Significant figures in mathematical operations

True or False For any mathematical operation performed on two measurements, the number of significant figures in the answer is the same as the least number of significant figmes in either of the measurements. Explain your answer. 

Compare and contrast the multiplication/division significant figure rule to the significant figure rule applied for addition/subtraction in mathematical operations. 

Solve. This is the easiest part of solving a problem. Once you set up the problem properly and devise a solution map, you follow the map to solve the problem. Carry out mathematical operations (pa5ung attention to the rules for significant figures in calculations) and cancel units as needed. 

Dimensional analysis, sometimes called the factor label (unit conversion) method, is a method for setting up mathematical problems. Mathematical operations are conducted with the units associated with the numbers, and these units are cancelled until only the unit of the desired answer is left. This results in a setup for the problem. Then the mathematical operations can efficiently be conducted and the final answer calculated and rounded off to the correct number of significant figures. For example, to determine the number of centimeters in 2.3 miles ... 

Most likely at one point, the numbers obtained in one s measurements will be used within mathematical operations. What does one do if each number has a different amount of significant figures If one adds 2.0 litres of liquid with 1.000252 litres, how much does one have afterwards What would 2.45 times 223.5 get ... 

Now we will put together the things you have learned about significant figures by considering some mathematical operations in the following examples. 

In general, the number of significant figures that you will report in your calculation will be determined by the /east precise measured value. What values qualify as the least precise me2isurement will vary depending on the mathematical operations involved. 

When we use measured quantities in calculations, the results of the calculation must reflect the precision of the measured quantities. We should not lose or gain precision during mathematical operations. Follow these rules when carrying significant figures through calculations. 

Note that in spite of the mathematical definitions cited, detection limits are rather nebulous quantities. Because they depend on many variables, a factor of 2—3 times uncertainty in the values can be anticipated. They can vary significantly between various manufacturers instrumentation and are especially sensitive to different modes of sample introduction. They can also be modified by the optimization for the determination of specific elements. When performing multielement analyses, a compromise of optimization must be tolerated. This compromise usually results in the achievement of optimal detection limits for only a few elements, with the remainder often being a factor of 2—3 times their optimized values. Also, because detection limits are so dependent on operating parameters, it is prudent to frequently (i.e., with each batch of samples analyzed) compute detection limits to reliably report ultra-trace concentration levels. Care must be taken to not report too many significant figures when stating detection limits, so as to be consistent with the probability level selected in the computation. Typical published detection limits for various types of instrumentation are tabulated in Table 10.1. 

---