Ch 1a
Guidelines for Significant Figures
Updated 11/1/2012 by M. K. Sprague
Introduction
No measurement made in chemistry/physics has infinite precision. For the experimentalists in
the crowd: can you measure the velocity of a beach ball to 10−60 m/s? For the theorists in the
crowd: do you trust your computer program enough to believe computed energies to 10−60 J? The
answer to both questions is no!
You’ll learn all about formal error analysis in your future classes (Ph 77, Ch 6, etc.). For now,
we use significant figures not only to indicate the precision of a measurement, but to estimate the
uncertainty on a value derived from many measurements.
The Cardinal Rule for Significant Figures in Ch 1a
Follow the guidelines given in the first homework set. These rules hold for all problem sets,
quizzes, and examinations in Ch 1a, and may differ from those you find in other sources.
Additional Guidance
•
There is a brief section on significant figures in OGC (6th edition) or OGN (5th edition),
both Appendix A.3
•
It is good practice to keep more digits than necessary in intermediate steps, and round to
the correct number of significant figures at the end of a calculation (when you are
reporting the final result)
•
You should NEVER truncate known constants in such a way that the significant figures
in your calculation are reduced. As an example, don’t use h = 6.626×10-34 J s if the
numbers provided all have 5 or more significant figures.
Specific Mathematics of Significant Figures
•
Trailing and leading zeros: In Ch 1, leading zeros NEVER count as significant figures.
However (and this is different from what a lot of you are used to), trailing zeros
ALWAYS count as significant figures. In scientific notation, it’s simple to figure out the
number of significant figures: it’s simply the number of digits in the significand.
Thus, in Ch 1a:
0.035 has 2 significant figures
1970 has 4 significant figures (NOT 3!!!)
1.97×103 has 3 significant figures
Ch 1a
Guidelines for Significant Figures
Updated 11/1/2012 by M. K. Sprague
•
Sums and Differences: The number of decimal places in your result is equal to the
smallest number of decimal places in the original numbers.
Examples:
•
1 + 0.1 + 0.01 + 0.001 = 1
2 + 0.0078 + 0.1−10 = −8
Products and Quotients: The number of significant figures in your result is equal to the
smallest number of significant figures in the original numbers. Although this is not
rigorously correct when performing more advanced error analysis, it is the rule for Ch 1a.
Examples:
(36.0 ) ( 2.500 ×102 )
9010.0
(37.0 ) ( 2.500 ×102 )
= 0.999
= 1.03
9010.0
That second result should feel unsatisfying (one less decimal place…), but too bad. In Ch
1a, those are the rules we live with. Take Ch 6 or Ph 77 to figure out a better way to deal
with this.
•
Logarithms and Antilogarithms: For y = log ( x ) (the logarithm), the number of digits
after the decimal point in y equals the number of significant figures in x. Conversely, for
x = 10 y (the antilogarithm), the number of significant figures in x equals the number of
digits after the decimal of y.
Examples:
log (3.000 ×10−4 ) = −3.5229
antilog ( −8.1) = 10−8.1 = 8 ×10−9
•
Rounding data: If the last digit is 0, 1, 2, 3, or 4, round down. If the last digit is 5, 6, 7,
8, or 9, round up. This ensures an even chance of rounding your number up or down.
3 SF
1.665 ⎯⎯⎯
→1.67
4 SF
40.732 ⎯⎯⎯
→ 40.73
5 SF
27.96104 ⎯⎯⎯
→ 27.961
4 SF
10.730 ⎯⎯⎯
→10.73
Yes, that last one does count as rounding a number!
Examples: