Name: Kevin O.
How would the following be rounded to three
significant figures using significant figure rules for multiplication
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? Or.....do 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.
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.
The number of significant figures is an incorrigibly ragged measure of the resolution of a
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
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
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
sorry, but "Phooey" and "Hrrmph"
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,
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Update: June 2012