5.1 Introduction to Decimals, Place Value, and Rounding


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1 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 Some examples of decimal fractions are,, and ,000 In Chapter 1, we talked about the idea of place value. Recall that in our decimal placevalue system, each place has onetenth 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 Hundreds Tens Ones The ones place value is onetenth of the tens place value; the tens place value is onetenth 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 wholenumber part and the fractional part of a decimal fraction Example 2 Writing a Number in Decimal Form Write the mixed number 3 2 in decimal form. 10 Tenths Ones The decimal point 391
2 392 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 onetenth of the value before it. The second place value is hundredths The next place is thousandths, the 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 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 ? 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 is read as five and three hundredths. Hundredths Writing a Decimal Number in Words The rightmost digit, 3, is in the hundredths position.
3 INTRODUCTION TO DECIMALS, PLACE VALUE, AND ROUNDING SECTION is read as twelve and fiftyseven 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 can be read two point five eight can be read zero point six eight nine. Thousandths The rightmost digit, 7, is in the thousandths position is read as five thousand three hundred twentyone ten thousandths. When the decimal has no wholenumber 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 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 This fraction can then be simplified to 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 commonfraction form ,000 Three places Three zeros This mixed number can 29 be simplified to CHECK YOURSELF 5 Write as common fractions or mixed numbers. Do not simplify. (a) (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 is the same as Look at the fractional form: ,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.
4 394 CHAPTER 5 DECIMALS OBJECTIVE 4 Example 6 Which is larger? Comparing the Sizes of Two Decimal Numbers 0.84 or Write 0.84 as Then we see that (or 842 thousandths) is greater than (or 840 thousandths), and we can write CHECK YOURSELF 6 Complete the statement, using the symbol or 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 The number 4.6 is located sixtenths of the distance from 4 to 5. Since each tick mark represents onetenth, we count to the sixth tick mark and draw a dot The number 4.68 is eighttenths of the distance from 4.6 to 4.7. We might estimate its location as: CHECK YOURSELF 7 Plot the number 8.3 on the number line. Then estimate the location for 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.
5 INTRODUCTION TO DECIMALS, PLACE VALUE, AND ROUNDING SECTION Example 8 Rounding to the Nearest Tenth NOTE 3.74 is closer to 3.7 than it is to is closer to 3.8 than it is to is rounded down to the nearest tenth, 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 to the nearest tenth. NOTE Many students find it easiest to mark this digit with an arrow 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 is rounded to CHECK YOURSELF 9 Round to the nearest tenth. Example 10 Round to the nearest hundredth 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 is rounded to Rounding to the Nearest Hundredth
6 396 CHAPTER 5 DECIMALS CHECK YOURSELF 10 Round to the nearest hundredth. Example 11 Rounding to a Specified Decimal Place Round to four decimal places. NOTE The fourth place to the right of the decimal point is the ten thousandths place 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 is rounded to CHECK YOURSELF 11 Round to three decimal places. READING YOUR TEXT The following fillintheblank 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 Thousandths Thousands Ones Hundreds Tens 4. Two and fiftyeight hundredths (a) ; (b) 5 8 1,
7 5.1 Exercises Boost your GRADE at ALEKS.com! For the decimal : 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 SelfTests NetTutor Name Section eprofessors Videos Date Write in decimal form. ANSWERS , , , , Write in words Write in decimal form. 17. Fiftyone thousandths 18. Two hundred fiftythree ten thousandths Seven and three tenths 20. Twelve and two hundred fortyfive thousandths Write each as a common fraction or mixed number SECTION
8 ANSWERS Complete each statement, using the symbol,, or Arrange in order from smallest 34. Arrange in order from smallest to largest. to largest , 0.072,, 0.007, ,, , , , , , , 2.052, Round to the indicated place tenths hundredths hundredths tenths hundredths thousandths thousandths tenths tenths ten thousandths ten thousandths thousandths 398 SECTION 5.1
9 ANSWERS two decimal places three decimal places four decimal places two decimal places Round 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 SECTION
10 ANSWERS In exercises 61 to 64, shade the portion of the square that is indicated by the given decimal Plot (draw a dot) 3.2 and 3.7 on the number line. Then estimate the location for Plot and on the number line. Then estimate the location for Plot and on the number line. Then estimate the location of Plot 5.73 and 5.74 on the number line. Then estimate the location for Estimate, to the tenth of a degree, the reading of the Fahrenheit thermometer shown. 400 SECTION 5.1
11 ANSWERS 70. Estimate, to the tenth of a centimeter, the length of the pencil shown (a) What is the difference in these values: 0.120, , and ? (b) Explain in your own words why placing zeros to the right of a decimal point does not change the value of the number Lula wants to round to the nearest hundredth. She first rounds to and then rounds to 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 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
12 Answers 1. Hundredths 3. Ten thousandths Twentythree hundredths 13. Seventyone thousandths 15. Twelve and seven hundredths or , 0.007, ,, , , 0.072,, F mg 75. Always 77. Always 402 SECTION 5.1
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