5.1 Introduction to Decimals, Place Value, and Rounding

5.1 Introduction to Decimals, Place Value, and Rounding 5.1 OBJECTIVES 1. Identify place value in a decimal fraction 2. Write a decimal in words 3. Write a decimal as a fraction or mixed number 4. Compare the size of several decimals 5. Round a decimal to any specified decimal place In Chapter 4, we looked at common fractions. We will turn now to a special kind of fraction, a decimal fraction, which is a fraction whose denominator is a power of 10. 3 45 123 Some examples of decimal fractions are,, and. 10 100 1,000 In Chapter 1, we talked about the idea of place value. Recall that in our decimal placevalue system, each place has one-tenth the value of the place to its left. Example 1 Identifying Place Values RECALL The powers of 10 are 1, 10, 100, 1,000, and so on. You might want to review Section 1.7 before going on. Label the place values for the number 538. 5 3 8 Hundreds Tens Ones The ones place value is one-tenth of the tens place value; the tens place value is one-tenth of the hundreds place value; and so on. CHECK YOURSELF 1 Label the place values for the number 2,793. We now want to extend this idea to the right of the ones place. Write a period to the right of the ones place. This is called the decimal point. Each digit to the right of that decimal point will represent a fraction whose denominator is a power of 10. The first place to the right of the decimal point is the tenths place: NOTE The decimal point separates the whole-number part and the fractional part of a decimal fraction. 0.1 1 10 Example 2 Writing a Number in Decimal Form Write the mixed number 3 2 in decimal form. 10 Tenths 3 2 10 3.2 Ones The decimal point 391

392 CHAPTER 5 DECIMALS CHECK YOURSELF 2 Write 5 3 in decimal form. 10 As you move farther to the right, each place value must be one-tenth of the value before it. The second place value is hundredths 0.01. The next place is thousandths, the 100 1 fourth position is the ten thousandths place, and so on. The figure illustrates the value of each position as we move to the right of the decimal point. Ones Tenths Hundredths Thousandths Ten thousandths Hundred thousandths 2. 3 4 5 6 7 Decimal point OBJECTIVE 1 Example 3 Identifying Place Values NOTE For convenience we will shorten the term decimal fraction to decimal from this point on. What are the place values for 4 and 6 in the decimal 2.34567? The place value of 4 is hundredths, and the place value of 6 is ten thousandths. CHECK YOURSELF 3 What is the place value of 5 in the decimal of Example 3? Understanding place values will allow you to read and write decimals by using these steps. Step by Step: Reading or Writing Decimals in Words NOTE If there are no nonzero digits to the left of the decimal point, start directly with Step 3. Step 1 Step 2 Step 3 Read the digits to the left of the decimal point as a whole number. Read the decimal point as the word and. Read the digits to the right of the decimal point as a whole number followed by the place value of the rightmost digit. OBJECTIVE 2 Example 4 Write each decimal number in words. 5.03 is read as five and three hundredths. Hundredths Writing a Decimal Number in Words The rightmost digit, 3, is in the hundredths position.

INTRODUCTION TO DECIMALS, PLACE VALUE, AND ROUNDING SECTION 5.1 393 12.057 is read as twelve and fifty-seven thousandths. NOTE An informal way of reading decimals is to simply read the digits in order and use the word point to indicate the decimal point. 2.58 can be read two point five eight. 0.689 can be read zero point six eight nine. Thousandths The rightmost digit, 7, is in the thousandths position. 0.5321 is read as five thousand three hundred twenty-one ten thousandths. When the decimal has no whole-number part, we have chosen to write a 0 to the left of the decimal point. This simply makes sure that you don t miss the decimal point. However, both 0.5321 and.5321 are correct. CHECK YOURSELF 4 Write 2.58 in words. NOTE The number of digits to the right of the decimal point is called the number of decimal places in a decimal number. So, 0.35 has two decimal places. One quick way to write a decimal as a common fraction is to remember that the number of decimal places must be the same as the number of zeros in the denominator of the common fraction. OBJECTIVE 3 Example 5 Writing a Decimal Number as a Mixed Number Write each decimal as a common fraction or mixed number. 0.35 35 100 This fraction can then be simplified to 7 20. Two places Two zeros The same method can be used with decimals that are greater than 1. Here the result will be a mixed number. NOTE The 0 to the right of the decimal point is a placeholder that is not needed in the common-fraction form. 2.058 2 58 1,000 Three places Three zeros This mixed number can 29 be simplified to 2. 500 CHECK YOURSELF 5 Write as common fractions or mixed numbers. Do not simplify. (a) 0.528 (b) 5.08 RECALL By the Fundamental Principle of Fractions, multiplying the numerator and denominator of a fraction by the same nonzero number does not change the value of the fraction. It is often useful to compare the sizes of two decimal fractions. One approach to comparing decimals uses this fact: Writing zeros to the right of the rightmost digit does not change the value of a decimal. 0.53 is the same as 0.530. Look at the fractional form: 53 100 530 1,000 The fractions are equivalent. We have multiplied the numerator and denominator by 10. We will see how this is used to compare decimals in Example 6.

394 CHAPTER 5 DECIMALS OBJECTIVE 4 Example 6 Which is larger? Comparing the Sizes of Two Decimal Numbers 0.84 or 0.842 Write 0.84 as 0.840. Then we see that 0.842 (or 842 thousandths) is greater than 0.840 (or 840 thousandths), and we can write 0.842 0.84 CHECK YOURSELF 6 Complete the statement, using the symbol or. 0.588 0.59 When working with a decimal, it may be helpful to picture the location of the decimal on a number line. Example 7 Plotting Decimals on a Number Line Plot the number 4.6 on the given number line. Then estimate the location for 4.68. 4 5 The number 4.6 is located six-tenths of the distance from 4 to 5. Since each tick mark represents one-tenth, we count to the sixth tick mark and draw a dot. 4 5 4.6 The number 4.68 is eight-tenths of the distance from 4.6 to 4.7. We might estimate its location as: 4.68 4 4.6 4.7 5 CHECK YOURSELF 7 Plot the number 8.3 on the number line. Then estimate the location for 8.51. 8 9 Whenever a decimal represents a measurement made by some instrument (a rule or a scale), the decimals are not exact. They are accurate only to a certain number of places and are called approximate numbers. Usually, we want to make all decimals in a particular problem accurate to a specified decimal place or tolerance. This will require rounding the decimals. We can picture the process on a number line.

INTRODUCTION TO DECIMALS, PLACE VALUE, AND ROUNDING SECTION 5.1 395 Example 8 Rounding to the Nearest Tenth 3.78 3.7 3.8 NOTE 3.74 is closer to 3.7 than it is to 3.8. 3.78 is closer to 3.8 than it is to 3.7. 3.74 3.74 is rounded down to the nearest tenth, 3.7. 3.78 is rounded up to 3.8. CHECK YOURSELF 8 Use the number line in Example 8 to round 3.77 to the nearest tenth. Rather than using the number line, this rule can be applied. Step by Step: To Round a Decimal Step 1 Step 2 Step 3 Find the place to which the decimal is to be rounded. If the next digit to the right is 5 or more, increase the digit in the place you are rounding by 1. Discard the remaining digits to the right. If the next digit to the right is less than 5, just discard that digit and any remaining digits to the right. OBJECTIVE 5 Example 9 Rounding to the Nearest Tenth Round 34.58 to the nearest tenth. NOTE Many students find it easiest to mark this digit with an arrow. 34.58 Locate the digit you are rounding to. The 5 is in the tenths place. Because the next digit to the right, 8, is 5 or more, increase the tenths digit by 1. Then discard the remaining digits. 34.58 is rounded to 34.6. CHECK YOURSELF 9 Round 48.82 to the nearest tenth. Example 10 Round 5.673 to the nearest hundredth. 5.673 The 7 is in the hundredths place. The next digit to the right, 3, is less than 5. Leave the hundredths digit as it is and discard the remaining digits to the right. 5.673 is rounded to 5.67. Rounding to the Nearest Hundredth

396 CHAPTER 5 DECIMALS CHECK YOURSELF 10 Round 29.247 to the nearest hundredth. Example 11 Rounding to a Specified Decimal Place Round 3.14159 to four decimal places. NOTE The fourth place to the right of the decimal point is the ten thousandths place. 3.14159 The 5 is in the ten thousandths place. The next digit to the right, 9, is 5 or more, so increase the digit you are rounding to by 1. Discard the remaining digits to the right. 3.14159 is rounded to 3.1416. CHECK YOURSELF 11 Round 0.8235 to three decimal places. READING YOUR TEXT The following fill-in-the-blank exercises are designed to assure that you understand the key vocabulary used in this section. Each sentence comes directly from the section. You will find the correct answers in Appendix C. Section 5.1 (a) A fraction is a fraction whose denominator is a power of 10. (b) The period to the right of the ones place is called the point. (c) (d) The number of digits to the right of the decimal point is called the number of decimal. When a decimal represents a measurement made by some instrument, it is called an number. CHECK YOURSELF ANSWERS 1. 2 7 9 3 2. 5 3 3. Thousandths 10 5.3 Thousands Ones Hundreds Tens 4. Two and fifty-eight hundredths 528 5. (a) ; (b) 5 8 1,000 100 6. 0.588 0.59 7. 8. 3.8 9. 48.8 10. 29.25 11. 0.824 8.3 8.51 8 9

5.1 Exercises Boost your GRADE at ALEKS.com! For the decimal 8.57932: 1. What is the place value of 7? 2. What is the place value of 5? 3. What is the place value of 3? 4. What is the place value of 2? Practice Problems Self-Tests NetTutor Name Section e-professors Videos Date Write in decimal form. ANSWERS 23 5. 6. 100 209 7. 10,000 8. 9. 23 56 1,000 10. 371 1,000 3 5 10 7 431 10,000 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. Write in words. 11. 11. 0.23 12. 0.371 13. 0.071 14. 0.0251 12. 13. 15. 12.07 16. 23.056 14. Write in decimal form. 17. Fifty-one thousandths 18. Two hundred fifty-three ten thousandths 15. 16. 17. 18. 19. Seven and three tenths 20. Twelve and two hundred forty-five thousandths Write each as a common fraction or mixed number. 21. 0.65 22. 0.00765 23. 5.231 24. 4.0171 19. 20. 21. 22. 23. 24. SECTION 5.1 397

ANSWERS 25. 26. 27. Complete each statement, using the symbol,, or. 25. 0.69 0.689 26. 0.75 0.752 28. 29. 27. 1.23 1.230 28. 2.451 2.45 30. 31. 29. 10 9.9 30. 4.98 5 32. 31. 1.459 1.46 32. 0.235 0.2350 33. 34. 35. 36. 33. Arrange in order from smallest 34. Arrange in order from smallest to largest. to largest. 7 25 0.71, 0.072,, 0.007, 0.0069 2.05,, 2.0513, 2.059 10 10 7 251, 0.0701, 0.0619, 0.0712, 2.0515, 2.052, 2.051 100 100 37. 38. 39. Round to the indicated place. 35. 53.48 tenths 36. 6.785 hundredths 40. 41. 37. 21.534 hundredths 38. 5.842 tenths 42. 43. 44. 39. 0.342 hundredths 40. 2.3576 thousandths 45. 46. 41. 2.71828 thousandths 42. 1.543 tenths 43. 0.0475 tenths 44. 0.85356 ten thousandths 45. 4.85344 ten thousandths 46. 52.8728 thousandths 398 SECTION 5.1

ANSWERS 47. 6.734 two decimal places 48. 12.5467 three decimal places 49. 6.58739 four decimal places 50. 503.824 two decimal places Round 56.35829 to the nearest: 51. Tenth 52. Ten thousandth 53. Thousandth 54. Hundredth In exercises 55 to 60, determine the decimal that corresponds to the shaded portion of each decimal square. Note that the total value of a decimal square is 1. 55. 56. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 57. 58. 59. 60. SECTION 5.1 399

ANSWERS 61. 62. 63. In exercises 61 to 64, shade the portion of the square that is indicated by the given decimal. 61. 0.23 62. 0.89 64. 65. 66. 67. 68. 63. 0.3 64. 0.30 69. 65. Plot (draw a dot) 3.2 and 3.7 on the number line. Then estimate the location for 3.62. 3 4 66. Plot 12.51 and 12.58 on the number line. Then estimate the location for 12.537. 12.5 12.6 67. Plot 7.124 and 7.127 on the number line. Then estimate the location of 7.1253. 7.12 7.13 68. Plot 5.73 and 5.74 on the number line. Then estimate the location for 5.782. 5.7 5.8 69. Estimate, to the tenth of a degree, the reading of the Fahrenheit thermometer shown. 400 SECTION 5.1

ANSWERS 70. Estimate, to the tenth of a centimeter, the length of the pencil shown. 70. 71. 72. 73. 74. 71. (a) What is the difference in these values: 0.120, 0.1200, and 0.12000? (b) Explain in your own words why placing zeros to the right of a decimal point does not change the value of the number. 75. 76. 77. 72. Lula wants to round 76.24491 to the nearest hundredth. She first rounds 76.24491 to 76.245 and then rounds 76.245 to 76.25 and claims that this is the final answer. What is wrong with this approach? 73. Allied Health A nurse calculates a child s dose of Reglan to be 1.53 milligrams (mg). Round this dose to the nearest tenth of a milligram. 74. Allied Health A nurse calculates a young boy s dose of Dilantin to be 23.375 mg every 5 min. Round this dose to the nearest hundredth of a milligram. In exercises 75 to 77, indicate whether the given statement is always true, sometimes true, or never true. 75. A decimal can be written as a fraction or a mixed number. 76. A decimal written to the thousandth is greater than a decimal written to the hundredth. 77. Zeros can be written to the right of the rightmost decimal place without changing the size of the number. SECTION 5.1 401

Answers 1. Hundredths 3. Ten thousandths 5. 0.23 7. 0.0209 9. 23.056 11. Twenty-three hundredths 13. Seventy-one thousandths 15. Twelve and seven hundredths 17. 0.051 19. 7.3 65 13 21. 23. 5 231 100 or 20 1000 25. 0.69 0.689 27. 1.23 1.230 29. 10 9.9 31. 1.459 1.46 7 7 33. 0.0069, 0.007, 0.0619,, 0.0701, 0.0712, 0.072,, 0.71 100 10 35. 53.5 37. 21.53 39. 0.34 41. 2.718 43. 0.0 45. 4.8534 47. 6.73 49. 6.5874 51. 56.4 53. 56.358 55. 0.44 57. 0.28 59. 0.3 61. 63. 65. 67. 3 4 7.12 7.13 69. 98.6 F 71. 73. 1.5 mg 75. Always 77. Always 402 SECTION 5.1