MEASUREMENTS 49 



measurements can never be quite as precise and, therefore, never quite 

 as accurate as the physicist's. 



One might suggest that better instruments would detect smaller differ- 

 ences and be able to give more precise values. This is true only to a cer- 

 tain point. Eventually random fluctuations in atomic or molecular struc- 

 ture, kinetic activity of particles, or other such changes become as large 

 as the differences we are trying to detect. The human ear, for example, 

 is an extremely sensitive detector of slight variations in pressure. The 

 natural kinetic or thermal movement of air molecules causes very small 

 changes in pressure when the molecules strike the tympanic membrane. 

 If the ear were only a trifle more sensitive it would detect the bombard- 

 ment by random movement of single air molecules, and all sound would 

 be superimposed on a steady rumble. This delicate instrument is the 

 product of evolution, but the instruments built by man are subject to 

 the same limitations. There is a limit beyond which attempts to refine 

 measurements are pointless. 



Significant Digits: The number of significant figures resulting from 

 a measurement is an indication of the precision of the instrument. If 

 we weigh a pebble on a triple beam balance we find that it weighs 4.7 g. 

 We realize that this number means that the actual mass is between 4.65 

 g and 4.74 g. When we use a torsion balance to weigh the same pebble, 

 we find a weight of 4.72 g. This obviously represents a value between 

 4.715 and 4.724, but we cannot tell whether the pebble actually weighs 

 slightly more or less than 4.72 g. On a good analytical balance, we might 

 weigh to the nearest tenth of a milligram, expressing the result as 4.7208 

 g. These five figures are significant, revealing the precision of the meas- 

 urement. It would be a mistake to weigh on the triple beam balance, 

 obtain a weight of 7.2 g, and then write the weight as 7.2000 g. Zeroes 

 placed after the decimal point are counted as significant figures indica- 

 tive of precision. In the case of large numbers, like 563,000,000, only the 

 first three figures have any meaning. It is interesting to speculate on the 

 precision required in a financial institution like a bank or the steps re- 

 quired to preserve accuracy and precision in taking the decennial United 

 States Census. 



More significant digits become available only by improving the meas- 

 uring technique. Calculation, especially multiplication and division, 

 tends to increase the number of figures, but the final result can be no 

 more precise than the least precise of the individual measurements. 



Measurements can never be more precise nor more accurate than the 

 standards used in the measurement. I once saw a bottle of a standard 



