1. 7 Chapter Decimals: Rational Numbers and Percent
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2. 7-2 Operations on Decimals
Multiplying Decimals
Scientific Notation
Dividing Decimals
Mental Computation
Rounding Decimals
Estimating Decimal Computations Using Rounding
Round-off Errors
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Adding Decimals
Add 2.16 and 1.73.
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Adding Decimals
We can change the problem to one we already know how to solve, that is, to a sum involving fractions.
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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.
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Example 7-4
Compute each of the following: a. (6.2)(1.43) b. (0.02)(0.013)
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Example 7-4 (cont)
Compute each of the following: c. (1000)(3.6)
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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 · 1012.
The mass of an electron, atomic mass units, is expressed as · 10−4.
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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.
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Example 7-5
Write each of the following in scientific notation:
a. 413,682,000
· 108 b
2.31 · 10−5
c. 83.7
8.37 · 101
d. −10,000,000
−(1 · 107)
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Example 7-6
Convert each of the following to standard numerals:
a · 10−5
b · 107
31,200,000
c. −(4.08 · 104)
−40,800
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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
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Dividing Decimals
Divide by 4.
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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.
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Dividing Decimals
When the divisor is not a whole number, as in ÷ 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 This corresponds to rewriting the problem in form (a) as an equivalent problem in form (b), as follows:
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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.
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Mental Computation
Some tools for doing mental computations with whole numbers can be used for decimal numbers:
Breaking and bridging = = =
= 9.68
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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
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Mental Computation
3. Making compatible numbers
9.27
+ 3.79 = =
= 13.06
4. Balancing with decimals in subtraction
4.63
− 1.97 =
= − ( )
= 4.66
= − 2.00
2.66
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Mental Computation
5. Balancing with decimals in division
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Rounding Decimals
Rounding can be done on some calculators using the key.
FIX
To round the number to thousandths, enter
FIX 3
The display will show
Then enter and press the key. The display will show =
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Example 7-9
Round each of the following numbers:
a to the nearest hundredth
7.46
b to the nearest tenth
7.5
c to the nearest unit 7
d to the nearest thousand
7000
e. 745 to the nearest ten
750
f to the nearest ten 70
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23. 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
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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 decimal places than the number used with the fewest decimal places. That is, an answer can be no more accurate than the least accurate number used to find it.
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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.
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