7 shows that we must add 100 , so the interval from .3 to .4 is divided into 10 equal parts, 7 which are hundredths. To add 100 (.07), we begin at .3 and go to the end of the seventh interval. This is the point for .37. Finally, the interval from .37 to .38 is divided into 10 2 equal parts, which are thousandths. To add 1000 , we begin at .37 and go to the end of the second interval. This is the point for .372. Common Core State Standards Mathematics 0 .1 .2 .3 .4 .5 .6 .37 .7 .8 .9 .38 .4 .3 .38 .37 .372 Figure 6.7 E XA M P LE D Sketch a number line and mark the approximate location of each decimal. 1. .46 2. 1.75 3. 2.271 Solution .46 0 1.75 1 2.271 2 1 ben35651_ch06_301-402.indd Page 310 9/19/14 7:08 PM f-w-166 310 Chapter 6 /201/MH02188/ben35651_disk1of1/0078035651/ben35651_pagefiles Decimals: Rational and Irrational Numbers TRY IT! 6.1.3 Using the given points for scale, precisely mark the points for 0, 2, and 1.25 on the number line. 1.75 2.25 Decimals can represent negative as well as positive numbers. For every decimal, whether positive or negative, there is a corresponding decimal called its opposite (or inverse for addition) such that the sum of the two decimals is zero. Several decimals and their opposites are shown on the number line in Figure 6.8. - 1.2 and 1.2 are opposites - .6 and .6 are opposites Figure 6.8 - 1.4 - 1.2 - 1 - .8 - .6 - .4 - .2 0 .2 .4 .6 .8 1 1.2 1.4 EQUALITY OF DECIMALS Equality of decimals can be illustrated visually by comparing the shaded amounts in their Decimal Squares. Figure 6.9 shows that 4 parts out of 10, 40 parts out of 100, and 400 parts out of 1000 are all represented by the same amount of shading—in each Decimal Square, four columns are shaded. This illustrates that .4 5 .40 5 .400 Figure 6.9 E X A M P LE E .4 .40 .400 Complete each equation by writing the indicated decimal, and describe the Decimal Square representing each decimal in the equation. 1. .35 5 _______ (thousandths) 3. .600 5 _______ (tenths) 2. .670 5 _______ (hundredths) Solution 1. .35 5 .350 (35 parts out of 100 and 350 parts out of 1000 are shaded). 2. .670 5 .67 (670 parts out of 1000 and 67 parts out of 100 are shaded). 3. .600 5 .6 (600 parts out of 1000 and 6 parts out of 10 are shaded). ben35651_ch06_301-402.indd Page 311 8/13/14 9:21 AM f-w-166 /201/MH02188/ben35651_disk1of1/0078035651/ben35651_pagefiles 311 Section 6.1 Decimals and Rational Numbers Decimal Squares also give us a visual model for place value. Consider the Decimal Square for .475 in Figure 6.10. The 4 full columns that are shaded (400 thousandths) repre400 sent 104 or .4 ( 1000 5 104 ) ; the 7 small squares that are shaded (70 thousandths) 7 70 7 re present 100 or .07 ( 1000 5 100 ) ; and the 5 small parts that are shaded (5 thousandths) 5 represent 1000 or .005. Thus, the decimal .475 can be thought of as 4 tenths, 7 hundredths, and 5 thousandths. .475 5 .4 1 .07 1 .005 5 1000 7 70 = 1000 100 Figure 6.10 4 400 = 1000 10 INEQUALITY OF DECIMALS Content Standards 4.NF.7 Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols ., 5, or ,, and justify the conclusions, e.g., by using a visual model. Research indicates that students from elementary school through college often have difficulty determining inequalities for decimals. One source of confusion is to think of the digits in the decimal as representing whole numbers (see Example F on the following page). Figure 6.11 shows that .47 , .6. Even though 47 is greater than 6, a smaller amount of the square is shaded for .47 than for .6. Common Core State Standards Mathematics Figure 6.11 .47 < .6 We can also see that .47 , .6 by noting that in the Decimal Square for .47, 4 full columns and part of another are shaded, whereas in the Decimal Square for .6, 6 full columns are shaded. In other words, the digit in the tenths place for .47 is less than the digit in the tenths place for .6. In general, the following place value test determines inequalities for decimals. Place Value Test for Inequality of Decimals The greater of two positive decimals that are both less than 1 will be the decimal with the greater digit in the tenths place. If these digits are equal, this test is applied to the hundredths digits, etc. ben35651_ch06_301-402.indd Page 312 8/13/14 9:21 AM f-w-166 312 Chapter 6 /201/MH02188/ben35651_disk1of1/0078035651/ben35651_pagefiles Decimals: Rational and Irrational Numbers The question in the next example is from a test given as part of a nationwide testing program in schools every 4 years.* Over one-half of the 13-year-olds who took the test selected an incorrect answer. E X A MP LE F Which number is the greatest? .19 .036 .195 .2 Solution One approach is to use the place value test for inequality of decimals. Since 2 is the greatest of the digits in the tenths place of these four decimals, .2 is the greatest number. Another approach is to change each decimal to thousandths. This will show that 200 thousandths is the greatest number of thousandths among these four decimals. .190 .036 .195 .200 A visual approach with Decimal Squares illustrates that 2 full columns of shading (or 2 parts shaded out of 10) is more than 19 parts shaded out of 100 or 195 parts shaded out of 1000. .2 .19 .195 Note: Of the 13-year-olds tested, 47 percent selected .195 for the answer. This error may be due to the fact that 195 is greater than 19, 36, or 2. RATIONAL NUMBERS Up to this point, numbers written in the form ab, where a and b are integers with b ? 0, have been called fractions. This is the terminology commonly used in elementary and middle schools. However, the word fraction has a more general meaning and includes the quotient of any two numbers, integers or not, as long as the denominator is not zero. Specifically, fractions ab, where a and b are integers and b ? 0, are called rational numbers. Rational Number Any number that can be written in the form ab , where b ? 0 and a and b are integers, is called a rational number. 2 2 For example, 19 , 73 , 15 , 107 , and 31 are rational numbers. When the denominator of a ratio2 nal number equals 1, the rational number equals an integer: 61 5 6, 41 5 2 4, 121 5 12, etc. Therefore, integers are also rational numbers. *T. P. Carpenter, M. K. Corbitt, H. S. Kepner, M. M. Lindquist, and R. E. Reys, Results from the Second Mathematics Assessment of the National Assessment of Educational Progress. ben35651_ch06_301-402.indd Page 313 8/13/14 9:21 AM f-w-166 /201/MH02188/ben35651_disk1of1/0078035651/ben35651_pagefiles Section 6.1 Decimals and Rational Numbers 313 Rational Numbers as Terminating or Infinite Repeating Decimals Rational numbers can be expressed by many different number symbols or numerals. For example, 3 3 10 is a rational number and 10 5 .3, so .3 is also a rational number. In the following paragraphs we will show that all rational numbers ab can be written as decimals. We have seen that it is easy to convert a fraction to a decimal if the denominator is a power of 10: 64 7283 54 5 .64 5 7.283 5 .0054 100 1000 10,000 Sometimes when the denominator is not a power of 10, the fraction can be replaced by an 25 equal fraction whose denominator is a power of 10. For example, 14 can be replaced by 100 because 100 is a multiple of 4: 1 25 3 1 25 5 5 5 .25 4 25 3 4 100 Since 10 5 2 3 5, we know by the Fundamental Theorem of Arithmetic that any power of 10 will have only 2 and 5 as prime factors. 100 5 102 5 22 3 52 10 5 2 3 5 1000 5 103 5 23 3 53 etc. Consider replacing 38 by a fraction whose denominator is a power of 10. Since 8 5 23, we need to multiply the numerator and denominator of 38 by 53. 3 3 3 3 53 375 5 35 3 5 3 5 .375 3 8 2 2 35 10 E XA M P LE G Convert each fraction to a decimal by first writing a fraction whose denominator is a power of 10. 1. 7 20 2. 3 25 3. 11 16 4. 3 40 7 35 3 12 11 11 11 3 54 6875 5 5 .35 2. 5 5 .12 3. 5 4 5 4 5 5 .6875 20 100 25 100 16 2 2 3 54 104 3 75 3 3 3 52 4. 5 3 5 .075 5 3 5 3 40 2 35 2 3 53 10 Solution 1. In the examples on the preceding pages, all the decimals had a finite number of digits. Such decimals are called terminating (or finite) decimals. However, there are decimals that are not terminating. There is no power of 10 that has 3 as a factor, so 13 cannot be written as a fraction whose denominator is a power of 10. In general, we have the following rule. Terminating Decimal If a nonzero rational number ab is in simplest form, it can be written as a terminating decimal if and only if b has only 2s and/or 5s in its prime factorization. ben35651_ch06_301-402.indd Page 314 8/13/14 9:21 AM f-w-166 314 Chapter 6 E X A M P LE H /201/MH02188/ben35651_disk1of1/0078035651/ben35651_pagefiles Decimals: Rational and Irrational Numbers Which of these rational numbers can be written as terminating decimals? 1. 5 6 2. 1 80 3. 9 15 4. 3 14 5 cannot be written as a terminating decimal. 2. 80 has only 6 9 3 1 can be written as a terminating decimal. 3. 5 , and since the denomfactors of 2 and 5, so 80 15 5 9 can be written as a terminating decimal. inator of the fraction in simplest form has only 5 as a factor, 15 3 cannot be written as a terminating decimal. 4. 14 has a factor of 7, so 14 Solution 1. 6 has a factor of 3, so Let’s consider finding a decimal for 13 . We know from the fraction-quotient concept in Section 5.2 that 13 5 1 4 3. Figure 6.12 illustrates the first few steps in dividing 1 by 3. Part a shows a unit square with 10 tenths and illustrates that 1 4 3 is .3 with .1 remaining. In part b, the remaining .1 is replaced by 10 hundredths, and dividing by 3 produces .03 with .01 remaining. In part c, the remaining 1 hundredth is replaced by 10 thousandths, and dividing by 3 produces .003 with .001 remaining. The three steps of the division process in Figure 6.12 produce .3, .03, and .003, which give a total shaded amount of .333 with .001 remaining. Dividing 10 tenths by 3 .3 Dividing 10 hundredths by 3 .03 1 tenth = 10 hundredths Dividing 10 thousandths by 3 .003 1 hundredth = 10 thousandths .1 remaining Figure 6.12 (a) .01 remaining .001 remaining (b) (c) Continuing this process of splitting the remaining part by 10 and dividing by 3 shows that the decimal for 13 has a repeating pattern of 3s. When a decimal does not terminate and contains a repeating pattern of digits, it is called a repeating decimal or an infinite decimal. The step-by-step visual illustration in Figure 6.12 has a corresponding numerical division algorithm. The first step is to divide 10 tenths by 3. The result is .3 with .1 remaining. The second step is to divide 10 hundredths by 3. The result of the first two steps is .33 with .01 remaining. In the third step, 10 thousandths are divided by 3. The result of the first three steps is a quotient of .333 with .001 remaining. This process can be continued to obtain any number of 3s in the decimal approximation for 13 . ben35651_ch06_301-402.indd Page 315 8/13/14 9:21 AM f-w-166 /201/MH02188/ben35651_disk1of1/0078035651/ben35651_pagefiles 315 Section 6.1 Decimals and Rational Numbers Step 1 .3 3q1.0 9 One-tenth → 1 Content Standards 7.NS.2d Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats. Common Core State Standards Mathematics Step 2 .33 3q1.00 9 10 9 One-hundredth →1 Step 3 .333 3q1.000 9 10 9 10 9 One-thousandth → 1 The division algorithm can be used to obtain a terminating or repeating decimal for any rational number. For example, when the numerator of 38 is divided by its denominator, the division algorithm shows that the decimal terminates after three digits. .375 8q3.000 2400 60 56 40 40 On the other hand, when the numerator of 47 is divided by its denominator, the quotient does not terminate but repeats the same arrangement of six digits (571428) over and over. In this case the decimal is repeating. The reason for this can be seen by looking at the remainders 5, 1, 3, 2, 6, and 4, which are circled below. These six numbers, plus 0, are all the possible remainders when a number is divided by 7. So after six steps in the process of dividing 4 by 7, the numbers in the decimal quotient repeat. Notice the use of the bar above the six digits in the quotient to indicate the repeating pattern. The block of digits that is repeated over and over is called the repetend. .5714285 7q4.0000000 35 50 49 10 7 30 28 20 14 60 56 40 35 The preceding example illustrates why every rational number rs can be represented by either a repeating decimal or a terminating decimal. When r is divided by s, the remainders are always less than s (see the Division Algorithm Theorem, page 152). If a remainder of 0 occurs in the division process, as it does when we divide 3 by 8, then the decimal terminates. If there is no zero remainder, then eventually a remainder will be repeated, in which case the digits in the quotient will also start repeating. ben35651_ch06_301-402.indd Page 316 8/13/14 9:21 AM f-w-166 316 Chapter 6 Calculator Connection /201/MH02188/ben35651_disk1of1/0078035651/ben35651_pagefiles Decimals: Rational and Irrational Numbers Calculators are convenient for finding decimal representations of fractions. For fractions represented by repeating decimals, such as 127 5 .5833333333 . . . , the number in the calculator view screen is an approximation because it shows only a few of the digits. For many applications this is sufficient accuracy. Most calculators use several more digits than are shown in their view screens. When the number of digits in a decimal exceeds the space in a calculator’s view screen, the calculator may keep several hidden digits that are used internally for greater accuracy. To determine if your calculator uses hidden digits, try the following keystrokes to obtain a decimal approximation for 171 . Keystrokes 1 View Screen ÷ 17 = 0.0588235 × 100,000 = 5882.3529 − 5882 = 0.3529411 Notice that the final view screen shows five hidden digits 2, 9, 4, 1, and 1 that were not in the first view screen. The purpose of multiplying by a power of 10 and subtracting the whole number part of the product is to move the digits in the view screen to the left to make room for digits that may be hidden. Did you “uncover” hidden digits beyond the last digit that was initially displayed for 171 by your calculator? E X A MP LE I Write the decimal for each rational number. Use a bar to show the repetend (repeating digits). 1. 3 11 2. 5 6 3. 5 12 Solution 1. .27 2. .83 3. .416 Notice in the solutions to Example I that the repeating pattern in .83 does not begin until the hundredths digit and the repeating pattern in .416 does not begin until the thousandths digit. Writing Terminating and Infinite Repeating Decimals as Quotients of Integers Every rational number is the quotient of two integers, and we have seen that such numbers can be written as a terminating or repeating decimal. Conversely, every terminating or repeating decimal can be written as the quotient of two integers. An example of a terminating decimal written as a quotient of two integers is shown below. Terminating decimals can be written as fractions whose denominators are powers of 378 10. The following equations show why .378 equals 1000 . 3 7 8 1 1 10 100 1000 300 70 8 378 5 1 1 5 1000 1000 1000 1000 .378 5 Similarly, .47 5 47 3802 643 3.802 5 and 64.3 5 100 1000 10 ben35651_ch06_301-402.indd Page 317 9/6/14 2:03 PM f-w-166 /201/MH02188/ben35651_disk1of1/0078035651/ben35651_pagefiles Section 6.1 Decimals and Rational Numbers 317 At first it may appear to be a difficult task to write an infinite repeating decimal, such as the following, as a rational number in the form ab . Consider: N 5 .37 5 .373737 . . . However, using a technique that allows us to eliminate the infinite repeating decimal will make the process of writing a rational number in the form ab easier. Notice that if we multiply N by 100, the decimal portion of 100N looks the same as the decimal portion of N: 100N 5 37.37 . . . N 5 .37 . . . Subtract the second equation from the first equation, and the infinite repeating decimal portion is eliminated. 100N 5 37.37 . . . 2(N 5 .37 . . .) 99N 5 37 N5 Thus, 37 99 With certain adjustments this technique will enable us to express any infinite repeating decimals in the form ab . E XA M P LE J Write the infinite repeating decimal N 5 .572 5 .57272 . . . as a rational number in the form ab . Solution First, multiply N by 10 so the repeating decimal portion begins at the decimal point: 10N 5 5.7272 . . . Then multiply N by 1000 so that the infinite repeating decimal portion of 10N and 1000N are the same. Subtract: 1000N 5 572.72 . . . 2(10N 5 5.72 . . .) 990N 5 572 2 5 5 567 567 N5 990 TRY IT! 6.1.4 Write each of the following rational numbers as the quotient of integers in the form ab: 1. L 5 2.7 5 2.777 . . . 3. N 5 2.12 5 2.121212 . . . 2. M 5 1.067 5 1.0676767 . . . ben35651_ch06_301-402.indd Page 318 8/13/14 9:21 AM f-w-166 318 Chapter 6 /201/MH02188/ben35651_disk1of1/0078035651/ben35651_pagefiles Decimals: Rational and Irrational Numbers DENSITY OF RATIONAL NUMBERS In Chapter 5, on page 255, we saw examples of the fact that the rational numbers, when written as fractions, are dense on the real number line. That is, between any two rational numbers there is always another rational number. To show this, we looked at a method for finding a fraction between two given fractions. In the following example, we will consider ways of finding a decimal between any two given decimals. E X A M P LE K Sketch a number line and mark the location of each pair of decimals. Then find another decimal between them. 1. .124 and .125 2. .47 and .621 3. 1.1 and 1.2 Solution 1. One method of finding decimals between two given decimals is to express both decimals with a greater number of decimal places. For example, .124 5 .1240 and .125 5 .1250, and the 9 four-place decimals .1241, .1242, . . . , .1249 are between .124 and .125. Also, .124 5 .12400 and .125 5 .12500 and the 99 five-place decimals .12401, .12402, . . . , .12499 are between .124 and .125. Similarly, the process can be continued as there are 999 six-place decimals between .124000 and .125000, etc. 2. Since .47 5 .470, any of the three-place decimals between .47 and .621 may be selected, and by increasing the number of decimal places for .47 and .621, more decimals can be found between these numbers. 3. 1.1 5 1.10 and 1.2 5 1.20, so the 9 decimals 1.11, 1.12, . . . , 1.19 are between 1.1 and 1.2. Also, 1.1 5 1.100 and 1.2 5 1.200, so the 99 decimals 1.101, 1.102, . . . , 1.199 are between 1.1 and 1.2, etc. (a) .124 (b) .125 0 .47 .5 .1243 .500 (c) 1.1 .621 1.2 1 1.15 APPROXIMATION Often a calculator view screen will be filled with the digits of a decimal, when some approximate value is all that is necessary. Rounding to a given place value is the most common method of obtaining decimal approximations. Rounding Decimals can be rounded to the nearest whole number, nearest tenth, nearest hundredth, etc. Before we look at a rule for rounding decimals, let’s consider some visual illustrations. Squares for three decimals are shown in Figure 6.13 on the next page. Consider rounding these decimals to the nearest tenth. The decimal .648 rounds to .6 because 6 full columns and less than one-half of the next column are shaded; .863 rounds to .9 because 8 full columns and more than one-half of the next column are shaded; .35 can be either rounded up to .4 or rounded down to .3, because 3 full columns and one-half of the next column are shaded. In this text, we adopt the policy of rounding up. Next consider rounding the decimals in Figure 6.13 to the nearest hundredth. The square for .648 shows that 64 hundredths are shaded (6 full columns and 4 hundredths ben35651_ch06_301-402.indd Page 319 8/13/14 9:21 AM f-w-166 /201/MH02188/ben35651_disk1of1/0078035651/ben35651_pagefiles 319 Section 6.1 Decimals and Rational Numbers Figure 6.13 .648 .863 .35 in the next column); the remaining 8 thousandths are more than one-half of the next hundredth, so .648 rounds to .65. The Decimal Square for .863 shows that 86 hundredths are shaded (8 full columns and 6 hundredths in the next column); the remaining 3 thousandths are less than one-half of the next hundredth, so .863 rounds to .86. E XA M P LE L Round each decimal to the nearest tenth and to the nearest hundredth. 1. .283 2. .068 3. 14.649 Solution 1. .283 rounded to the nearest tenth is .3 and to the nearest hundredth is .28. 2. .068 rounded to the nearest tenth is .1 and to the nearest hundredth is .07. 3. 14.649 rounded to the nearest tenth is 14.6 and to the nearest hundredth is 14.65. Content Standards 5.NBT.4 Use place value understanding to round decimals to any place. Common Core State Standards Mathematics Notice in Example L that 14.649 rounded to the nearest tenth is not 14.7. You can confirm this by visualizing a Decimal Square for .649: 6 full columns are shaded and less than one-half of the next column (49 thousandths) is shaded. The preceding examples are special cases of the following general rule for rounding decimals. Notice that this is similar to the rule that was stated for rounding whole numbers on page 91 of Chapter 3. Rule for Rounding Decimals 1. Locate the place value to which the number is to be rounded, and check the digit to its right. 2. If the digit to the right is 5 or greater, then all digits to the right are dropped and the digit with the given place value is increased by 1. 3. If the digit to the right is 4 or less, then all digits to the right of the digit with the given place value are dropped. TRY IT! 6.1.5 Round 1.6825 to the given number of decimal places. 1. Two decimal places (round to hundredths). 2. One decimal place (round to tenths). 3. Three decimal places (round to thousandths). ben35651_ch06_301-402.indd Page 320 8/13/14 9:21 AM f-w-166 320 Chapter 6 Calculator Connection /201/MH02188/ben35651_disk1of1/0078035651/ben35651_pagefiles Decimals: Rational and Irrational Numbers Some calculators automatically round decimals that exceed the space in the view screen. On such calculators, if 2 is divided by 3, the decimal 0.66 . . . 667 will be displayed, where the last digit in the view screen is rounded from 6 to 7. Almost all calculators that round off at a digit that is followed by 5 will increase this digit, as described in the preceding rule for rounding numbers. For example, 55 99 is equal to the repeating decimal .5555 . . . . If 55 is divided by 99 on a calculator that rounds, 0.55 . . . 556 will show in the display. Try this on your calculator. PROBLEM-SOLVING APPLICATION Problem The price of a single pen is 39 cents. This price is reduced if pens are purchased in quantity. The price per pen is always a whole number of cents and never less than 2 cents. If all of the pens in a box were purchased at a single reduced rate, and the total paid was $22.91, then what is the number of pens in the box? Understanding the Problem We don’t know the reduced price per pen nor the number of pens. But, we do know the price is a whole number (in cents) and that the price per pen times the number of pens is $22.91. Question 1: What are the possible reduced prices per pen when bought in quantity? Devising a Plan The possible price in cents-per-pen ranges from 2 to 38 cents (why?). The number of pens is an unknown whole number that we can call N (for number of pens). Whatever number N is, (cents-per-pen) 3 N pens 5 2291 cents. So, at least one of the following statements is true. 2 3 N 5 2291, 3 3 N 5 2291, 4 3 N 5 2291, . . . , 37 3 N 5 2291, 38 3 N 5 2291 The cost per pen cannot be an even number because 2291 is an odd number. That reduces our list to 18 possibilities. 3 3 N 5 2291, 5 3 N 5 2291, 7 3 N 5 2291, . . . , 37 3 N 5 2291 Because neither 3 nor 5 are factors of 2291, this further reduces the list of possible values for the cost per pen. 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37 Question 2: Why are we able to eliminate these nine numbers from the list? Carrying Out the Plan Because the price of each pen is less than 39 cents, we need only check the primes 7, 11, 13, 17, 19, 23, 29, 31, and 37. Question 3: Which of these primes divide 2291, and what is the number of pens in the box? Looking Back Suppose we kept the conditions of the problem the same but changed the reduced price for the entire box to $15.17. Question 4: How many pens would be in the box?