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Calculations using Significant Figures

from http://chemistry.about.com/library/weekly/aa082701a.htm

Uncertainty in Calculations

Measured quantities are often used in calculations. The precision of the calculation is limited by the precision of the measurements on which it is based.

• Addition and Subtraction
  When measured quantities are used in addition or subtraction, the uncertainty is determined by the absolute uncertainty in the least precise measurement (not by the number of significant figures). Sometimes this is considered to be the number of digits after the decimal point.

  Example
  32.01 m
  5.325 m
  12 m
  
  Added together, you will get 49.335 m, but the sum should be reported as '49' meters.

   

  Example #1                                2.311		Example #2                                37.438
  -2.11		-6.50
  Answer not rounded-->	0.201	Answer not rounded-->	30.938
  Rounded to sig figs-->	0.20	Rounded to sig figs-->	30.94

   

  Multiplication and Division
  When experimental quantities are multiplied or divided, the number of significant figures in the result is the same as that in the quantity with the smallest number of significant figures. If, for example, a density calculation is made in which 25.624 grams is divided by 25 mL, the density should be reported as 1.0 g/mL, not as 1.0000 g/mL or 1.000 g/mL.

  Advanced Sig. Fig. Calcs. ***Combined Operations

  Remember to follow the order of operations. Be sure to remember to include only the sig. figs. before going on to the next operation.

Losing Significant Figures

Sometimes significant figures are 'lost' while performing calculations. For example, if you find the mass of a beaker to be 53.110 g, add water to the beaker and find the mass of the beaker plus water to be 53.987 g, the mass of the water is 53.987-53.110 g = 0.877 g
The final value only has three significant figures, even though each mass measurement contained 5 significant figures.

Rounding and Truncating Numbers

There are different methods which may be used to round numbers. The usual method is to round numbers with digits less than '5' down and numbers with digits greater than '5' up (some people round exactly '5' up and some round it down).

Example:
If you are subtracting 7.799 g - 6.25 g your calculation would yield 1.549 g. This number would be rounded to 1.55 g, because the digit '9' is greater than '5'.

In some instances numbers are truncated, or cut short, rather than rounded to obtain appropriate significant figures. In the example above, 1.549 g could have been truncated to 1.54 g.

Exact Numbers

Sometimes numbers used in a calculation are exact rather than approximate. This is true when using defined quantities, including many conversion factors, and when using pure numbers. Pure or defined numbers do not affect the accuracy of a calculation. You may think of them as having an infinite number of significant figures. Pure numbers are easy to spot, because they have no units. Defined values or conversion factors, like measured values, may have units. Practice identifying them!

Example:
You want to calculate the average height of three plants and measure the following heights: 30.1 cm, 25.2 cm, 31.3 cm; with an average height of (30.1 + 25.2 + 31.3)/3 = 86.6/3 = 28.87 = 28.9 cm. There are three significant figures in the heights; even though you are dividing the sum by a single digit, the three significant figures should be retained in the calculation.

Practice Problems

Metric System  Metric conversions  Accuracy Precision Dimensional Analysis Scientific Notation  Significant Figures  Significant Figures in Calculations  Density

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