### Significant Figures What is a significant figure?

```DIFFERENTIATE:
ACCURACY
AND
PRECISION
Can you hit the bull's-eye?
Three targets
with three
arrows each to
shoot.
How do
they
compare?
Both
accurate
and precise
Precise
but not
accurate
Neither
accurate
nor precise
Can you define accuracy and precision?
What would a bullseye with accuracy and
no precision look like?
Significant Figures
What is a significant figure?
• There are 2 kinds of numbers:
APPROXIMATE (OR MEASURED):
weight, height, mass, distance—
anything MEASURED needs
significance attached.
 No measurement is perfect.
EXACT (counted items and
conversion factors have unlimited
significant digits): the amount of
money in your account, jelly beans
in a jar, etc. And #m in a km or in
in a ft
. l2. . . . I . . . . I3 . . . .I . . . . I4. .
First digit (known)
=2
cm
2.?? cm
Second digit (known) = 0.7
2.7? cm
Third digit (estimated) between 0.05- 0.07
Length reported
=
2.75 cm
or
2.74 cm
or
2.76 cm
Known + Estimated Digits
In 2.76 cm…
• Known digits 2 and 7 are 100% certain
• The third digit 6 is estimated (uncertain)
• In the reported length, all three digits
(2.76 cm) are significant including the
estimated one
When to use Significant figures
To a mathematician 21.70,
or 21.700 is the same.
But, to a scientist
21.7cm and 21.70cm
is NOT the same
When to use Significant figures
•When a measurement
is recorded only
those digits that are
dependable are
written down.
Always estimate ONE place past the smallest mark!
Zero as a Measured Number
. l 3. . . . I . . . . I 4 . . . . I . . . . I 5. .
What is the length of the line?
First digit
Second digit
Last (estimated) digit is
cm
5.?? cm
5.0? cm
5.00 cm
Enter question text...
. l8. . . . I . . . . I9. . . .I . . . . I10. .
cm
What is the length of the line?
1) 9.6 cm
2) 9.62 cm
3) 9.63 cm
0%
3
1
compare with your neighbor’s
answer? Why or why not?
0%
2
0%
Significant Figures
The numbers reported in a
measurement are limited by the
measuring tool
Significant figures in a
measurement include the known
digits plus one estimated digit
Counting Significant Figures
RULE 1. All non-zero digits in a measured number
are significant. Only a zero could indicate that
rounding occurred.
Number of Significant Figures
38.15 cm
5.6 m
65.6 kg
122.55 m
4
2
___
___
How many sig figs in 65.8 kg
0%
0%
5
0%
3
0%
2
1
0%
4
1. 1
2. 2
3. 3
4. 4
5. 5
How many sig figs in 122.55 m
0%
0%
5
0%
3
0%
2
1
0%
4
1. 1
2. 2
3. 3
4. 4
5. 5
RULE 2. Leading zeros in decimal numbers are
NOT significant.
Number of Significant Figures
0.008 mm
1
0.0156 mg
3
0.0042 kg
____
0.000262 mL
____
How many sig figs in 0.0042 kg
0%
0%
5
0%
3
0%
2
1
0%
4
1. 1
2. 2
3. 3
4. 4
5. 5
How many sig figs in
0.000262 mL
0%
0%
5
0%
3
0%
2
1
0%
4
1. 1
2. 2
3. 3
4. 4
5. 5
Sandwiched Zeros
RULE 3. Zeros between nonzero numbers are significant.
(They can not be rounded unless they are on an end of a
number.)
Number of Significant Figures
50.8 mm
3
2001 min
4
0.702 kg
____
0.00405 m
____
How many sig figs in 0.702 kg
0%
0%
5
0%
3
0%
2
1
0%
4
1. 1
2. 2
3. 3
4. 4
5. 5
How many sig figs in 0.00405 m
0%
0%
5
0%
3
0%
2
1
0%
4
1. 1
2. 2
3. 3
4. 4
5. 5
Trailing Zeros
RULE 4. Trailing zeros in numbers without
decimals are NOT significant. They are only
serving as place holders.
Number of Significant Figures
25,000 cm
2
200. yr
3
48,600 L
____
25,005,000 g
____
How many sig figs in 48,600 L
0%
0%
5
0%
3
0%
2
1
0%
4
1. 1
2. 2
3. 3
4. 4
5. 5
How many sig figs in 48,600. L
0%
0%
5
0%
3
0%
2
1
0%
4
1. 1
2. 2
3. 3
4. 4
5. 5
How many sig figs in
25,005,000 g
0%
0%
5
0%
3
0%
2
1
0%
4
1. 1
2. 2
3. 3
4. 4
5. 5
Which answers contain 3
significant figures?
1. ) 0.4760
2. ) 0.004706
3. ) 4760
47
60
0%
)
0.
00
47
06
0%
)
)
0.
47
60
0%
All the zeros are significant in
.00370
25,300
2.050 x 103
0%
2.
05
0
x1
0.
..
0%
25
30
0
0%
.0
03
7
1.
2.
3.
534,675 rounded to 3
significant figures is
1. 535
2. 535,000
3. 5.35 x 105
0%
5.
35
x1
05
0%
53
50
00
53
5
0%
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
0%
nd
15
a
0.
00
00
an
d
40
1.
..
0%
40
0.
0
22
.0
an
d
22
.0
...
0%
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
The answer has the same number of decimal
places as the measurement with the fewest
decimal places.
25.2
one decimal place
+ 1.34 two decimal places
26.54
answer 26.5 one decimal place
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
Multiplying and Dividing
Round (or add zeros) to the calculated
answer until you have the same number
of significant figures as the measurement
with the fewest significant figures.
Learning Check
A. 2.19 X 4.2 =
1) 9
B.
C.
2) 9.2
3) 9.198
4.311 ÷ 0.07 =
1) 61.58
2) 62
3) 60
2.54 X 0.0028
=
0.0105 X 0.060
1) 11.3
2) 11
3) 0.041
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
How many sig
figs?
•7
• 40
• 0.5
• 0.00003
5
• 7 x 10
•1
•1
•1
•1
•1
How many sig figs
here?
• 1.2
• 2100
• 56.76
• 4.00
• 0.0792
• 7,083,000,00
0
•2
•2
•4
•3
•3
•4
How many sig figs
here?
• 3401
• 2100
• 2100.0
• 5.00
• 0.00412
• 8,000,050,000
•4
•2
•5
•3
•3
•6
Sig Figs?? 640
0%
0%
6
0%
4
0%
3
0%
2
0%
5
1
2
3
4
5
6
1
a)
b)
c)
d)
e)
f)
Sig Figs?? 200.0
0%
0%
6
0%
4
0%
3
0%
2
0%
5
1
2
3
4
5
6
1
a)
b)
c)
d)
e)
f)
Sig Figs?? 0.5200
0%
0%
6
0%
4
0%
3
0%
2
0%
5
1
2
3
4
5
6
1
a)
b)
c)
d)
e)
f)
Sig Figs?? 1.005
0%
0%
6
0%
4
0%
3
0%
2
0%
5
1
2
3
4
5
6
1
a)
b)
c)
d)
e)
f)
Sig Figs?? 10,000
0%
0%
6
0%
4
0%
3
0%
2
0%
5
1
2
3
4
5
6
1
a)
b)
c)
d)
e)
f)
How many sig
figs?
700
700.
700.00
.007
.00700
•1
•3
•5
•1
•3
How many sig figs
here?
• 1.02 x 2.3
• 210 x 200
• 210. x 210
• .070 x .910
• 0.0791 x 33.1
• 2.3x105 x 200
•2
•1
•2
•2
•3
•1
How many sig figs
here?
• 23 x 2 x 231
• 455 x 21 x 25.2
• 2100.0 x .0005
• 5.00 x 311.22
• 0.00412 x 9.1
•1
•2
•1
•3
•2
Metric Prefixes
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