7 Chapter 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.

Adding Decimals Add 2.16 and 1.73.

Adding Decimals We can change the problem to one we already know how to solve, that is, to a sum involving fractions.

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.

Example Compute each of the following: a. (6.2)(1.43) b. (0.02)(0.013)

Example (cont) Compute each of the following: c. (1000)(3.6)

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.

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.

Example Write each of the following in scientific notation: 4.13682 · 108 b. 0.0000231 2.31 · 10−5 c. 83.7 8.37 · 101 d. −10,000,000 −(1 · 107)

Example Convert the following to standard numerals: a. 6.84 · 10−5 0.0000684 b. 3.12 · 107 31,200,000 c. −(4.08 · 104) −40,800

Scientific Notation Calculators with an key can be used to represent numbers in scientific notation. EE For example, to find (5. 2 · 1016) (9.37 · 104), press

Dividing Decimals Divide 128.6 by 4.

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.

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:

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.

Mental Computation Some tools for doing mental computations with whole numbers can be used for decimal numbers: 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

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

Mental Computation 3. Making compatible numbers 9.27 + 3.79 + 3.79 = 9.25 + 0.02 = + 3.75 + 0.04 13.00 + 0.06 = 13.06 4. Balancing with decimals in subtraction 4.63 − 1.97 = 4.63 + 0.03 = − (1.97 + 0.03) = 4.66 = − 2.00 2.66

Mental Computation 5. Balancing with decimals in division

Rounding Decimals Rounding can be done on some calculators using the key. FIX To round the number 2.3669 to thousandths, enter FIX 3 The display will show 0.000. Then enter 2.3669 and press the key. The display will show 2.367. =

Example Round each of the following numbers: 7.46 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

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

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.

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 © 2013, 2010, and 2007, Pearson Education, Inc.