# A Detailed Investigation of the Standard Rounding for Multiplication and Division

Christopher L. Mulliss (cmulliss@worldnet.att.net)
Department of Physics and Astronomy, The University of Toledo
Toledo, Ohio, USA

and

Wei Lee (wlee@phys730.cycu.edu.tw)
Department of Physics, Chung Yuan Christian University
Chung-Li, Taiwan

Popular Version of Paper C24.03
Saturday, April 18, 1998, 11:24am
1998 Joint (April) APS/AAPT Meeting, Columbus, Ohio

## Introduction

The use of rounding rules and significant figures is taught to students in virtually all high school and introductory college-level science courses. Despite its widespread use in the education of science students, there is still much confusion about the origin, accuracy, and safety-of-use of these "rules". In a recent note to The Physics Teacher, R. H. Good [1] raises serious questions about the validity and safety of standard rounding rules by pointing out a division problem where the rule causes valuable information to be lost in the calculation. In his note, Good describes a fictional situation where a physicist receives a large grant to determine an important physical constant more precisely than previously known. After much work, the data are given to a technician who, using the standard rounding rules, proceeds to throw away some of the hard-earned information in the calculation of the constant. This situation is clearly unacceptable!

The purpose of this work is to test the accuracy and safety of the standard rounding rule for multiplication and division. While it has been shown that this rounding rule can be inaccurate (Schwartz [2]), this work is the first to quantify its accuracy in a reliable way. Our investigation will show that the standard rounding rule in highly inaccurate, predicting the correct number of significant figures in the result less than 50% of the time! When the rule does fails, it almost always predicts 1 less significant figure in the result than is warranted. Thus, the use of the standard rule often causes valuable information to be discarded in calculations.

## Background Information

The concept of a rounding rule is closely related to that of significant figures. When a number is written in significant figures, as they often are in the physical sciences, each digit is considered to be certain and the number has an implied uncertainty of ± 1/2 in the last decimal place. Because of this implied error, there is an approximate relationship between the number of significant figures and the precision (percent uncertainty) in the quantity. A number written with 1, 2, and 3 significant figures has a precision of approximately 10%, 1%, and 0.1% respectively. This approximate relationship is the justification for the standard rounding rule for multiplication and division.

The standard rounding rule states that the result from a multiplication or division should be written with the same number of significant digits as the least precisely known number used in the computation. For example, the product of a 2-significant-figure number and a 3-significant-figure number should be written with 2 significant figures according to the standard rule.

## Testing the Rounding Rule

A statistical method was used to investigate the accuracy of the standard rounding rule. This method involved the random generation of millions of multiplication and division problems. For each randomly generated problem, the result was calculated and the predictions of the standard rounding rule were applied to the result. If the standard rule predicted the minimum number of significant figures needed to contain all of the valuable information in the result, the rule was said to "work". If the standard rule predicted more or less than this number of digits, then the rule was said to "fail". This method was applied to 1 million multiplication and 1 million division problems and statistics were computed.

Besides the standard rounding rule, there is an often used alternate rounding rule. This alternate rule states that one should always use one more significant figure than suggested by the standard rule. In the division problem discussed by R. H. Good [1], this alternate rule would have protected against the loss of valuable information. In order to investigate this alternate rounding rule, for comparison to the standard rule, it was subjected to the same statistical method described above.

## Results

The application of the standard rule was found to work only 46.4% of the time. The standard rule is, indeed, highly inaccurate. The standard rule was found to predict 1 less significant digit than warranted 53.5% of the time. The standard rule is very dangerous to data, causing valuable information to be lost over half of the time. On very rare occasions (0.05% of the time), the standard rule was found to predict 1 digit too many. The fact that the standard rule can fail is due to its approximate nature, but this is the first work to quantify the accuracy of the standard rounding rule in a reliable way.

The accuracy of the alternate rule was found to be 58.9%, about 13% more accurate than the standard rule. The most important aspect of the alternate rule is, however, the fact that it never discards valuable information. This is supported by a mathematical analysis (not described in this presentation) that shows that the standard rule can, at its worst, be wrong by only ± 1 significant figure. The "extra" significant digit that the alternate rule calls for ensures that it never discards valuable information. Thus, the alternate rule is more accurate and completely safe for data.

## Conclusion

It is shown that the standard rounding rule is highly inaccurate, causing valuable information to be lost over 50% of the time. The alternate rounding rule is shown to be more accurate than the standard rule and completely safe. With no perfect rounding rule possible, the best rounding rule is the simplest rule that is relatively accurate and safe. The alternate rule is superior to the standard rule and should be adopted as the new standard.

This work has been submitted (received Feb. 23, 1998) and accepted (Mar. 11, 1998) for publication in the Chinese Journal of Physics.

## References

[1] R. H. Good, "Wrong Rounding Rule", The Physics Teacher, 34(3), p.192 (1996)

[2] L. M. Schwartz, "Propagation of Significant Figures", Journal of Chemical Education, 62, p.693 (1985)