Significant Figures
I. What are significant figures?
Consider how you would measure the length of the solid in the figure
shown. The right edge of the solid lies somewhere between the 1.30
and 1.40 cm marks. You can estimate the length as 1.38 cm, but there is
some uncertainty in the last digit. In chemistry, we record the digits we
know with certainty as well as the uncertain digit. For this number,
there are three significant figures, two of which are certain (1.3) and the
last one which is somewhat uncertain (8).
[https://chem.libretexts.org/@api/deki/files/28424/4ea
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II. Determining the Number of Significant Figures
When we work with numbers, we frequently must determine how many significant figures they contain. How do
we do that? First, we must remember that the non-zero digits are always significant, no matter where they occur.
The only problem in counting significant figures, then, is deciding whether a zero is significant. To do so, you can
use the following rules:
Significant Figure Rules
1. A zero alone in front of a decimal point is not significant; it is used simply to make sure we do not
overlook the decimal point (0.602)).
2. A zero to the right of the decimal point but before the first non-zero digit is a place marker and is
not significant (0.00602). If you write the number in scientific notation, leading non-significant
leading zeros are no longer part of the number: 6.02103.
3. A zero between non-zero digits is significant (602, 0.00602).
4. A zero at the end of a number and to the right of the decimal point is significant (0.006020,
602.0).
5. A zero at the end of a number and to the left of the decimal point (6020) may or may not be
significant. It may be precisely known, and thus significant, or it may simply be a place holder. To
communicate the number of significant figures in a number such as 6020, write it in scientific
notation: The number 6.02103 has three significant figures, while 6.020103 has four significant
figures.
The following table summarizes the significant figures in the numbers we just considered. The digits that are
significant are highlighted.
Number
Count of significant figures
0.602
3
0.00602
3
602
3
0.006020
4
602.0
4
6020
3 or 4
3
6.02103
3
4
6.02010
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Practice: Determine the number of significant figures in each measurement:
a) 40.70 m
b) 0.0100 L
c) 205 g
d) 200 kg
Solutions:
a) 4 significant figures; Rules 3 and 4
b) 3 significant figures; Rules 1, 2 and 4
c) 3 significant figures; Rule 3
d) 1, 2, or 3 significant figures; Rule 5
III. Significant Figures in Calculations
When you use measured values in calculations, it is important to express calculated values with the correct number
of significant figures. How do we know how many significant figures a calculated value should have? The rules you
should follow depend on the type of calculation:
Multiplication and Division: In a multiplication or division problem, the product or quotient must have the
same number of significant figures as the least precise number in the problem, i.e. the number with less
significant figures. Consider the following example:
3.6 cm × 5.18 cm = ?
The first number has two significant figures, and the second has three. Since the calculated answer must match
the number with the fewest significant figures, it should have two significant figures:
3.6 cm × 5.18 cm = 8.8 cm2 (not 8.78 cm2)
Practice: Express the answers to the following operations with the proper number of significant figures:
a) 1.52 m × 318.2 m
b)
9.85  103 g
1.4 104 cm3
c)
5.10 g
0.9775 g/cm3
Solutions:
a) 1.52 m × 318.2 m = 484 m2
9.85  103 g
= 0.70 g/cm3
b)
1.4 104 cm3
c)
5.10 g
= 5.22 cm3
0.9775 g/cm3
Addition and Subtraction: A sum or difference can only be as precise as the least precise number used in
the calculation. Thus, we round the sum or difference to the first uncertain digit. If we add 1.125 cm to 9.5
cm, for instance, we get 10.6 cm. There is uncertainty in the tenths position of 9.5, and this uncertainty
carries over to the answer. In this case, it is not the number of significant figures that is important but
rather the place of the last significant digit.
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Consider another example:
0.08044 + 12.01 = ?
The first number has four significant figures and the second number has five significant figures. However,
this information does not determine the number of significant figures in the answer. If we line these
numbers up, we can see which digits can be added to give an answer that has significance:
0.08044
+ 12.01
––––––––––––––––
12.09 044
The digits after the 8 in the first number have no corresponding digits in the second number; so it is not
possible to add them and obtain digits that are significant. The calculated value is 12.09, which has two
decimal places, dictated by the measurement with the greatest uncertainty (12.01).
Practice: Express the answers to the following operations with the proper number of significant figures:
a) 9.5015 g + 2.66 g
b) 10.0 mm – 0.922 mm
Solutions: Express the answers to the following operations with the proper number of significant figures:
a) 9.5015 g + 2.66 g = 12.16 g
b) 10.0 mm – 0.922 mm = 9.1 mm
Significant Figures in Unit Conversions: Exact Numbers
When converting a measurement value from one unit to another, we often use conversion factors that are
exact numbers. Examples of conversion factors that are exact relationships are
1 ft = 12 in
1 in = 2.54 cm
1 m = 100 cm
1 L = 1000 mL
Because conversion factors such these are exact, they have an infinite number of significant figures.
Therefore, when we use such a conversion factor to convert from one unit to another, the number of
significant figures is determined by the number of significant figures in the measurement.
For example, when we convert 3.00 feet to centimeters, the two conversion factors used are exact so the
result of the calculation should have three significant figures since 3.00 ft has three significant figures:
3.00 ft ×
12 in
2.54 cm
= 91.4 cm (3 significant figures)
×
1 ft
1 in
As another example, the conversion of 10.00 milliliters (mL) to liters (L) involves an exact conversion
factor so the result of the calculation should have four significant figures since 10.00 mL has four
significant figures:
10.00 mL ×
1L
= 0.01000 L (4 significant figures)
1000 mL
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