Significant Figures in Measurements
Scientists report measurements in significant figures. The significant figures in a measurement include all digits that can be known precisely plus a last digit that must be estimated. When you are taking a temperature using a thermometer calibrated in 1o intervals, it is easy to report the temperature to the nearest degree. With this thermometer you can also estimate the temperature to the nearest tenth of a degree. Suppose you estimate a temperature that lies between 35o and 36o to be 35.8o. This number has three significant figures. The first two digits (3 and 5) are known with certainty. The last digit (8) has been estimated. It involves some uncertainty. You are not sure the true temperature is 35o + 0.8o.
Rules for Determining Which Digits in a Measurement are Significant
1. Every nonzero digit in a recorded measurement is significant. The measurements 24.7 m, 0.743 m, and 714 m all express a measure of length to three significant figures. The abbreviation, m, stands for the unit of length, meter.
2. Zeros appearing between nonzero digits are significant. The measurements 7003 m, 40.79 m, and 1.503 m all have four significant figures.
3. Zeros appearing in front of all nonzero digits are not significant. They are acting as place-holders. The measurements 0.0071 m, 0.42 m, and 0.000099 m all have two significant figures.
4. Zeros at the end of a number and to the right of a decimal point are significant. The measurements 43.00 m, 1.010 m, and 9.000 m all have four significant figures.
5. Zeros at the end of a measurement and to the left of the decimal point can be confusing. They are not significant if they just serve as place markers to show the magnitude of the number. The zeros in the measurements 300 m, 7000 m, and 27210 m are probably not significant, but some of them may be. We cannot tell any difference. If these zeros were measured then they are significant. To avoid ambiguity, the measurements should then be written in standard exponential form: 3.00 x 102 m, 7.000 x 103 m, and 2.7210 x 104. In these examples the number of significant figures is three, four, and five, respectively.
Example 1
How many significant figures are in each of these measurements?
a. 123 m d. 4.5600 m
b. 0.123 m e. 0.078 m
c. 40506 m f. 98000 m
Solution:
All nonzero digits are significant. Use rules 2-5 to decide about zeros.
a. 3 (rule 1) d. 5 (rule 4)
b. 3 (rule 3) e. 2 (rule 3)
c. 5 (rule 2) f. 2 (rule 5)
Significant Figures in Calculations
When calculations are done with scientific measurements, we sometimes end up with an answer with more digits than can be justified as significant. Such numbers must be rounded off to make them consistent with the data they represent. An answer cannot be more precise than the least precise measurement. To round off a number, we must first decide how many significant figures it should have. Our decision will depend on the given measurements and on the arithmetic operation used. Once we know the number of significant figures our answer should have, we count that many digits starting on the left. If the digit immediately following the last significant digit is less than 5, all the digits after the last significant place are dropped. If the digit is 5 or greater, the value of the digit in the last significant place is increased by 1. Rounding off 56.212 m to four significant figures gives us 56.21 m, and 56.216 m becomes 56.22 m.
Example 2
Round off each of these measurements to the number of significant figures shown in the parentheses.
a. 314.721 m (4)
b. 0.001775 m (2)
c. 8792 m (2)
Solution:
The underlined digit immediately follows the last significant digit. (Number of significant figures are shown in parenthesis.)
a. 314.721 m; 2 is less than 5; 314.7 m (4)
b. 0.001775 m; 7 is greater than 5; 0.0018 m (2)
c. 8792 m; 9 is greater than 5; 8800 m (2)
Addition and Subtraction. The result of an addition or subtraction can have no more digits to the right of the decimal point than are contained in the measurement with the least number of digits to the right of the decimal point.
Example 3
Do the following operations and give the answer to the correct number of significant figures.
a. 12.52 m + 349.0 m + 8.24 m b. 74.626 m - 28.34 m
Solution:
a. Align the decimal points and add the numbers.
12.52 m
349.0 m
8.24 m
369.76 m
The answer must be rounded off to one digit after the decimal point. The answer is
369.8 m or 3.698 x 102 m
b. Align the decimal points and subtract the numbers.
74.626 m
-28.34 m
46.286 m
The answer must be rounded off to two digits after the decimal point. The answer is
46.29 m or 4.629 x 101 m.
Multiplication and Division. In calculations involving multiplication and division, the answer must contain no more significant figures than the measurement with the least number of significant figures. The position of the decimal point has nothing to do with the number of significant figures.
Example 4
Do the following operations and give the answer to the correct number of significant figures.
a. 7.55 m x 0.34 m c. 2.4526 m ¸ 8.4
b. 2.10 m x 0.70 m d. 0.365 m ¸ 0.0200
Solution:
The calculated answer is given, then rounded off.
a. 2.567 m2 = 2.6 m2 (0.34 m has two significant figures)
b. 1.47 m2 = 1.5 m2 (0.70 m has two significant figures)
c. 0.291976 m = 0.29 m (8.4 has two significant figures)
d. 18.25 m = 18.3 m (both numbers have three significant figures)
Adapted from: Robertson, C. (Ed.) 1987. Chemistry. Addison-Wesley Publishing Company, Inc.
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