Significant Figures Help
Introduction to Significant Figures
When multiplication or division is done using power-of-10 notation, the number of significant figures (also called significant digits ) in the result cannot legitimately be greater than the number of significant figures in the least-exact expression.
Consider the two numbers x = 2.453 × 10 4 and y = 7.2 × 10 7 . The following is a valid statement in pure arithmetic:
But if x and y represent measured quantities, as they would in experimental science or engineering, the above 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 quantities in scientific notation, count the number of individual numerals (digits) in the decimal portions of each quantity. Then identify the quantity with the smallest number of digits, and count the number of individual numerals in it. That’s the number of significant figures you can claim in the final answer or solution.
In the above example, there are four single digits in the decimal part of x, and two single digits in the decimal part of y. So 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:
No More Squigglies
In science and engineering, approximation is the rule, not the exception. If you want to be rigorous, therefore, you must use squiggly equals signs whenever you round off any quantity, or whenever you make any observation. But writing these squigglies can get tiresome. Most scientists and engineers are content to use ordinary equals signs when it is understood there is an approximation or error involved in the expression of a quantity. From now on, let us do the same. No more squigglies!
Suppose we want to find the quotient x/y in the above situation, instead of the product xy. Proceed as follows:
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:
This looks like it’s exactly equal to 2. In pure mathematics, 2.00000 = 2. But not in physics! (This is the sort of thing that drove the purist G.H. Hardy to write that mathematicians are in better contact with reality than are physical scientists.) Those five zeros are important. They indicate how near the exact number 2 we believe the resultant to be. We know 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 that. 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:
But in other instances, 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 , while y = 9.22 × 10 –7 . The process of finding the sum goes like this:
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 appreciably change in practical terms, 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
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