Use of appropriate significant figures
While it is important to know how accurate a measurement is, it
is also important to convey one's degree of confidence in the accuracy
of a number when making an observation or carrying out a calculation.
This can be accomplished by noting numerical observations or quoting
results to the correct number of significant figures.
e.g. weighing the same object with different balances:
Analytical balance: Weight = 2.1234 ± 0.0001g
Triple beam balance: Weight = 2.1 ± 0.1g
Usually the ± is dropped, and it is understood that the number quoted has an uncertainty of at least 1 unit in the last digit. All digits quoted are called significant figures. Note that the last digit included is the one with an uncertainty of ±1.
Rules
- All non-zero digits are significant.
For example:
123 (3 sig figs)
- Zeros between non-zero digits are significant.
For example:
12.507 (5 sig figs)
- Zeros to the left of the first non-zero digit are not significant.
For example:
1.02 (3 sig figs)
0.12 (2 sig figs)
0.012 (2 sig figs)
- If a number ends in zeros to the right of the decimal point, those zeros are significant.
For example:
2.0 (2 sig figs)
2.00 (3 sig figs) {This signifies greater accuracy.}
- If a number ends in zeros to the left of the decimal point,
those zeros may or may not be significant.
For example:
If we make a statement that the weight of an object is 120 g,
how do we convey our knowledge of whether the balance was accurate
to ± 1 g or ± 10 g?
Answer: The ambiguity can be removed by using exponential notation.
The weight can be expressed as 12 x 101 g or 1.2 x 102 g if we wish to quote unambiguously to 2 sig figs, and 12.0 x 101 g or 1.20 x 102 g if we have a confidence level extending to 3 sig figs.
Note: We cannot write 120.0 g since this requires known accuracy
of ± 0.1 g.
These rules apply to measured quantities which are non-exact. If
you are told that a number is exact then there is no uncertainty; i.e. the number is good to an infinite number of sig figs. Exact integers fall in this category.
An important example of an exact quantity is the coefficient of a reagent in a chemical reaction. This number, called the stoichiometric coefficient, expresses the specific number of molecules of reagent A which undergo reaction with a specific number of molecules of reagent B. The stoichiometric coefficients are exact.
2 H2 + O2 --------> 2 H2O
Calculations with numbers having different accuracies
Multiplication or Division: the result can have no more sig figs than the least accurate number. For example:
If an object has mass of 29.1143 g and a volume of 25.0 cm3, then its density is given by
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Density = 29.1
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143 g = 1.164572 g cm-3 = 1.16 g cm-3
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25.0
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cm3
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Addition or Subtraction: the result must be reported to the same number of decimal places as the number with the fewest decimal places. For example:
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19.
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2 g
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0.
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4745 g
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127.
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g
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SUM = 146.
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6745 g = 147 g because one weight is known only to the
nearest 1 g!
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NOTE: Round off numbers only at the END of calculations; otherwise, errors may be inadvertently carried through.
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