Reading, Writing, Comparing, and Rounding Decimal Numbers

Section 2.1 PRE-ACTIVITY PREPARATION Reading, Writing, Comparing, and Rounding Decimal Numbers Financial transactions in dollars and cents are perhaps the most common examples of how you use decimal numbers in your daily life. Your understanding of decimal place values and the proper placement of the decimal point are very important when dealing with money, as you know. Imagine your frustration if you expect a $652.00 paycheck, only to receive one for $65.20! When you write your own personal checks, as well, you must write the amounts properly in both digits and words. There are other applications of decimals, of course. Your cumulative grade point average (GPA) and sports statistics such as batting averages or race times are common examples. Accuracy in presentation is also of great importance in the fields of science, medicine, engineering, manufacturing, and architectural design for which measurements are most often given in the form of decimal numbers. You may come across rounded decimal numbers as well advertised prices $5 for 3 boxes of crackers or, rounded to the nearest cent, $1.67 to purchase a single box estimated weight measurements your scanned purchase of 3.2 pounds of apples from the produce section or 1.94 pounds of cheese from the deli census statistics presenting larger numbers, such as Michigan s estimated population of 10.1 million residents or the 17.5 million college students in the United States Before using the four basic operations of addition, subtraction, multiplication, and division to compute with decimal numbers, you should have a working knowledge of how reading, writing, comparing, and rounding decimal numbers are made easier by the design of the decimal number system. LEARNING OBJECTIVES Master reading and writing decimal numbers. Put any set of decimal numbers in order. Master the process of rounding a decimal number to a given decimal place value. 125

126 Chapter 2 Decimal Numbers TERMINOLOGY PREVIOUSLY USED base ten system decimal system expanded form greater than symbol > less than symbol < midpoint number line place digit place value round down round up standard form NEW TERMS TO LEARN compare decimals decimal number decimal places decimal place digit decimal point hundredths hundred-thousandths order standard decimal notation tenths ten-thousandths thousandths trailing zeros BUILDING MATHEMATICAL LANGUAGE From your study of whole numbers, you know that when you write a number in its standard form in the decimal (base ten) system, every position in the number has a corresponding and specific place value. Note the partial decimal system place value chart on the next page. The chart for whole numbers is extended to also include the place values to the right of the ones place places used to represent numbers less than one (1), places representing fractional parts of one whole unit. What do the phrases a number less than one (1) and fractional parts mean? If you think in terms of money, you know, for instance, that $2.38 is more than $2 and less than $3. Similarly, when the gas pump displays your purchase of 15.29 gallons of gas, you know that you have put in more than 15 gallons but less than 16 gallons. That $.38 represents less than $1 or part of one dollar and the.29 gallons represents a fractional part of one gallon of gas. When a decimal number is written in standard form (standard decimal notation), a decimal point separates its whole number part from its fractional part. The places to the right of the decimal point, beginning with the tenths, are commonly referred to as the decimal places. Because they are fractional parts, their place names all end with ths.

Section 2.1 Reading, Writing, Comparing, and Rounding Decimal Numbers 127 The number 15.29 ( fifteen and twenty-nine hundredths ) is positioned properly in the chart below. Place name Thousands Hundreds Tens Ones. Tenths Hundredths Thousandths Ten-Thousandths Place value 1000 100 10 1..1.01.001.0001 Example number 1 5. 2 9 It is important to note that, as you move from left to right in the chart, each place value is equal to the place value to its left divided by 10. The place values grow smaller as you move to the right. The pattern is: 1000 10 = 100 one hundred 100 10 = 10 ten 10 10 = 1 one 1 10 =.1 one tenth.1 10 =.01 one hundredth.01 10 =.001 one thousandth, and so on as you move left to right in place value. You can think of this pattern in its reverse as well. As you move from right to left, the place values grow larger, as each place value represents 10 times the place value to its immediate right: 10.001 =.01 10.01 =.1 10.1 = 1 10 1 = 10 10 10 = 100 10 100 = 1000, and so on as you move left in place value When there is no whole number part, a decimal number may be written with or without a zero (0) as the placeholder digit for the ones place. For example,.275 can also be written as 0.275 Both are read, two hundred seventy-five thousandths. Both denote a number less than one (1). Writing a zero in the ones place helps to assure that you will not overlook the decimal point when working with decimal numbers. For a whole number, the decimal point is understood to be to the immediate right of the ones place digit. A whole number is usually written without the decimal point, unless it is necessary to use it for an operation that involves other decimal numbers. For example, 29 can be written as 29. or 29.0 if necessary.

128 Chapter 2 Decimal Numbers You might find it helpful to visualize the parts of one whole unit represented by the place values to the right of the decimal point (the decimal places). The shaded areas in the figures below represent the place values 1,.1, and.01: Dividing 1 into 10 equal parts Dividing 1 into 100 equal parts and so on, as the fractional parts of one whole units become smaller when each part is sub-divided into 10 equal parts. 1 1.1 or 10 "one tenth" 1.01 or 100 "one hundredth" How might you visually represent.8 (eight tenths) in the figure below? How might you visually represent.29 (twenty-nine hundredths) in the figure below?

Section 2.1 Reading, Writing, Comparing, and Rounding Decimal Numbers 129 This book will refer to the zeros appended to the right of the digits after the decimal point as trailing zeros. For example, 8.36 = 8.360 = 8.3600 = 8.36000, and so on. Trailing zeros do not change the value of the number. Why? Recall that a zero used as a placeholder signifies a value of zero for that place. For example, 8.360 in its expanded form is 8 ones + 3 tenths + 6 hundredths + 0 thousandths or 8 ones + 3 tenths + 6 hundredths + 0 or 8.36 A number line can help you visualize the comparison of decimal numbers. As with whole numbers, the farther you move to the right on a number line, the larger the number will be. On the number line below, 2.1 is larger than 2.0, 2.9 is larger than 2.1, and 3.0 is larger than 2.9. 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 To indicate the decimal numbers between 2.1 and 2.2, zoom in on that section and divide it into appropriate equal intervals. For example, 2.11, 2.15, and 2.18 are properly indicated on the number line below. 2.10 2.11 2.12 2.13 2.14 2.15 2.16 2.17 2.18 2.19 2.20 The word and is used to signify the placement of the decimal point when reading or writing a decimal number in its word form. For example, 48.63 in word form is forty-eight and sixty-three hundredths. The term decimals is often used interchangeably with the term decimal numbers as in the sentence, In some applications, you might think it easier to compute with decimals than with fractions. Reading and Writing Decimal Numbers You may be accustomed to hearing or reading aloud the common translation of a decimal number, say 91.0815, as ninety-one point zero eight one five. However, keeping in mind the decimal system place value chart, you can use the following two techniques to translate between the standard form and word form of decimal numbers in correct math terminology. The first provides the proper way to express a decimal number in words, and both techniques underscore what the fractional part of a decimal number actually means. To verify your translated answer, you can use the opposite technique.

130 Chapter 2 Decimal Numbers TECHNIQUE Translating a Decimal Number from Standard Form to Word Form Technique Step 1: If there is a whole number part (left of the decimal point), say or write out its word name and translate the decimal point as and. If there is no whole number part, skip to Step 2. Step 2: Say or write the number formed by the digits to the right of the decimal point and attach the place value name of the final digit (farthest to the right). Special Case: Zero (0) as the whole number (see page 131, Model C) MODELS Translate each of the following numbers to its word form. A 502.467 Step 1: five hundred two and Step 2:. tenths hundredths thousandths 4 6 7 four hundred sixty-seven thousandths Answer: five hundred two and four hundred sixty-seven thousandths B 91.0815 Step 1: ninety-one and Step 2:. tenths hundredths thousandths ten-thousandths 0 8 1 5 eight hundred fifteen ten-thousandths Answer: ninety-one and eight hundred fifteen ten-thousandths

Section 2.1 Reading, Writing, Comparing, and Rounding Decimal Numbers 131 Special Case: Zero (0) as the Whole Number C 0.10275 Step 1: When the whole number is zero, it is not necessary to say or write zero and for the whole number name and decimal point. Skip to Step 2. Step 2:. tenths hundredths thousandths ten-thousandths hundred-thousandths 1 0 2 7 5 Answer: ten thousand two hundred seventy-five hundred-thousandths TECHNIQUE Translating a Decimal Number from Word Form to Standard Form Technique Step 1: Write the whole number (words before and ) in its standard form and substitute a decimal point for the word and. Step 2: Translate the word name of the fractional part to digits in standard form, aligning its final digit with the named decimal place value. Step 3: Use a zero (0) placeholder if a decimal place is missing. Special Case: No whole number part in the word form (see page 133, Model D)

132 Chapter 2 Decimal Numbers MODELS Translate each of the following numbers to its standard form. A four hundred six and twenty-one hundredths Step 1: 406. Step 2: twenty-one hundredths Step 3: all decimal places accounted for no zero decimal place holders needed. tenths hundredths 2 1 Answer: 406.21 B ninety-four and two hundred seven hundred-thousandths Step 1: 94. Step 2: two hundred seven hundred-thousandths Step 3: tenths hundredths thousandths ten-thousandths hundred-thousandths 0 0 2 0 7 zero placeholders for the missing decimal places Answer: 94.00207 C three and fifty hundredths Step 1: 3. Step 2: fifty hundredths Step 3: all decimal places accounted for Answer: 3.50 tenths hundredths 5 0

Section 2.1 Reading, Writing, Comparing, and Rounding Decimal Numbers 133 Special Case: No Whole Number Part in the Word Form D eighty-three thousandths Step 1: 0. If there is no whole number part in the word form, use zero (0) as the whole number, followed by a decimal point, to assure the decimal point is properly placed. Step 2: eighty-three thousandths tenths hundredths thousandths 0 8 3 Step 3: zero placeholders for the missing tenths place Answer: 0.083 Comparing the Values of Decimal Numbers You already know how to compare values of whole numbers; that is, determine which number is smaller and which is larger. The process is almost automatic. For example, to compare 461 and 428 you look to the tens place digits. Knowing that 6 tens are greater than 2 tens, you can readily say 461 > 428. You can also compare decimal numbers by comparing their corresponding place values. For example, which is smaller, 0.825 or 0.823? same same different You see that the digit 8 occupies the tenths place in each number and that there is a 2 in both hundredths places. However, in the thousandths place, the digits are different. The thousandths place digit 3 is less than the thousandths place digit 5. Now you can say with certainty that 0.823 < 0.825. That is, eight hundred twenty-three thousandths is less than eight hundred twenty-fi ve thousandths. How can you order, arrange from smallest to largest or largest to smallest, any list of decimal numbers? The following methodology offers a simple and effective way to do so that takes advantage of your ability to compare whole numbers.

134 Chapter 2 Decimal Numbers METHODOLOGY Comparing (Ordering) Decimal Numbers Order the following list of numbers from smallest to largest. Example 1: 3.25, 3.6, 4.3, 3, 3.384 Example 2: 6.219, 5.201, 6.3, 6.21, 6.02 Try It! Steps in the Methodology Example 1 Example 2 Step 1 List the numbers. Step 2 Sort by whole numbers. Write the numbers in a column, aligning place values and decimal points. Arrange the numbers in the desired order (smallest to largest or largest to smallest) according to their whole numbers (ignoring the decimal places). 3.25 3.6 4.3 3. 3.384 smallest to largest Rank 3.25 3.6 3. 3.384 4.3 5 Step 3 Append trailing zeros. For the set of numbers with the same whole number part, use trailing zeros so that each number ends with the same place value.????? Why can and why do you do this? Special Case: Two or more whole number sets to order in the list (see page 135, Model) 3.25 3.250 3.6 3.600 3. 3.000 3.384 3.384 4.3 4.3 Rank 5 Step 4 Order by fractional parts. Order the set according to the fractional parts. 3.25 3.250 THINK 0 < 250 < 384 < 600 3.6 3.600 3. 3.000 3.384 3.384 4.3 4.3 Rank 2 4 1 3 5 Step 5 Present the answer. Write the original numbers in the correct order. 3, 3.25, 3.384, 3.6, 4.3

Section 2.1 Reading, Writing, Comparing, and Rounding Decimal Numbers????? Why can and why do you do Step 3? Trailing zeros do not change the value of the decimal numbers. 135 When you make the decimal numbers all end with the same fractional place, you can easily compare fractional parts according to the entire series of digits occupying those places. In Example 1 you can now simply compare 250 thousandths to 600 thousandths to 000 thousandths to 384 thousandths. Arranging 250, 600, 000, and 384 is an easy task: 0 thousandths < 250 thousandths < 384 thousandths < 600 thousandths MODEL Special Case: Two or More Whole Number Sets to Order in the List Arrange the following list of numbers in order from largest to smallest. 1.7087, 0.078, 0.87, 1.781, 0.0877, 0.8 Step 2 largest to smallest Step 1 1.7087 1.7087 set of numbers with 1 as the whole number 0.078 1.781 0.87 0.078 1.781 0.87 set of numbers with 0 as the whole number 0.0877 0.0877 0.8 0.8 } } When there are two or more whole number sets in the list, order each set separately, using trailing zeros as appropriate. Then combine the rankings for the final order. Steps 3 & 4 Rank 1.7087 1.7087 2 1.781 1.7810 1 0.078 0.0780 6 0.87 0.8700 3 0.0877 0.0877 5 0.8 0.8000 4 THINK 7810 > 7087 THINK 8700 > 8000 > 877 > 780 Step 5 Answer: 1.781, 1.7087, 0.87, 0.8, 0.0877, 0.078

136 Chapter 2 Decimal Numbers Rounding Decimal Numbers The methodology for rounding decimal numbers uses the concept of a midpoint (middle) number to make the decision whether to round up or round down, as did the methodology for rounding whole numbers (refer to Section 1.2). Note that the methodology will refer to the digit in a specified decimal place as the decimal place digit. METHODOLOGY Rounding a Decimal Number Example 1: Round 43.9738 to the nearest hundredth. Example 2: Round 24.61809 to the nearest thousandths place. Try It! Steps in the Methodology Example 1 Example 2 Step 1 Determine final number of decimal places. Step 2 Identify the place digit. Determine the number of decimal places in the final answer.??? Why do you do this? Identify the digit in the specified place value (the place digit) by marking it with an arrow. Special Case: Rounding to the nearest whole number (see page 138, Model 1) hundredth The final answer will have two decimal places. 4 3. 9 7 3 8 Step 3 Identify the digit to the right of the place digit. Step 4 Compare to the number 5. Identify the digit occupying the decimal place immediately to the right of the place digit by circling it. 4 3. 9 7 3 8 Determine whether the circled digit is less than, equal to, or greater than 5. 3 < 5 Step 5 Round up or down. If the circled digit is less than 5, do not change the place digit. If the circled digit is 5 or greater, round up by adding one to the place digit. The hundredths place digit does not change 4 3. 9 7 x x

Section 2.1 Reading, Writing, Comparing, and Rounding Decimal Numbers 137 Steps in the Methodology Example 1 Example 2 Step 6 Present the answer. To present your answer, drop all decimal place digits to the right of the specified place value.??? Why can you do this? 43.97 Special Case: Presenting a zero in the specifi ed place value (see page 138, Model 2)??? Why do you do Step 1? The last decimal place value for a rounded answer will always be named, so the final answer must have only as many decimal places as specified to reflect the corresponding fractional part. In Example 1, the named hundredths place requires two and only two places to the right of the decimal point.??? Why can you do Step 6? As the result of rounding, the digits to the right of the specified place value digit become zeros (just as they did with whole numbers). They are trailing zeros, however, because of their positions in the decimal number. Therefore, they can be dropped without changing the value of the rounded decimal number.

138 Chapter 2 Decimal Numbers MODELS Model 1 Special Case: Rounding to the Nearest Whole Number Round 246.547 to the nearest whole number. Rounding to the nearest whole number means rounding to the ones place. Step 1 no decimal places (Round to the ones place.) Step 2 246.547 Step 3 246.547 Step 4 5 = 5 Step 5 The 6 changes to a 7. Step 6 Answer: 247. or 247 Pictured on a number line: 246.547 246 246.5 247 midpoint Model 2 Special Case: Presenting a Zero in the Specifi ed Place Value Round 12.3997 to the nearest hundredths place. Step 1 2 decimal places (Round to the hundredths place.) Step 2 12.3997 Step 3 12.3997 Step 4 9 > 5 Step 5 The 9 changes to 0 and carry the 1 to the tenths place, making it a 4. 12.40xx Step 6 Answer: 12.40 Pictured on a number line: 12.3997 After rounding up or down, if the specified decimal place digit is zero (0), it is necessary to present it in the answer to indicate that the original number has been rounded to that place. 12.39 12.395 midpoint 12.40

Section 2.1 Reading, Writing, Comparing, and Rounding Decimal Numbers 139 ADDRESSING COMMON ERRORS Issue Incorrect Process Resolution Correct Process Reading decimal numbers incorrectly when there are zeros as decimal place holders in the standard form 36.052 is read as thirty-six and fifty-two hundredths Use the place value chart and align the digits with the chart. tenths hundredths thousandths 3 6. 0 5 2 36.052 is read as thirty-six and fifty-two thousandths Comparing without regard to decimal place values Compare: 4.0387 and 4.38 THINK 387 > 38 It is not the number of digits after the decimal point that determines order, but the place value of those digits. Compare: 4.0387 and 4.38 4.0387 4.0387 4.38 4.3800 4.0387 is larger than 4.38 438 Use trailing zeros (that is, append zeros so the numbers have the same place values) so you can make a true comparison of their fractional parts. THINK 3800 > 387 3800 ten-thousandths > 387 ten-thousandths Therefore, 4.38 > 4.0387 Incorrectly identifying decimal places when rounding Round 87.6349 to the nearest hundredth. Thinking, ones, tenths, hundredths the 4 is in the hundredths place. The whole number is to the left of the decimal point, starting with the ones. The fractional part of a whole is to the right, starting with the tenths. In 87.6349, the 3 is in the hundredths place. 87.6349 87.6349 Answer: 87.63 Answer: 87.635 Incorrectly presenting the final rounded decimal number 32.9179 rounded to the tenths place is 32.9000 Rounding decimals requires dropping the digits to the right of the specified place. For example, rounding to tenths means there will be only one number to the right of the decimal point. 32.9179 rounded to the tenths place is 32.9

140 Chapter 2 Decimal Numbers Issue Incorrect Process Resolution Correct Process Not presenting zero as a decimal place holder to indicate designated accuracy in a rounded answer Round 73.0348 to the nearest tenth. 73.0348 Answer: 73 When rounding to a designated place value, the final answer must present the specified number of decimal places to indicate that level of accuracy. Round 73.0348 to the nearest tenth. Rounding to tenths means there must be one decimal place in the final answer. 73.0348 rounded to the nearest tenth is 73.0 PREPARATION INVENTORY Before proceeding, you should have an understanding of each of the following: the terminology and notation associated with reading, writing, comparing, and rounding decimal numbers the identification of place values the value of a digit with respect to its particular position in a decimal number the placement of the decimal point in a whole number the use of trailing zeros as an effective method for comparing (ordering) decimal numbers the correct presentation of a rounded decimal number

Section 2.1 ACTIVITY Reading, Writing, Comparing, and Rounding Decimal Numbers PERFORMANCE CRITERIA Translating between standard decimal notation and words correct identification and interpretation of each given place value correct use of and or the decimal point correct use of place value names Arranging a set of decimal numbers in order from smallest to largest or largest to smallest appropriate use of trailing zeros correct comparisons Rounding decimal numbers to specified place values correct identification of the specified place value consistent documentation and presentation with appropriate notation accuracy in the rounding process CRITICAL THINKING QUESTIONS 1. What are three real-world situations that use decimal numbers? 2. How is the whole number part separated from the decimal fraction part in reading and writing a decimal number? 141

142 Chapter 2 Decimal Numbers 3. How can any whole number be expressed as a decimal number? 4. What are the names of the three decimal place values to the right of the thousandths place? 5. What does it mean to use zero (0) as a decimal placeholder? 6. Why can you add trailing zeros to a decimal number without changing the value of the number? 7. What is the relationship between ones and tenths? between tenths and hundredths?

Section 2.1 Reading, Writing, Comparing, and Rounding Decimal Numbers 143 8. How can you make sure that you order a set of decimal numbers correctly? 9. What is the most significant difference between rounding whole numbers and rounding decimal numbers? 10. When would you present a zero (0) as the final decimal place digit in a rounded answer? 11. In the U.S. monetary system, why are dollar amounts rounded to two decimal places?

144 Chapter 2 Decimal Numbers TIPS FOR SUCCESS Know the place value chart. To assure your translation is correct, cover up the original representation of the decimal number. Then convert back to see if you can get the same number, in digits or words. Use trailing zeros when comparing decimal numbers. Use consistent notation for the rounding process. Drawing a number line may help to visualize the comparison to the midpoint number in the rounding process. DEMONSTRATE YOUR UNDERSTANDING 1. Identify the place indicated. a) 5.046 The 6 is in the place. b) 359.20 The 2 is in the place. c) 0.6974 The 0 is in the place. 2. Write the following numbers in standard decimal notation. a) Three thousand four hundred and six tenths b) Five hundred thirty-two thousandths c) Six thousand and forty-nine ten-thousandths d) Eight and three hundred seven hundred-thousandths 3. Write in words. a) 203.52 b) 48.0057 c) 0.906 d) 0.75201

Section 2.1 Reading, Writing, Comparing, and Rounding Decimal Numbers 145 4. Order the following numbers from smallest to largest: 2.046, 2.4, 1.06, 2, 2.46 Worked solution: Answer: 5. Order the following numbers from smallest to largest: 24.07, 24.007, 24.005, 24.058, 24.0059 Worked solution: Answer: 6. Order the following numbers from largest to smallest: 0.05, 1.03, 1.9, 0.1, 0.201 Worked solution: Answer:

146 Chapter 2 Decimal Numbers 7. Round 713.54973 to the indicated place. 713.54973 Rounding Process Answer a) tenth b) hundredth c) thousandth d) tenthousandth e) hundred f) nearest whole number TEAM EXERCISES 1. In the grids below, fill in the correct number of rectangles to represent the following decimal numbers. (Hint: you may find it helpful to use trailing zeros.) a) 0.17 Use a pencil. b) 0.04 Use a pen. c) 0.5 Use a highlighter. d) 1.4 Use a different color highlighter. e) How could you modify one of the grids to shade in a representation of the decimal number 0.006?

Section 2.1 Reading, Writing, Comparing, and Rounding Decimal Numbers 147 2. List fifteen decimal numbers between 0.25 and 0.26 IDENTIFY AND CORRECT THE ERRORS Identify and correct the errors in the following worked solutions. If the worked solution is correct, write Correct in the second column. If the worked solution is incorrect, solve the problem correctly in the third column. Worked Solution What is Wrong Here? 1) Round 62.3585 to the nearest hundredth. Identify the Errors Rounded to the thousandths place, not the specified hundredths place. Correct Process 62.3585 Answer: 62.36 2) Write in words: 5.036. 3) Round 5.6719 to the nearest hundredth. 4) Round 88.9673 to the nearest tenth. 5) Round 4.1357 to the nearest whole number.

148 Chapter 2 Decimal Numbers Worked Solution What is Wrong Here? 6) List in order from smallest to largest: 3.656, 3.67, 13.76, 3.1657 Identify the Errors Correct Process ADDITIONAL EXERCISES 1. Write each of the following numbers in its standard form. a) one and eighty-four thousandths b) five hundred two and thirteen hundredths c) twelve and one hundred two thousandths d) seventy-six ten-thousandths 2. Write each of the following numbers in words. a) 0.408 b) 1502.07 c) 94.0036 d) 8.020 3. List in order from smallest to largest. 1.056, 1.06, 10.005, 1.5, 1.504 4. List in order from smallest to largest. 0.61, 0.006, 0.0059, 0.0601, 0.0519 5. Order from largest to smallest. 5.304, 5.043, 5.0043, 5.034, 5.0344 6. Round each of the following decimal numbers to the indicated place. a) Round 115.2354 to the nearest hundredth. b) Round 14.299 to the nearest hundredth. c) Round 8.398 to the nearest whole number. d) Round 0.6142 to the nearest thousandth. e) Round 43.0709 to the nearest tenth. f) Round 20.1095 to the nearest tenth.