   # 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 ± 0.0001 g
Pan balance: Weight = 2.1 ± 0.1 g

Often the ± value 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 ("sig figs"). Note that the last digit included is the one with an uncertainty of ±1.

Rules

1. All non-zero digits are significant.

For example:
123. (3 sig figs)

2. Zeros between non-zero digits are significant.

For example:
12.507 (5 sig figs)

3. 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)

4. 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.}

5. 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

 Density = 29.1 143 g = 1.164572 g cm-3 = 1.16 g cm-3 25.0 cm3

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:

 19. 2      g 0. 4745 g 127. g SUM = 146. 6745 g = 147. g because one weight is known only to the nearest 1 g!

NOTE: Round off numbers only at the END of calculations; otherwise, errors may be inadvertently carried through.  Copyright © 2001, 2014 Trustees of Dartmouth College Last updated Friday, September 5, 2014 Please contact the site editor at genchem@webster.dartmouth.edu with questions, comments or suggestions 