Download Chapter 7

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.
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Slide
2
Adding Decimals
Add 2.16 and 1.73.
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Slide
3
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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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.
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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)
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0.013
 0.02
0.00026
Slide
6
Example (cont)
Compute each of the following:
c. (1000)(3.6)
3.6
 1000
3600.0
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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.
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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.
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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)
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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
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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
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Slide 12
Dividing Decimals
Divide 128.6 by 4.
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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.
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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:
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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.
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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
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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
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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
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Slide 19
Mental Computation
5. Balancing with decimals in division
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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.
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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
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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
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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.
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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.
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Slide 25