Primer on Significant Digits
It is important to properly utilize significant figures. One of the most clear
indications that a person does not understand the results that he or she is presenting is
when many more digits are used than are significant. Calculations involving measured
data should reflect the precision of the measurements. In general, it is fairly easy to
determine how many significant figures are present in a number by following these rules
(from Chemistry, R. Chang, 1994, McGraw-Hill, Inc., NY and CH111-CH113 General
Chemistry Lab Manual, L. Reilly and P. Long, Dept. of Chemistry, MTU):
• Any digit that is not zero is significant. Thus 834 cm has three significant figures,
1.345 kg has four significant figures, etc.
• Zeros between nonzero digits are significant. Thus, 606 m contains three significant
figures, 46,503 kg contains five significant figures, etc.
• Zeros to the left of the first nonzero digit are not significant. Their purpose is to
indicate the placement of the decimal point. Thus, 0.08 m contains one significant figure,
0.0000468 g contains three significant figures, and so on.
• If a number is greater than 1, then all the zeros written to the right of the decimal point
count as significant figures. For example, 2.0 mg has two significant figures, 30.053 mL
has five significant figures, and 2.050 cL has four significant figures. If a number is less
than 1, then only the zeros that are at the end of the number and the zeros that are
between nonzero digits are significant. For example, 0.090 kg has two significant
figures, 0.3005 L has four significant figures, 0.00420 min has three significant figures,
etc.
•For numbers that do not contain decimal points, the trailing zeros (that is, zeros after the
last nonzero digit) may or may not be significant. Thus 400 cm may have one significant
figure (the digit 4), two significant figures (40), or three significant figures (400). One
cannot know which is correct without more information. By using scientific notation,
however, this ambiguity is avoided. In the previous example, the number can be
expressed as 4 × 102 for one significant figure, 4.0 × 102 for two significant figures, or
4.00 × 102 for three significant figures.
•ADDITION AND SUBTRACTION:
The sum or difference should contain the same number of places to the right of the
decimal point as there are in the value that contains the fewest places to the right of the
decimal point.
Examples:
266.31
5.203
0.1223
(4 sig. figs., 4 places)
+ 823.1
- 0.00420
(3 sig. figs., 5 places)
1094.6
0.1181
(4 sig. figs., 4 places)
(one sig. fig.
after the decimal point)
1
•MULTIPLICATION AND DIVISION
The product or quotient should contain the same number of significant figures as the
value that contains the fewest number of significant figures.
Examples:
250.0/50.000 = 5.000
50.00 x 5.00 = 250. (2.50 × 102)
Once the correct number of significant figures that a result should contain has been
determined, the rounding off procedure is as follows: If the digit to the right of the
significant figure is less than 5, the preceding number is left unchanged. If the digit is 5
or greater, add 1 to the preceding number.
Example:
415.0/9.29 = 44.7
(44.671689 rounded off is 44.7)
If you use your calculator to perform a series of calculations, all digits can be entered and
the rounding off done at the end.
Example:
12.3350 x 3.0045 = 13.828547 = 13.8
2.68
Do not round off at every step of a series of calculations because error is introduced at
each step. This discussion of significant figures refers to measured quantities such as
those obtained in a laboratory setting. Exact numbers are those that are defined (such as
1 gram = 1000 mg) or are counted (such as 2 hydrogen atoms in a water molecule).
These exact numbers are considered to have a large number of significant figures in
calculations involving them.
RELATION BETWEEN INSTRUMENTAL PRECISION AND SIGNIFICANT
DIGITS
It is not meaningful to report numbers that are much smaller than the uncertainty
about that number. Few people would take the offer of a gift of $1.00 + $100 because
they would be as likely to lose money as to gain money. Similarly, to report a value of
1.054368 + 0.23 g is not meaningful. All digits to the right of the first decimal point are
small relative to the uncertainty. The rule to follow is that the smallest digit of the value
reported should occupy the same position relative to the decimal point as the largest digit
of the uncertainty. In the example above, the number would be reported as 1.1 + 0.23.
There are still numerous instances in which analog rather than digital instruments
are used. In general, values are reported with one digit smaller than the smallest
demarcation on the scale. For instance, a graduated cylinder marked every milliliter may
be read to tenths of a milliliter. It is accepted that the second smallest digit is certain, and
the uncertainty is of a magnitude comparable to the smallest digit reported.
2