Significant Figures and Rounding Rules

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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

1998 Paper: "On the Standard Rounding Rule for Multiplication and Division" by Christopher Mulliss & Wei Lee, Chinese Journal of Physics, 36(3), 479-487.

"Lay Language Paper" A non-technical description of the paper.

PDF Reprint Download the PDF version of the paper as it appeared in print.

HTML Reprint Includes corrections of several minor typos that appeared in print.

Referee's Report

Comments See what textbook writers have to say about the paper.

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.

PDF Reprint Download the PDF version of the paper as it appeared in print.

Relating the Rounding Rules for Addition and Multiplication.

Implications. The new insights provided by this research suggest that changes are needed in the way that rounding rules are taught.

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.

Portuguese Rounding Rule for Multiplication and Division More information on the Portuguese Rounding Rule that is studied, along with the standard and alternate rules, in this paper.

PDF Reprint Download the PDF version of the paper as it appeared in print.

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.

PDF Reprint Download the PDF version of the paper as it appeared in print.

Related Links

Math Skills Review: Significant Figures. Part of the Texas A&M University Department of Chemistry Site. A good site that covers all of the basics.

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!

Introductory Physics Students' Treatment of Measurement Uncertainty. This is a Ph.D. thesis by Duane Deardorff (Physics, North Carolina State University, 2001) that is focused on Science Education! I am encouraged to see "Physics Education" work being done from within Physics departments.

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.