The concept of significant
figures (sig figs, s.f.) arises from the concept of physical measuring of a
variable. Instruments we use have finite accuracy and reliability. For example,
a desk ruler has least readable unit (least count) of 1 mm, while a common
Vernier caliper can measure with accuracy of 0.02 mm. Thus the reasonably
reliable length of a steel plate measured with the ruler could be 26 mm (two s.f.),
while measured with the caliper would be 26.14 mm (four s.f.). If one has to
add two measures, let's say a plate measured with a ruler and a plate measured
with the caliper, the ±0.5 mm uncertainty of the first plate will dominate the
uncertainty in length of the second plate ±0.01 mm. When that person reports
the combined length of 52.14 mm he or she cannot defend the final precision up
to 0.02 mm because the ruler's precision of 1 mm dominates. Side-note: some technicians and researchers state
that many elementary measuring devices allow an (experienced) operator to
estimate the measured value one figure beyond the instrument's least count. In
our ruler example, we could estimate the plate's length as 26.1 or 26.2 mm
(three s.f.)
Very often we have to use the measured
values in various calculations which ordinarily give as answers with a long
trail of decimals. While it is perfectly fine to carry the trails during a
multi-step calculation, the final answer must be always rounded off and reported
with a correct number of s.f. In other words, the accuracy of the final answer
cannot exceed the accuracy of the least accurate measurements or data
provided. Four groups of different
arithmetical "operations" have different rules of maintaining s.f.
Those groups are Addition and Subtraction, Division and Multiplication,
Logarithms and Antilogarithms, and Trigonometric functions.
Addition and subtraction can
produce and answer with a higher or lesser s.f.
Division and multiplication
results are limited by the data with least s.f. I have heard respected opinions
that there's an exception to the division and multiplication rule. The opinion
is based on the notion of fractional uncertainty: when an answer begins with a
digit 1, the answer's accuracy would be preserved better if we maintain an
extra s.f. than the original data with lowest s.f. I personally adhere to that
opinion. However, I advice to proceed with caution - it appears that this
"rule of 1" is not widely discussed and often omitted. Please consult
relevant literature or specific industry's guidelines for further use.
In logarithm operations the
mantissa determines the number of s.f.
For purpose of real life applications
trig functions preserve the number of s.f. of the input variable.
A few words on rounding tie-breaks
(when the "digit" we want to cut off is "exactly" 5, e.g.
26.145 or 26.14500 or 26.145000). Many disciplines use the round half up or round half away
from zero rules. Both those rules are asymmetrical and lead to biases. In
science the most accepted tie-breaking rule is round half to even. As per 09/25/2013 1900 the Wikipedia's relevant
page has
a correct explanation with an example.
My favorite intro text on s.f. with
computational examples is Quantitative
Chemical Analysis by Daniel C. Harris.
References:
ASTM Standard E29, 2008, "Standard Practice for Using Significant Digits in Test Data to Determine Conformance with Specifications," ASTM International, West Conshohocken, PA, 2008, DOI: 10.1520/E0029-08, www.astm.org.
Harris, D. C. (2007), Quantitative Chemical Analysis, 7 ed., 663 pp., W. H. Freeman, New York.
Michener, B.; Scarlata, C.; Hames, B. (2008). Rounding and Significant Figures: Laboratory Analytical Procedure (LAP). 7 pp.; NREL Report No. TP-510-42626. http://www.nrel.gov/biomass/pdfs/42626.pdf
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