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# Significant Figures for Physics Help (page 2)

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By McGraw-Hill Professional
Updated on Sep 17, 2011

## In Addition And Subtraction

When measured quantities are added or subtracted, determining the number of significant figures can involve subjective judgment. The best procedure is to expand all the values out to their plain decimal form (if possible), make the calculation as if you were a pure mathematician, and then, at the end of the process, decide how many significant figures you can reasonably claim.

In some cases the outcome of determining significant figures in a sum or difference is similar to what happens with multiplication or division. Take, for example, the sum x + y , where x = 3.778800 × 10 −6 and y = 9.22 × 10 −7 . This calculation proceeds as follows:

x = 0.000003778800

y = 0.000000922

x + y = 0.0000047008

= 4.7008 × 10 −6

= 4.70 × 10 −6

In other instances, however, one of the values in a sum or difference is insignificant with respect to the other. Let’s say that x = 3.778800 × 10 4 , whereas y = 9.22 × 10 −7 . The process of finding the sum goes like this:

x = 37,788.00

y = 0.000000922

x + y = 37,788.000000922

= 3.7788000000922 × 10 4

In this case, y is so much smaller than x that it doesn’t significantly affect the value of the sum. Here it is best to regard y , in relation to x or to the sum x + y , as the equivalent of a gnat compared with a watermelon. If a gnat lands on a watermelon, the total weight does not change appreciably, nor does the presence or absence of the gnat have any effect on the accuracy of the scales. We can conclude that the “sum” here is the same as the larger number. The value y is akin to a nuisance or a negligible error:

x + y = 3.778800 × 10 4

G. H. Hardy must be thanking the cosmos that he was not an experimental scientist. However, some people delight in subjectivity and imprecision. A gnat ought to be brushed off a watermelon without giving the matter any thought. A theoretician might derive equations to express the shape of the surface formed by the melon’s two-dimensional geometric boundary with surrounding three-space without the gnat and then again with it and marvel at the difference between the resulting two relations. An experimentalist would, after weighing the melon, flick the gnat away, calculate the number of people with whom he could share the melon, slice it up, and have lunch with friends, making sure to spit out the seeds.

Practice problems for these concepts can be found at: Scientific Notation for Physics Practice Test

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