Significant Figures
47
Significant Figures in Results of Calculations
The results of measurements often are used to calculate some other result. In
such a case, the result of the calculation should be expressed with an appropriate number of significant figures to reflect the reliability of the original measurements. There are two rules for this procedure.
1. Addition or subtraction. The values to be added or subtracted should all be
expressed with the same units. If they are expressed in scientific notation, they
should all be expressed with the same power of 10. The result should be
rounded off so that it has only as many digits after the decimal point as the
number with the fewest digits after the decimal. For example, consider the sum
of the following weights:
13.8426 g
764.5
g
7.08 g
Sum = 785.4226 g
This mathematical sum is a very misleading statement because it contains seven
significant figures. It implies that the total weight is known to the nearest 0.0001 g
when, in fact, one of the weights being added is known only to the nearest 0.1 g
and another only to the nearest 0.01 g. The weight known least reliably (to the
fewest decimal places) limits the reliability of the sum. Therefore, the sum is
properly expressed as 785.4 g. (Note that the number of significant figures in the
weights is irrelevant in applying this rule for addition or subtraction. The proper
result has four significant figures, although one of the weights being added has
only three significant figures. It is the number with the fewest digits after the
decimal point that determines the position of the least reliable digit in the
answer.)
2. Multiplication or division. The product or quotient should be rounded off to
the same number of significant figures as the least accurate number involved in
the calculation. Thus, 0.00296 x 5845 = 17.3, but 0.002960 x 5845 = 17.30.
However, this rule should be applied with some discretion. For example, consider the following multiplication:
0.00296 x 5845 x 93
The rule indicates that the result should be rounded off to two significant
figures, so that the product would be 1600, or 1.6 x 10
3
. However, an error of
± 1 in the value of 93 is not much more significant than an error of ± 1 in 102; we
can say that 93 almost has three significant figures. Because the other numbers
involved all have at least three significant figures, it would be reasonable to
report the result of this multiplication as 1.61 x 10
3 (using three significant
figures). Obviously, such decisions must be made by common sense rather than
47
Significant Figures in Results of Calculations
The results of measurements often are used to calculate some other result. In
such a case, the result of the calculation should be expressed with an appropriate number of significant figures to reflect the reliability of the original measurements. There are two rules for this procedure.
1. Addition or subtraction. The values to be added or subtracted should all be
expressed with the same units. If they are expressed in scientific notation, they
should all be expressed with the same power of 10. The result should be
rounded off so that it has only as many digits after the decimal point as the
number with the fewest digits after the decimal. For example, consider the sum
of the following weights:
13.8426 g
764.5
g
7.08 g
Sum = 785.4226 g
This mathematical sum is a very misleading statement because it contains seven
significant figures. It implies that the total weight is known to the nearest 0.0001 g
when, in fact, one of the weights being added is known only to the nearest 0.1 g
and another only to the nearest 0.01 g. The weight known least reliably (to the
fewest decimal places) limits the reliability of the sum. Therefore, the sum is
properly expressed as 785.4 g. (Note that the number of significant figures in the
weights is irrelevant in applying this rule for addition or subtraction. The proper
result has four significant figures, although one of the weights being added has
only three significant figures. It is the number with the fewest digits after the
decimal point that determines the position of the least reliable digit in the
answer.)
2. Multiplication or division. The product or quotient should be rounded off to
the same number of significant figures as the least accurate number involved in
the calculation. Thus, 0.00296 x 5845 = 17.3, but 0.002960 x 5845 = 17.30.
However, this rule should be applied with some discretion. For example, consider the following multiplication:
0.00296 x 5845 x 93
The rule indicates that the result should be rounded off to two significant
figures, so that the product would be 1600, or 1.6 x 10
3
. However, an error of
± 1 in the value of 93 is not much more significant than an error of ± 1 in 102; we
can say that 93 almost has three significant figures. Because the other numbers
involved all have at least three significant figures, it would be reasonable to
report the result of this multiplication as 1.61 x 10
3 (using three significant
figures). Obviously, such decisions must be made by common sense rather than