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Name: Kevin O.
Status: educator
Age: 40s
Location: N/A
Country: N/A
Date: 3/22/2004

How would the following be rounded to three significant figures using significant figure rules for multiplication and division?

9.996701232 x 10-3 mol Ga

If you round between the final 9 and the 6 your answer ends up as 10.00 x 10-3 mol Ga which would then be written as 1.000 x 10-2 mol Ga as per significant figure rules for scientific notation. Would you then round it to 1.00 x 10-2 mol Ga to retain three significant figures? you get to now keep the extra digit?

Three significant figures would be 1.00 x 10-2. The apparent uncertainty is then 5 parts in 1000. If you had started with 9.9867 x 10-3, you would have arrived at 9.99 x 10-3, with an apparent uncertainty of 5 parts in 9990 -- a factor of nearly ten less. If this bothers you, and I think it should, then you see the problem with using the number of significant figures as the sole vehicle for an uncertainty estimate. If you're serious about uncertainty, you estimate the error directly, provide it with the number, and propagate it in calculations involving the number. There's a good body of work on propagation of error estimates; the thinking behind it is based on sampling from distributions. Usually, you can get a pretty good estimate of statistical uncertainty, and are limited in the end by your knowledge of systematic errors affecting your measurement.

Tim Mooney

To me 10.00 X 10-3 and 1.000 X 10-2 indicate four significant figures. The second would be proper scientific notation. And 1.00 X 10-2 would indicate three significant figures and be the proper answer to your question as well as proper notation.

Your question is subject to interpretation and opinion... and perhaps a bit of creative privilege. Perhaps we can get enough responses to have it put to a vote.

Larry Krengel

The number of significant figures is an incorrigibly ragged measure of the resolution of a number. It will never be a perfect academic gospel. Personally I would label it a temporary exercise, and be tolerant of reasonable choices.

Taken literally, both 1.00e-2 and 9.99e-3 are three significant figures, even though their self-proportional resolution is different by a factor of nearly ten. So perhaps you make an exception with yourself, and ignore leading "1.". Then your 10:1 discontinuity happens at the border of 1.999e-2 / 2.00e-2.

What we would like to have handy is a way to make the uncertainty be 1/1000 of the value, regardless of the value. If we expressed a value by its logarithm, (for log base 10: 100 as "2.000", and 300 as "2.477"), then we would have 1/1000 and 1/100 and 1/1,000,000 well expressed, but we couldn't conveniently get 1/50 or 1/200.

Log base 2 would give us resolutions about as far apart as the familiar 1-2-5-10 sequence, a real nice distance, but most of us do not want to be binary.

Any kind of shorthand description of uncertainty will form kinks. This is irritating to me, because I like language in the abstract and want it to work shiningly well. So I do not think your question is interesting, I think it is difficult and nearly irrelevant, being answerable only by arbitrary rules with common sense problems.

You might as well pick the answer dictated by simple consistent rules, regardless of unreasonable consequences.

sorry, but "Phooey" and "Hrrmph"

Jim Swenson

Hi Kevin,

The way I teach significant figures, I would consider 1.000 x 10-2 to be 4 significant digits. Remembering that the final zeroes may or may not be significant depending on how the measurement was made. If I knew how you got your 9.996 etc. I could give you a better answer. If three significant figures is all you get, stick with the 1.00 x 10-2.

Hope this helps,

Martha Croll

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