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Significant Digits

The significant digits (also called significant figures and abbreviated sig figs, sign.figs, sig digs or s.f.) of a number are those digits that carry meaning contributing to its precision. This includes all digits except:

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Use of appropriate significant figures

 

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

  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 102g 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 thestoichiometric 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.


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