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It is important to be honest when reporting a measurement, so that it = does=20 not appear to be more accurate than the equipment used to make the = measurement=20 allows. We can achieve this by controlling the number of digits, or=20 significant figures, used to report the = measurement.
Determining the Number of = Significant=20 Figures
The number of significant figures in a measurement, such as 2.531, is = equal=20 to the number of digits that are known with some degree of confidence = (2, 5, and=20 3) plus the last digit (1), which is an estimate or approximation. As we = improve=20 the sensitivity of the equipment used to make a measurement, the number = of=20 significant figures increases.
| Postage Scale | 3 =B11 g | 1 significant figure | ||
| Two-pan balance | 2.53 =B10.01 g | 3 significant figures | ||
| Analytical balance | 2.531 =B10.001 g | 4 significant = figures |
Rules for counting significant figures are summarized below.
Zeros within a number are always significant. Both 4308 and = 40.05=20 contain four significant figures.
Zeros that do nothing but set the decimal point are not significant. = Thus,=20 470,000 has two significant figures.
Trailing zeros that aren't needed to hold the decimal point are = significant.=20 For example, 4.00 has three significant figures.
If you are not sure whether a digit is significant, assume that it = isn't. For=20 example, if the directions for an experiment read: "Add the sample to = 400 mL of=20 water," assume the volume of water is known to one significant figure. =
Addition and Subtraction with = Significant=20 Figures
When combining measurements with different degrees of accuracy and = precision,=20 the accuracy of the final answer can be no greater than the least = accurate=20 measurement. This principle can be translated into a simple rule = for=20 addition and subtraction: When measurements are added or = subtracted, the=20 answer can contain no more decimal places than the least accurate = measurement.=20
| 150.0 g H2O | (using significant figures) |
| + 0.507 g salt | |
| 150.5 g solution=20 |
Multiplication and Division = With=20 Significant Figures
The same principle governs the use of significant figures in = multiplication=20 and division: the final result can be no more accurate than the least = accurate=20 measurement. In this case, however, we count the significant figures in = each=20 measurement, not the number of decimal places: When measurements = are=20 multiplied or divided, the answer can contain no more significant = figures than=20 the least accurate measurement.
Example: To illustrate this rule, let's calculate the cost of the = copper in=20 an old penny that is pure copper. Let's assume that the penny has a mass = of=20 2.531 grams, that it is essentially pure copper, and that the price of = copper is=20 67 cents per pound. We can start by from grams to pounds.
We then use the price of a pound of copper to calculate the cost of = the=20 copper metal.
There are four significant figures in both the = mass of the=20 penny (2.531) and the number of grams in a pound (453.6). But there are = only two=20 significant figures in the price of copper, so the final answer can only = have=20 two significant figures.
| Practice Problem 7
Calculate the length in inches of a piece of wood 1.245 feet = long.=20 Determine the correct number of significant figures. |
When the answer to a calculation contains too many significant = figures, it=20 must be rounded off.
There are 10 digits that can occur in the last decimal place in a=20 calculation. One way of rounding off involves underestimating = the=20 answer for five of these digits (0, 1, 2, 3, and 4) and = overestimating=20 the answer for the other five (5, 6, 7, 8, and 9). This approach to = rounding off=20 is summarized as follows.
If the digit is smaller than 5, drop this digit and leave the = remaining=20 number unchanged. Thus, 1.684 becomes 1.68.
If the digit is 5 or larger, drop this digit and add 1 to the = preceding=20 digit. Thus, 1.247 becomes 1.25.