Subtraction with significant figures

The isotopic molar masses are precise to five or more significant figures, so we are tempted to express the result with five significant figures. The mass defect is determined by addition and subtraction, however, and two of the isotopic molar masses are known to just three decimal places, so the mass defect is precise to three decimal places, and the... 

For addition or subtraction, the limiting term is the one with the smallest number of decimal places, so count the decimal places. For multiplication and division, the limiting term is the number that has the least number of significant figures, so count the significant figures. 

When we add or subtract numbers, the number of significant figures in the answer is limited by the number with the smallest number of significant figures to the right of the decimal, e.g.,... 

All values have a minimum of three significant figures, so the mass of P4O10 is correctly stated with three digits. The mass of excess O2 (17.7 g) is found by subtracting two numbers that are accurate to the first decimal place. Therefore, the mass of excess O2 correctly shows one decimal place. The sum of the oxygen that was consumed (32.3 g) and the given mass of phosphorus (25.0 g) is 57.3 g, the calculated mass of the product phosphorus decoxide. 

Note in Example 6-1 that the difference between and (Sx,) /iV is very small. If we had rounded these numbers before subtracting them, a serious error would have appeared in the computed value of s. To avoid this source of error, never round a standard deviation calculation until the very end. Furthermore, and for the same reason, never use Equation 6-5 to calculate the standard deviation of numbers containing five or more digits. Use Equation 6-4 instead. Many calculators and computers with a standard deviation function use a version of Equation 6-5 internally in the calculation. You should always be alert for roundoff errors when calculating the standard deviation of values that have five or more significant figures. 

When discussing significant figures earlier, we stated that the relative uncertainty in the answer to a multiplication or division operation could be no better than the relative uncertainty in the operator that had the poorest relative uncertainty. Also, the absolute uncertainty in the answer of an addition or subtraction could be no better than the absolute uncertainty in the number with the largest absolute uncertainty. Without specific knowledge of the uncertainties, we assumed an uncertainty of at least 1 in the last digit of each number. 

Check Note that in parts (a) and (c) we made the energy units in free energy changes (kJ) consistent with those in R (J). Based on the rules for significant figures in addition and subtraction, we retain one digit to the right of the decimal place in part (c). 

When carrying measured quantities through calculations, the least certain measurement limits the certainty of the calculated quantity and thereby determines the number ofsign -cant figures in the final answer. The final answer should be reported with only one uncertain digit. To keep track of significant figures in calculations, we will make frequent use of two rules, one for addition and subtraction, and another for multiplication and division. 

For addition and subtraction, the result has the same number of decimal places as the measurement with the fewest decimal places. When the result contains more than the correct number of significant figures, it must be rounded off. Consider the following example in which the uncertain digits appear in color ... 

Notice that we have neglected the very small concentration of H (aq) due to H2O autoionization. Notice also that the amount of HCOOH that ionizes is very small compared with the initial concentration of the acid. To the number of significant figures we are using, the subtraction yields 0.10 M ... 

The guidelines used to determine the number of significant figures in addition and subtraction are different from those for multipfication and division. In addition and subtraction the result can have no more deciinal places than the measurement with the fewest nianber of decimal places. In the following example the uncertain digits appear in color ... 

The answer obtained by multiplication or division must contain the same number of significant figures as the quantity with the fewest significant figures used in the calculation. The answer obtained by addition or subtraction must contain the same number of places to the right of the decimal as the quantity in the calculation with the fewest number of places to the right of the decimal. 

Do the following additions and subtractions, and write the answers with the correct number of significant figures ... 

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