Transcript Significant Figures

INTRODUCTION
IV. Significant Figures
1
A. Purpose of Sig Figs
Units of Measurement:
• Measurements indicate the magnitude
of something
• Must include:
– A number
– A unit
A. Purpose of Significant
Figures
• There are 2 different types of numbers in our
world
– Exact numbers (counting numbers)
– Measured numbers
• Measured numbers are measured with a tools;
these numbers have ERRORS and are not
exact.
• Errors (called the degree of uncertainty) are a
result of:
– Human error reading the instrument (not
intentional error)
– Limitation in precision of instruments
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1. Exact Numbers
An exact number is obtained when you
count objects or use a defined
relationship.
Counting objects are always exact
2 soccer balls
4 pizzas
Exact relationships, predefined values, not
measured
1 foot = 12 inches
1 meter = 100 cm
For instance is 1 foot = 12.000000000001 inches?
No
1 ft is EXACTLY 12 inches.
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Learning Check
A. Exact numbers are obtained by
1. using a measuring tool
2. counting
3. definition
B. Measured numbers are obtained by
1. using a measuring tool
2. counting
3. definition
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Solution
A. Exact numbers are obtained by
2. counting
3. definition
B. Measured numbers are obtained by
1. using a measuring tool
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Learning Check
Classify each of the following as an exact or
a measured number.
1. 1 yard = 3 feet
2. The diameter of a red blood cell is 7.8 um.
3. There are 6 hats on the shelf.
3. Gold melts at 1064°C.
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Solution
Classify each of the following as an exact (1)
or a measured(2) number.
1. This is a defined relationship.
2. A measuring tool is used to determine
length.
3. The number of hats is obtained by
counting.
4. A measuring tool is required.
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2. Measurement and Significant
Figures
• Every measurement has
a degree of uncertainty.
• The volume, V, at right
is certain in the 1’s
place: 17mL<V<18mL
• A best guess is needed
for the tenths place.
• Two people may
estimate to the tenth
differently
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• We can see the measurement is between 1 and 2
cm
• We can’t see the markings between
• We must guess between 1 and 2
• I estimate the measurement to be 1.6 cm
• The last digit in a measurement is an estimate
• You may estimate differently
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What is the Length?
• We can see the measurement is between 1.61.7cm
• We can’t see the markings between 1.6-1.7
• We must guess between .6 & .7
• I estimate the measurement to be 1.67 cm
• The last digit in a measurement is an estimate
• You cannot make a more precise estimate than
11 to the hundreth using this ruler
Learning Check
What is the length of the wooden stick?
1) 4.5 cm
2) 4.54 cm
3) 4.547 cm
?
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Measured Numbers
• Measurements have a degree of uncertainty that
results from
– _________________
– _________________
• Significant figures are the numbers represented in a
measurement. “Sig Figs” include:
– All of the numbers recorded are known with
certainty
– One estimated digit.
• To indicate the precision of a measurement, the
value recorded should use all the digits known with
certainty.
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The mass of an object is measured using two
different scales. The same quantity is being
described at two different levels of precision
or certainty.
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B. Using Significant Figures
Timberlake lecture plus
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1. Counting Significant Figures
Non Zero Numbers
Number of Significant Figures
38.15 cm
5.6 ft
65.6 lb
122.55 m
4
2
___
___
All non-zero digits in a measured number are
significant
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Leading Zeros
Number of Significant Figures
0.008 mm
1
0.0156 oz
3
0.0042 lb
____
0.000262 mL
____
Leading zeros in decimal numbers (less than
1) are not significant.
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Sandwiched Zeros
Number of Significant Figures
50.8 mm
3
2001 min
4
0.702 lb
____
0.00405 m
____
Zeros between nonzero numbers are significant
Zeros between a non zero and followed by a decimal are
significant
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Trailing Zeros
Number of Significant Figures
25,000 in.
2
200 yr
1
48,600 gal
3
25,005,000 g
____
Trailing zeros in numbers without decimals
are not significant
They are significant if followed by or
preceded by a decimal
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Learning Check
A. Which answers contain 3 significant figures?
1) 0.4760
2) 0.00476
3) 4760
B. All the zeros are significant in
1) 0.00307
2) 25.300
3) 2.050 x 103
C. 534,675 rounded to 3 significant figures is
1) 535
2) 535,000
3) 5.35 x 105
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Solution
A. Which answers contain 3 significant figures?
2) 0.00476
3) 4760
B. All the zeros are significant in
2) 25.300
3) 2.050 x 103
C. 534,675 rounded to 3 significant figures is
2) 535,000
3) 5.35 x 105
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Learning Check
In which set(s) do both numbers contain
the same number of significant figures?
1) 22.0 and 22.00
2) 400.0 and 40
3) 0.000015 and 150,000
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Solution
In which set(s) do both numbers contain
the same number of significant figures?
3) 0.000015 and 150,000
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Learning Check
State the number of significant figures in each of
the following:
A. 0.030 m
1
2
3
B. 4.050 L
2
3
4
C. 0.0008 g
1
2
4
D. 3.00 m
1
2
3
E. 2,080,000 bees
3
5
7
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Solution
A. 0.030 m
2
B. 4.050 L
4
C. 0.00008 g
1
D. 3.00 m
3
E. 2,080,000 bees
3
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2. Significant Numbers in
Calculations
 A calculated answer cannot be more precise
than the measuring tool.
 A calculated answer must match the least
precise measurement.
 Significant figures are needed for final
answers from
1) adding or subtracting
2) multiplying or dividing
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a. Adding and Subtracting
The answer has the same number of
decimal places as the measurement with
the fewest decimal places. The uncertain
digit must reflect precision of
instruments.
25.2
one decimal place
+ 1.34 two decimal places
26.54
answer 26.5 one decimal place
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Learning Check
In each calculation, round the answer to the
correct number of significant figures.
A. 235.05 + 19.6 + 2.1 =
1) 256.75
2) 256.8
3) 257
B.
58.925 - 18.2
=
1) 40.725
2) 40.73
3) 40.7
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Solution
A. 235.05 + 19.6 + 2.1 =
2) 256.8
B.
58.925 - 18.2
3) 40.7
=
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b. Multiplying and Dividing
In carrying out a multiplication or division,
the answer cannot have more significant
figures than either of the original numbers.
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Learning Check
A. 2.19 X 4.2 =
1) 9
2) 9.2
B.
4.311 ÷ 0.07 =
1) 61.58
2) 62
C.
2.54 X 0.0028 =
0.0105 X 0.060
1) 11.3
2) 11
3) 9.198
3) 60
3) 0.041
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Solution
A.
2.19 X 4.2
=
2) 9.2
B.
4.311 ÷ 0.07 =
3) 60
C.
2.54 X 0.0028 =
0.0105 X 0.060
2) 11
Continuous calculator operation =
2.54 x 0.0028  0.0105  0.060
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Review
When reading a measured value, all nonzero
digits should be counted as significant.
There is a set of rules for determining if a
zero in a measurement is significant or not.
• RULE 1. Zeros in the middle of a number are
like any other digit; they are always
significant. Thus, 94.072 g has five
significant figures.
• RULE 2. Zeros at the beginning of a number
are not significant; they act only to locate the
decimal point. Thus, 0.0834 cm has three
significant figures, and 0.029 07 mL has four.
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• RULE 3. Zeros at the end of a number and
after the decimal point are significant. It is
assumed that these zeros would not be
shown unless they were significant. 138.200
m has six significant figures. If the value were
known to only four significant figures, we
would write 138.2 m.
• RULE 4. Zeros at the end of a number and
before an implied decimal point may or may
not be significant. We cannot tell whether
they are part of the measurement or whether
they act only to locate the unwritten but
implied decimal point.
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Practice Rule #1 Zeros
45.8736
•All digits count
0.000239 6
0.000239003
•Leading 0’s don’t
48000.
5
•0’s count in decimal form
5
•0’s don’t count w/o decimal
48000
3.982106 2
1.00040 4
6
•Trailing 0’s do
•All digits count
•0’s between digits count as
well as trailing in decimal form
Practice Rounding
Make the following into a 3 Sig Fig
number
1.5587
0.0037421
1367
128,522
1.6683 106
Your Final number
must be of the same
value as the number
you started with,
129,000 and not 129
Practice Rounding
Make the following into a 3 Sig Fig
number
1.5587
1.56
0.0037421
0.00374
1367
1370
128,522
129,000
1.6683 106
1.67 106
Your Final number
must be of the same
value as the number
you started with,
129,000 and not 129
Examples of Rounding
For example you want a 4 Sig Fig number
4965.03
780,582
1999.5
Examples of Rounding
For example you want a 4 Sig Fig number
0 is dropped, it is <5
4965.03
4965
780,582
780,600 8 is dropped, it is >5; Note
you must include the 0’s
1999.5
2000.
5 is dropped it is = 5; note
you need a 4 Sig Fig
Multiplication and division
32.27  1.54 =
3.68  .07925 =
1.750  .0342000 =
3.2650106  4.858 =
6.0221023  1.66110-24 =
Multiplication and division
32.27  1.54 = 49.6958
49.7
3.68  .07925 = 46.4353312
46.4
1.750  .0342000 = 0.05985
.05985
3.2650106  4.858 = 1.586137  107
1.586 107
6.0221023  1.66110-24 = 1.000000
1.000
Addition and Subtraction
.56
__ + .153
___ =
82000 + 5.32 =
10.0 - 9.8742 =
10 – 9.8742 =
__
Look for the
last
important
digit
Addition and Subtraction
.56
__ + .153
___ = .713
.71
__
82000 + 5.32 = 82005.32
82000
10.0 - 9.8742 = .12580
.1
10 – 9.8742 = .12580
0
Look for the
last
important
digit