Transcript Chapter 3

3
Numbers in
the Real
World
Copyright © 2015, 2011, 2008 Pearson Education, Inc.
Chapter 3, Unit C, Slide 1
Unit 3C
Dealing with
Uncertainty
Copyright © 2015, 2011, 2008 Pearson Education, Inc.
Chapter 3, Unit C, Slide 2
Significant Digits
Type of Digit
Significance
Nonzero digits
Always significant
Zeros that follow a nonzero digit
and lie to the right of the decimal
point (as in 4.20 or 3.00)
Always significant
Zeros between nonzero digits (as in Always significant
4002 or 3.06) or other significant
zeros (such as the first zero in 30.0)
Zeros to the left of the first nonzero
digit (as in 0.006 or 0.00052)
Never significant
Zeros to the right of the last
Not significant unless stated
nonzero digit but before the decimal otherwise
point as in (40,000 or 210)
Copyright © 2015, 2011, 2008 Pearson Education, Inc.
Chapter 3, Unit C, Slide 3
Example
State the number of significant digits and the implied
meaning of the following numbers.
a. a time of 11.90 seconds
b. a length of 0.000067 meter
c. a weight of 0.0030 gram
d. a population reported as 240,000
e. a population reported as 2.40 * 105
Copyright © 2015, 2011, 2008 Pearson Education, Inc.
Chapter 3, Unit C, Slide 4
Example (Solution)
a. a time of 11.90 seconds
 4 significant digits and implies a measurement to
the nearest 0.01 second
b. a length of 0.000067 meter
 2 significant digits and implies a measurement to
the nearest 0.000001 meter. Note that we can
rewrite this number as 67 micrometers, showing
clearly that it has only 2 significant digits
Copyright © 2015, 2011, 2008 Pearson Education, Inc.
Chapter 3, Unit C, Slide 5
Example (Solution)
c. a weight of 0.0030 gram
 2 significant digits. The leading zeros are not
significant because they serve only as
placeholders, as we can see by rewriting the
number as 3.0 milligrams. The final zero is
significant because there is no reason to include it
unless it was measured.
d. a population reported as 240,000
 the number has 2 significant digits and implies a
measurement to the nearest 10,000 people
Copyright © 2015, 2011, 2008 Pearson Education, Inc.
Chapter 3, Unit C, Slide 6
Example (Solution)
e. a population reported as 2.40 × 105
3 significant digits. Although this number means
240,000, the scientific notation shows that the first
zero is significant, so it implies a measurement to
the nearest 1000 people.
Copyright © 2015, 2011, 2008 Pearson Education, Inc.
Chapter 3, Unit C, Slide 7
Example
For each of the following operations, give your
answer with the specified number of significant
digits.
a. 7.7 mm × 9.92 mm; give your answer with 2
significant digits
b. 240,000 × 72,106; give your answer with 4
significant digits
Solution
a. 7.7 mm × 9.92 mm = 76.384 mm2. Because we
are asked to give the answer with 2 significant
digits, we round to 76 mm2.
Copyright © 2015, 2011, 2008 Pearson Education, Inc.
Chapter 3, Unit C, Slide 8
Example
b. 240,000 × 72,106; give your answer with 4
significant digits
Solution
b. 240,000 × 72,106 = 1.730544 × 1010. Because
we are asked to give the answer with 4 significant
digits, we round to 1.731 × 1010.
Copyright © 2015, 2011, 2008 Pearson Education, Inc.
Chapter 3, Unit C, Slide 9
Types of Measurement Error
Random errors occur because of random and
inherently unpredictable events in the
measurement process.
Systematic errors occur when there is a
problem in the measurement system that
affects all measurements in the same way,
such as making them all too low or too high by
the same amount.
Copyright © 2015, 2011, 2008 Pearson Education, Inc.
Chapter 3, Unit C, Slide 10
Size of Errors
The absolute error describes how far a measured
(or claimed) value lies from the true value.
absolute error = measured value – true value
The relative error compares the size of the error to
the true value.
absolute error
relative error 
true value

measured value  true value
 100%
true value
Copyright © 2015, 2011, 2008 Pearson Education, Inc.
Chapter 3, Unit C, Slide 11
Example
Find the absolute and relative error: A projected
budget surplus of $17 billion turns out to be $25
billion at the end of the fiscal year.
absolute error = measured value – true value
= $25 billion – $17 billion = $8 billion
relative error
absolute error

true value
= $8 billion / $17 billion ≈ 0.471 = 47.1%
Copyright © 2015, 2011, 2008 Pearson Education, Inc.
Chapter 3, Unit C, Slide 12
Describing Results
Accuracy describes how closely a
measurement approximates a true value. An
accurate measurement has a small relative
error.
Precision describes the amount of detail in a
measurement.
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Chapter 3, Unit C, Slide 13
Combining Measured Numbers
Rounding rule for addition or subtraction:
Round the answer to the same precision as the
least precise number in the problem.
Rounding rule for multiplication or division:
Round the answer to the same number of
significant digits as the measurement with the
fewest significant digits.
To avoid errors, round only after completing
all the operations, not during the intermediate
steps.
Copyright © 2015, 2011, 2008 Pearson Education, Inc.
Chapter 3, Unit C, Slide 14
Example
A book written in 1985 states that the oldest Mayan
ruins are 2000 years old. How old are they now?
Solution
A book written in 1985 is roughly 30 years old, so
we might be tempted to add 30 years to 2000 years
to get 2030 years for the age of the ruins. However,
2000 years is the less precise of the two numbers: It
is precise only to the nearest 1000 years, while 30
years is precise to the nearest 10 years. Therefore,
the answer also should be precise only to the
nearest 1000 years:
Copyright © 2015, 2011, 2008 Pearson Education, Inc.
Chapter 3, Unit C, Slide 15
Example (cont)
A book written in 1985 states that the oldest Mayan
ruins are 2000 years old. How old are they now?
2000 yr
precise to nearest 1000

30 yr
precise to nearest 10

 2000 yr
2030 yr
must round to nearest 1000
Given the precision of the age of the ruins, they are
still 2000 years old, despite the 30-year age of the
book.
Copyright © 2015, 2011, 2008 Pearson Education, Inc.
Chapter 3, Unit C, Slide 16
Example (cont)
The government in a town of 82,000 people plans to
spend $41.5 million this year. Assuming all this
money must come from taxes, what average
amount must the city collect from each resident?
Solution Find the average tax by dividing the $41.5
million, which has 3 significant digits, by the
population of 82,000, which has 2 significant digits.
The population has the fewest significant digits, so
the answer should be rounded to match its 2
significant digits.
Copyright © 2015, 2011, 2008 Pearson Education, Inc.
Chapter 3, Unit C, Slide 17
Example (cont)
$41,500,000  82,000 
3 sign. digits
2 sign. digits
$506.10
round to 2 sign digits
 $510 per person
The average resident must pay about $510 in taxes.
Copyright © 2015, 2011, 2008 Pearson Education, Inc.
Chapter 3, Unit C, Slide 18