Significant Figures in Arithmetic

We now consider how many digits to retain in the answer after you have performed arithmetic operations with your data. Rounding should only be done on the final answer (not intermediate results), to avoid accumulating round-off errors. 

If the numbers to be added or subtracted have equal numbers of digits, the answer goes to the same decimal place as in any of the individual numbers  

The number of significant figures in the answer may exceed or be less than that in the original data. 

If the numbers being added do not have the same number of significant figures, we are limited by the least-certain one. For example, the molecular mass of KrF2 is known only to the third decimal place, because we only know the atomic mass of Kr to three decimal places  

In the addition or subtraction of numbers expressed in scientific notation, all numbers should first be expressed with the same exponent  

Addition and subtraction Express all numbers with the same exponent and align all numbers with respect to the decimal point. Round off the answer according to the number of decimal places in the number with the fewest decimal places. 

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Ouch. A bit of careful arithmetic shows that the speed in the larger tube is 0.144 m/s (we ll keep one additional significant figure in this intermediate answer to prevent a rounding error). If the radius of the large tube is 0.010 m, the area is 3.14 x 10 4 m2. Therefore, the flow rate is ... 

Care is required to determine the appropriate number of significant figures in the result of an arithmetic combination of two or more numbers. ... 

Arithmetic operations that involve measurements are done the same way as operations involving any other numbers. However, the results must correctly indicate the uncertainty in the calculated quantities. Perform all of the calculations, and then round the result to the least number of significant figures in any of the measurements used in the calculations. To round a number, use the following rules. 

The ratio 1 L/10 cm is called a conversion factor because it is a factor equal to 1 that converts a quantity expressed in one unit to a quantity expressed in another unit. Note that the numbers in this conversion factor are exact, because 1 L equals exactly 1000 cm. Such exact conversion factors do not affect the number of significant figures in an arithmetic result. In the previous calculation, the quantity 5.00 cm (the measured length of the side) does determine or limit the number of significant figures. 

Using significant figures in calculations Given an arithmetic setup, report the answer to the correct number of significant figures and round it properly. (EXAMPLE 1.2)... 

How does one determine the significant figures in a number obtained from an arithmetic operation Follow this algorithm for addition and subtraction ... 

The use of the arithmetic functions is fairly obvious, but you should use powers of ten except in trivial cases. To enter 96 500 for instance, consider it 9.65 x 104 and enter 9.65 EE4. (On most calculators F.F4 means x 104). The calculator keeps track of the decimal point and provides an answer between one and ten times the appropriate power of ten. It will usually display many more figures than are significant, and you will have to round off the final result. If at least one factor was entered as a power of ten, the power-of-ten style will prevail in the display, and you need not fear running off scale, nor will any significant figures disappear off scale. 

In the three simple arithmetic operations we have performed, the number combination generated by an electronic calculator is not the answer in a single case The correct result of each calculation, however, can be obtained by rounding off. The rules of significant figures tell us where to round off. 

In arithmetic operations, the final results must not have more significant figures than the least well-known measurement. 

One check we can always do for this type of equilibrium problem is to put the final values back into the equilibrium expression and make sure that they produce a reasonable value for K. In this case, that gives K = (0.0088)/(0.006)(0.041) = 36. This number is not quite 33, but because all of our data have only two significant figures, we should not expect an exact match. Being this close to the original value of K is reassuring and means that we have probably done the arithmetic correctly. 

Example A1.2 shows how significant figures are handled in arithmetic operations. 

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