Significant Figures for Physics Help (page 2)
When multiplication or division is done using power-of-10 notation, the number of significant figures in the result cannot legitimately be greater than the number of significant figures in the least-exact expression. You may wonder why, in some of the preceding examples, we come up with answers that have more digits than any of the numbers in the original problem. In pure mathematics, this is not an issue, and up to this point we haven’t been concerned with it. In physics, however, things are not so clear-cut.
Consider the two numbers x = 2.453 × 10 4 and y = 7.2 × 10 7 . The following is a perfectly valid statement in arithmetic:
xy = 2.453 × 10 4 × 7.2 × 10 7
= 2.453 × 7.2 × 10 11
= 17.6616 × 10 11
= 1.76616 × 10 12
However, if x and y represent measured quantities, as they would in experimental physics, the preceding statement needs qualification. We must pay close attention to how much accuracy we claim.
How Accurate Are We?
When you see a product or quotient containing a bunch of numbers in scientific notation, count the number of single digits in the decimal portions of each number. Then take the smallest number of digits. This is the number of significant figures you can claim in the final answer or solution. In the preceding example, there are four single digits in the decimal part of x , and two single digits in the decimal part of y . Thus we must round off the answer, which appears to contain six significant figures, to two. It is important to use rounding and not truncation! We should conclude that
xy = 2.453 × 10 4 × 7.2 × 10 7
= 1.8 × 10 12
In situations of this sort, if you insist on being 100 percent rigorous, you should use squiggly equals signs throughout because you are always dealing with approximate values. However, most experimentalists are content to use ordinary equals signs. It is universally understood that physical measurements are inexact, and writing squiggly lines can get tiresome.
Suppose that we want to find the quotient x/y instead of the product xy? Proceed as follows:
x/y = (2.453 × 10 4 )/(7.2 × 10 7 )
= (2.453/7.2) × 10 −3
= 0.3406944444 ··· × 10 −3
= 3.406944444 ··· × 10 −4
= 3.4 × 10 −4
What About Zeros?
Sometimes, when you make a calculation, you’ll get an answer that lands on a neat, seemingly whole-number value. Consider x = 1.41421 and y = 1.41422. Both of these have six significant figures. The product, taking significant figures into account, is
xy = 1.41421 × 1.41422
This looks like it’s exactly equal to 2. In pure mathematics, 2.00000 = 2. However, not in physics! (This is the sort of thing that drove the famous mathematician G. H. Hardy to write that mathematicians are in better contact with reality than are physicists.) There is always some error in physics. Those five zeros are important. They indicate how near the exact number 2 we believe the resultant to be. We know that the answer is very close to a mathematician’s idea of the number 2, but there is an uncertainty of up to ±0.000005. If we chop off the zeros and say simply that xy = 2, we allow for an uncertainty of up to ±0.5, and in this case we are entitled to better than this. When we claim a certain number of significant figures, zero gets as much consideration as any other digit.
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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