Significant Figures and Rounding Rules
"This web site is dedicated to the
discussion of issues relating to the proper teaching
of significant figures and rounding rules in high school and college science
courses." - Christopher Mulliss (page Author)
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Rounding rules for arimethic operations are not perfect. We therefor recommend
that rounding rules are taught in a way that conveys the following information:
1. The fact that rounding rules are useful approximations but that they can fail.
2. The accuracy of the rounding rule.
3. The ways that a given rounding rule can fail (called failure modes).
4. How often each failure mode occurs.
In order to live up to our own standards, we have created tables that summarize our body of work and present
educators and students alike with the information that they need to apply and fully understand the rounding rules that they
use everyday.
Standard Rounding Rules Table showing the "standard" rounding rules
for many common operations: multiplication and division, addition and subtraction, common (base-10) and natural
logarithms, and common (base-10) and natural exponential functions.
Recommended Rounding Rules Table showing our recommended rounding rule
for each operation. The recommended rules were chosen based on their simplicity, accuracy, and "safety". A "safe" rounding rule does not
predict fewer significant digits than are justified because doing so causes valuable information to be discarded in
the rounding process.
Higher-Accuracy Rounding Rules Table showing the several higher-accuracy
rounding rules. These rounding rules emphasis accuracy over simplicity.
The Work of Christopher Mulliss and Dr. Wei Lee’s Team1
In 1996, I read a note in The Physics Teacher
by R. H. Good. In this note, Good describes a division problem that
he discovered by accident in which the standard rounding rule caused precision
to be lost in the result of the calculation. In other words, the
standard rounding rule caused a digit to be discarded even though we were
justified in keeping it! Thus, the "rule" that we all use is potentially
dangerous to our data - an unacceptable situation!!
This note caused me to question the standard rounding
"rule" for multiplication and division. Where did this rule comes
from? How often does it fail? How can it fail? Is there
a better alternative? How is the rule for multiplication and division
related to the rule for addition and subtraction? With the help of Dr. Wei Lee, I began a detailed investigation this so-called
rule that resulted in the answers to these fundamental and important questions.
The results prove to be quite surprising!
1 Department of Physics, Chung Yuan Christian University in Chung-Li, Taiwan
Links Related to Our Work
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1998 Paper: "On the Standard Rounding Rule for
Multiplication and Division" by Christopher Mulliss & Wei
Lee, Chinese Journal of Physics, 36(3), 479-487.
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PDF Reprint Download the
PDF version of the paper as it appeared in print.
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HTML Reprint Includes corrections
of several minor typos that appeared in print.
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Comments See what textbook
writers have to say about the paper.
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2000 Paper: "On the Standard Rounding Rule for
Addition and Subtraction" by Wei Lee, Christopher Mulliss, &
Hung-Chih Chiu, Chinese Journal of Physics, 38(1), 36-41.
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PDF Reprint Download the
PDF version of the paper as it appeared in print.
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Implications. The new insights provided
by this research suggest that changes are needed in the way that rounding
rules are taught.
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2004 Follow-up Paper: "On Various Kinds of Rounding Rules for Multiplication and Division"
by Wei-Da Chen, Wei Lee, and Christopher Mulliss, Chinese Journal of Physics, 42(4-I), 335-346.
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PDF Reprint Download the
PDF version of the paper as it appeared in print.
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2005 Paper: "On the Rounding Rules for Logarithmic and Exponential Operations"
by Wei-Da Chen, Wei Lee, and Christopher Mulliss, Chinese Journal of Physics, 43(6), 1017-1034.
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PDF Reprint Download the
PDF version of the paper as it appeared in print.
Related Links
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Wikipedia's Significance Arithmetic Entry. (a.k.a. rounding rules)
This page is fairly balanced and accurate and it lists some alternate approaches such as "interval arithmetic", the "crank three times" method, and the "Monte Carlo" method.
The external links are, however, entirely negative to any use of significant figures. I will trust readers to make up their own minds!
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NIST.
National Institute of Standards and Technology (NIST) Technical Note 1297 (1994 Edition): Guidelines for Evaluating
and Expressing the Uncertainty of NIST Measurment Results.
Suggestions are more than
welcome!
This page is maintained by Christopher
Mulliss who is solely responsible for its contents.
