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Chapter 7 Rational Numbers as Decimals and Percent Copyright © 2016, 2013, and 2010, Pearson Education, Inc. 7-2 Operations on Decimals • How concrete models, drawings, and strategies can be used to develop efficient algorithms for decimal operations. • Exponential and scientific notation for decimals. • Strategies for decimal mental computations and estimations. Copyright © 2016, 2013, 2010 Pearson Education, Inc. Slide 2 Adding Decimals Add 2.16 and 1.73. Copyright © 2016, 2013, 2010 Pearson Education, Inc. Slide 3 Adding Decimals We can change the problem to one we already know how to solve, that is, to a sum involving fractions. Copyright © 2016, 2013, 2010 Pearson Education, Inc. Slide 4 Multiplying Decimals If there are n digits to the right of the decimal point in one number and m digits to the right of the decimal point in the second number, multiply the two numbers ignoring the decimals, and then place the decimal point so that there are m + n digits to the right of the decimal point in the product. Copyright © 2016, 2013, 2010 Pearson Education, Inc. Slide 5 Example Compute each of the following: a. (6.2)(1.43) b. (0.02)(0.013) 1.43 2 digits 6.2 1 digit 286 858 8.866 (2+1=3 digits) Copyright © 2016, 2013, 2010 Pearson Education, Inc. 0.013 0.02 0.00026 Slide 6 Example (cont) Compute each of the following: c. (1000)(3.6) 3.6 1000 3600.0 Copyright © 2016, 2013, 2010 Pearson Education, Inc. Slide 7 Scientific Notation Scientists use scientific notation to handle either very large or very small numbers. For example, the distance light travels in one year is 5,872,000,000,000 miles, called a light year, is expressed as 5.872 · 1012. The mass of an electron, 0.00054875 atomic mass units, is expressed as 5.4875 · 10−4. Copyright © 2016, 2013, 2010 Pearson Education, Inc. Slide 8 Definition Scientific Notation In scientific notation, a positive number is written as the product of a number greater than or equal to 1 and less than 10 and an integer power of 10. To write a negative number in scientific notation, treat the number as a positive number and adjoin the negative sign in front of the result. Copyright © 2016, 2013, 2010 Pearson Education, Inc. Slide 9 Example Write each of the following in scientific notation: a. 413,682,000 4.13682 · 108 b. 0.0000231 2.31 · 10−5 c. 83.7 8.37 · 101 d. −10,000,000 −(1 · 107) Copyright © 2016, 2013, 2010 Pearson Education, Inc. Slide 10 Example Convert the following to standard numerals: a. 6.84 · 10 −5 b. 3.12 · 107 c. −(4.08 · 104) 0.0000684 31,200,000 − 40,800 Copyright © 2016, 2013, 2010 Pearson Education, Inc. Slide 11 Scientific Notation Calculators with an EE key can be used to represent numbers in scientific notation. For example, to find (5. 2 · 1016) (9.37 · 104), press Copyright © 2016, 2013, 2010 Pearson Education, Inc. Slide 12 Dividing Decimals Divide 128.6 by 4. Copyright © 2016, 2013, 2010 Pearson Education, Inc. Slide 13 Dividing Decimals When the divisor is a whole number, the division can be handled as with whole numbers. The decimal point can be placed directly over the decimal point in the dividend. Copyright © 2016, 2013, 2010 Pearson Education, Inc. Slide 14 Dividing Decimals When the divisor is not a whole number, as in 1.2032 ÷ 0.32, we can obtain a whole-number divisor by expressing the quotient as a fraction, and then multiplying the numerator and denominator of the fraction by 100. This corresponds to rewriting the problem in form (a) as an equivalent problem in form (b), as follows: Copyright © 2016, 2013, 2010 Pearson Education, Inc. Slide 15 Dividing Decimals In elementary school texts, this process is usually described as “moving” the decimal point two places to the right in both the dividend and the divisor. Copyright © 2016, 2013, 2010 Pearson Education, Inc. Slide 16 Mental Computation Some tools for doing mental computations with whole numbers can be used for decimal numbers: 1. Breaking and bridging 1.5 + 3.7 + 4.48 1.5 + 3 = 4.5 + 0.7 + 4.48 4.5 + 0.7 = 5.2 + 4.48 5.2 + 4 = 9.2 + 0.48 = 9.68 9.2 + 0.48 Copyright © 2016, 2013, 2010 Pearson Education, Inc. Slide 17 Mental Computation 2. Using compatible numbers Decimal numbers are compatible when their sum is a whole number. 7.91 3.85 4.09 + 0.15 12 + 4 16 Copyright © 2016, 2013, 2010 Pearson Education, Inc. Slide 18 Mental Computation 3. Making compatible numbers 9.27 = 9.25 + 0.02 + 3.79 = + 3.75 + 0.04 13.00 + 0.06 = 13.06 4. Balancing with decimals in subtraction 4.63 = 4.63 + 0.03 = 4.66 − 1.97 = − (1.97 + 0.03) = − 2.00 2.66 Copyright © 2016, 2013, 2010 Pearson Education, Inc. Slide 19 Mental Computation 5. Balancing with decimals in division Copyright © 2016, 2013, 2010 Pearson Education, Inc. Slide 20 Rounding Decimals Rounding can be done on some calculators using the FIX key. To round the number 2.3669 to thousandths, enter FIX 3 The display will show 0.000. Then enter 2.3669 and press the = display will show 2.367. Copyright © 2016, 2013, 2010 Pearson Education, Inc. key. The Slide 21 Example Round each of the following numbers: a. 7.456 to the nearest hundredth 7.46 b. 7.456 to the nearest tenth 7.5 c. 7.456 to the nearest unit 7 d. 7456 to the nearest thousand 7000 e. 745 to the nearest ten 750 f. 74.56 to the nearest ten 70 Copyright © 2016, 2013, 2010 Pearson Education, Inc. Slide 22 Estimating Decimal Computations Using Rounding Rounded numbers can be useful for estimating answers to computations. When computations are performed with rounded numbers, the results may be significantly different from the actual answer. Other estimation strategies, such as front-end, clustering, and grouping to nice numbers, that were investigated with whole numbers also work with decimals Copyright © 2016, 2013, 2010 Pearson Education, Inc. Slide 23 Round-off Errors Round-off errors are typically compounded when computations are involved. When computations are done with approximate numbers, the final result should not be reported using more significant digits than the number used with the fewest significant digits. Copyright © 2016, 2013, 2010 Pearson Education, Inc. Slide 24 Round-off Errors Non-zero digits are always significant. Zeroes before other digits are non-significant. Zeroes between other non-zero digits are significant. Zeroes to the right of a decimal point are significant. To avoid uncertainty, zeroes at the end of a number are significant only if to the right of a decimal point. Copyright © Copyright 2013, 2010,©and 2007,2013, Pearson Education, Inc.Education, 2016, 2010 Pearson Inc. Slide 25