Christopher L. Mulliss^{1} and Wei Lee^{2}
^{1}Department of Physics and Astronomy,
University of Toledo,
Toledo, Ohio 43606, U.S.A.
^{2}Department of Physics, Chung Yuan Christian University,
ChungLi, Taiwan 320, R.O.C.
(Received February 23, 1998)
PACS. 01.30.Pp Textbooks for undergraduates.
PACS. 01.55.+b General physics.
I. Introduction
Introductory physics textbooks often describe a standard
rounding rule for multiplication and division. This rule states that the
proper number of significant figures in the result of multiplication or
division is the same as the smallest number of significant figures in any
of the numbers used in the calculation. Over the years, college students
and teachers of physics have come to rely on this rule. In a recent publication
by Good [1], a simple division problem is discussed where the application
of the standard rounding rule leads to a loss of precision in the result.
Even more disturbing in his note is the long list of popular physics textbooks
at the firstyear university level which advocate the standard rule without
warning its readers that data may be jeopardized because valuable information
can be lost. The fact that rounding rules can lead to such a loss is due
to their approximate nature and has been well documented by Schwartz [2].
Some researchers feel that the approximate nature of significant figures
and rounding rules precludes the need for a detailed investigation of their
effects on error propagation [3]. Other researchers including the authors,
however, recognize the importance of rounding rules as common and convenient
(even if approximate) tools in error analysis [1, 4], especially for students
of introductory Physics and Chemistry. The purpose of this work is, therefore,
to investigate the standard rounding rule for multiplication and division
in detail and to use the results to show that an equally simple rounding
rule exists which is more accurate than the standard rule and always preserves
precision.
In this paper, the theoretical basis for the standard
rounding rule (from a fundamental assumption) is presented. The standard
rounding rule, as applied to the simple multiplication and division problems
x = y · z and x = y / z, is then considered.
A statistical test is applied to quantify the accuracy of the standard
rule, determining the percentage of multiplication and division problems
that fall into the following three categories: those where the true uncertainty
is as large or larger than that predicted by the standard rule but of the
same order of magnitude, those where the true uncertainty is less than
predicted, and those where the true uncertainty is an order of magnitude
larger than predicted. In the first case, the standard rule is said to
"work" because it predicts the minimum number of significant digits that
can be written down without losing precision, and therefore valuable information,
in the result. In the second and third cases the standard rule clearly
"fails," predicting fewer or more significant figures than are needed and,
therefore, losing or overstating precision.
The same analysis is then applied to the often suggested
alternate rule advocating the use of an additional significant figure over
that required by the standard rule. It is shown that it is not possible
to obtain an a priori rule that always works because the proper
number of significant figures depends critically on the result of the calculation.
The alternate rule is, however, shown to be significantly more accurate
than the standard rule for multiplication and nearly as accurate as the
standard rule for division. Most important, the alternate rule is shown
never to lead to a loss of precision.
II. Simple derivation of the standard rounding rule
The standard rounding rule for the multiplication and division of two numbers can be inferred from one very simple assumption. This assumption states that the precision (percentage error) of a number is approximately related to the number of significant figures in that number [5]. The fundamental principles that lead to this derivation are discussed in earlier literature, but not explicitly developed in a rigorous mathematical manner [6]. Written in mathematical form, this assumption expresses the precision in a number x with Nx significant figures in the form:
Precision (x) » 10^{(2 Nx)} %. (1)
As an illustration of this relationship, consider the number 52.37 written
to 1, 2, 3, and 4 significant digits. Following Bevington and Robinson
[7], the absolute error in this number is taken to be ±
½ in the least significant decimal place. See Table I, where the
corresponding absolute error and precision are also included.
While Eq. (1) is only an approximate relationship,
it often gives the correct order of magnitude for the precision. In reality,
the precision is given by the following modified form of Eq. (1):
Precision (x) = Cx · 10^{(2 Nx)} %, (2)
where Cx is a constant that can range from approximately 0.5
to exactly 5 depending on the actual value of the number x. The
value of Cx is almost entirely determined by the value of the first
(and the second, if present) significant digit in the number x;
for example, if x equals 10, 5.0, or 90.0 then the constant Cx
equals 5, 1.0, and approximately 0.556, respectively. The effects of the
leading constant in Eq. (2) will be critical in understanding why the standard
rounding rule sometimes fails.
For the derivation of the standard rounding rule,
consider the simple division problem x = y / z. Through differentiation,
one can easily show that the uncertainty in the ratio x is described
by dx / x = dy / y dz / z. The errors dy
and dz can be directed in the same direction (i.e. having the same
sign) or in the opposite direction. In the simplest approximation, it is
customary to take the maximum uncertainty to be the one quoted, leading
to
max(dx / x) = abs(dy / y) + abs(dz
/ z),
(3)
TABLE I. An illustration of the relationship between
the number of significant figures and the
precision of a quantity.




















or
Precision (x) = Precision (y) + Precision (z). (4)
Substituting the approximate relationship Eq. (1) into Eq. (4) yields
10^{Nx = }10^{Ny }+ 10^{Nz}. (5)
Differentiation of a simple multiplication problem x = y ·
z leads directly to the same relationships as those given for division
in Eqs. (3), (4), and (5).
At this point in the derivation, two separate cases
must be considered: the case where y and z do not have the
same number of significant figures and the case where they do.
Case 1. (Ny ¹ Nz) When Ny does not equal Nz, they must be different by at least 1. Even when they differ by this minimum amount, the term in Eq. (5) involving the smaller value of Ny and Nz is 10 times larger than the other term and thus completely dominates. One is, therefore, justified in replacing Eq. (5) by
10^{Nx = }10^{min(Ny, Nz)}, (6)
implying that Nx = min(Ny, Nz). This is, clearly, a statement of the standard rounding rule.
Case 2. (Ny = Nz) When Ny equals Nz, both terms in Eq. (5) become equally important. Under this condition, Eq. (5) reduces to
10^{Nx = }10^{N + log(2)}, (7)
where N = Ny = Nz. One can see that the integer Nx = Int[N log(2)]. Because log(2) is smaller than 0.5, it can never cause Nx to be rounded down to N 1. Thus Nx = N = min(Ny, Nz), which is, again, the standard rounding rule.
III. A statistical study of the standard rule
III1. The method
To investigate the statistical properties of the
standard rounding rule, a MonteCarlo procedure was used. A computer code
was written in Version 4.0.1 of Research Systems Inc.'s Interactive
Data Language (IDL). The code uses IDL's uniform, floatingpoint, random
number generator [8] to create two numbers. Each of these numbers have
a randomly determined number of significant figures ranging from 1 to 5
and a randomly determined number of places to the left of the decimal point
ranging from 0 to 5. Each digit in these numbers is randomly assigned a
value from 0 to 9, except for the leading digit that is randomly assigned
a value from 1 to 9. Each resulting number can range from the smallest
and least precise value of 0.1 to the largest and most precise value 99999.
The program calculates the product or ratio of the two generated numbers
and determines the number of significant figures which should be kept according
to the standard rule. It then uses Eqs. (2) and (4) to compute the true
precision and converts it into a true absolute error. The absolute error
in the product or ratio, as predicted by the standard rule, is taken to
be ± ½ in the least significant
decimal place [7]. The true absolute error and the value predicted by the
standard rule are compared and the multiplication or division problem is
counted as one of the three cases described previously. The program repeats
the calculation for one million multiplications or divisions and computes
statistics.
III2. Properties of the standard rule
Table II shows the results for the standard rule
as applied to simple multiplication and division problems. Clearly the
standard rule is not accurate. On average, the application of the standard
rule works only 46.4% of the time. Simple multiplication was found to preserve
precision 38% of the time. This is consistent with the work of Schwartz
[2] who estimated that the standard rule predicts the correct number of
significant figures (to an order of magnitude) 37% to 63.3% of the time,
based on a small grid of multiplications of the form x = y^{n}
where n is an integer. Of the times when the standard rule fails,
it has an overwhelming tendency to predict one less significant digit than
needed and, therefore, lead to a loss of precision. Despite this, many
researchers still incorrectly claim that a product or quotient can not
have more significant figures than the smallest number of significant figures
in any of the numbers used in the calculation [9]. The results also show
that it is possible for the standard rule to predict one more significant
digit than is needed a possibility that has been shown by Schwartz [2]
for some multiplications of the form x = y^{n}. While
the fact that the standard rule can fail is well established [1, 2], this
is the first work known to the authors to quantify the success rate of
the standard rounding rule for the general case of simple multiplication
and division.
To illustrate the application of the standard rounding
rule, examples are displayed in Table III from the output of the MonteCarlo
program. For multiplication and division problems, one example is given
for each of the three categories described earlier.
III3. Why the standard rule fails
To see why the rounding rule fails, one must find
the true precision that results from a multiplication or division problem.
To do this, one must substitute Eq. (2) into Eq. (4). Let N = min(Ny,
Nz) and N' = max(Ny, Nz). Let C and
C' be the constants from Eq. (2) that correspond to the numbers
(y or z) with N
TABLE II. The statistical results of the application
of the standard rounding rule to simple
multiplication and division problem. It shows the statistical likelihood
that the
application of the standard rounding rule will fall into each of the three
categories
described in the text.




Worked 



1 More Digit Needed 



1 Too Many Digit 



and N', respectively. If Ny = Nz, then N = Ny = Nz and one can arbitrarily take C to be Cy and C' to be Cz. One can show that this substitution yields
Nx = N + { log(Cx) log[C + C' · 10^{(N' N)}] }. (8)
The two bracketed terms in the above equation determine whether the
standard rule will work for any given problem. If one insists that the
"correct" number of significant digits be the minimum number needed to
preserve precision, then Eq. (8) must be evaluated in such a way that Nx
= N 1 when 2 < SUM £
1, Nx = N when 1 < SUM £
0, and Nx = N + 1 when 0 <
SUM £ 1, where SUM is the sum of the bracketed
terms in Eq. (8).
A careful examination of Eq. (8) provides much insight
into the standard rounding rule. There are several important points that
must be emphasized. Notice that log(Cx) appears in Eq. (8). This
fact alone proves that there is no a priori rule that can be used
to accurately predict the correct number of significant digits to be used
in all cases. In order to know how many significant digits should be used,
one must first know log(Cx) and, therefore, the result of the calculation.
The presence of log(Cx) in Eq. (8) also explains
why the standard rule behaves differently for multiplication and division
problems. The reason lies in the different relationship between Cx,
C, and C' for the two types of problems. As an example, consider
the multiplication and division of the numbers y = 2.7 and z
= 2.6. The results are approximately 7.0 and 1.0, respectively. In both
problems the values of C and C' are approximately 1.9, but
the value of Cx is approximately 0.7 for the multiplication problem
and 5.0 for the division problem. Thus the multiplication and division
of the same two numbers can lead to a very different value for Cx
which can, as a result, affect the evaluation of Eq. (8). There is a second
difference between multiplication and division dealing with order. In the
division of two numbers, switching the numbers can lead to very different
results. These different results can also lead to very different values
for Cx and, therefore, to different evaluations of Eq. (8). Obviously,
this complication does not occur in multiplication problems.
Eq. (8) also shows that the number of significant
digits predicted by the standard rounding can never be more than one digit
away from the correct value. This fact can be seen by studying the bracketed
terms in Eq. (8). Table IV shows the minimum and maximum values of the
sum of the bracketed terms in Eq. (8) as a function of N' N.
When Eq. (8) is evaluated in the manner previously described, the only
possible values for Nx are N 1, N, or N +
1. Thus, the standard rounding rule is never "wrong" by more than one significant
digit.
TABLE III. Comparison of standard rounding rule predictions
(x_{SR}) with true results (x_{true}).
For
simple multiplication and division, one example problem is chosen randomly
from the
output of the MonteCarlo program for each of the categories described
in the text.
The true result of each problem is obtained by assuming an uncertainty
of ± ½ in the
rightmost digit and then propagating the errors.
Multiplication

x = y · z  




Worked 



1 More Digit Needed 



1 Digit Too Many 



Division

x = y / z  




Worked 



1 More Digit Needed 



1 Digit Too Many 



III4. Properties of the alternate rule
Besides the standard rounding rule, there is an
often used alternate rounding rule. This rule requires one to use an extra
significant digit above that suggested by the standard rule. In order to
test the alternate convention, the MonteCarlo procedure described earlier
was applied to the alternate rounding rule.
Table V shows the results for the alternate rule
as applied to simple multiplication and division problems. The alternate
rule is almost as accurate as the standard rule for division, but is significantly
more accurate for multiplication. The average accuracy of the alternate
rule is 58.9% compared with 46.4% for the standard rule. The most significant
aspect of the alternate rule is that it never leads to a loss of precision;
this can be seen in Table V. The reason for this comes from the fact that
the standard rule can, at its worst, predict only one less significant
digit than actually needed. In these cases, the "extra" significant digit
that the alternate rule provides comes to the rescue. Thus, the alternate
rule is more accurate than the standard rule and completely safe for data.
The only problem with the alternate rule is that the results of calculations
may have one or, in rare cases, two too many significant digits. This disadvantage
is minor when compared with the standard rule where precision is lost over
50% of the time.
TABLE IV. The minimum and maximum values of the values
of the sum of the bracketed terms in
Eq. (8) as a function of N N.
N' N  




















TABLE V. The statistical results of the application
of the alternate rounding rule to simple
multiplication and division problems.




Worked 



More Digits Needed 



Too Many Digits 



IV. Summary
Although the best expression for the result of a
calculation includes the precise description of the uncertainty in terms
of the absolute or percentage error, this is often only possible for experimental
data. Many problems encountered by physics students in daily life, including
those in textbooks, do not deal with quantities where the uncertainties
are explicitly stated. In these cases, the number of significant figures
is the only available information upon which to base an error estimate
and a rounding rule becomes useful.
The original purpose of the standard rounding rule
was to provide a method for quoting the results of calculations without
grossly overstating the precision contained therein. This rounding convention
is conservative because it tries to ensure that the true result of a calculation
is included within the error bars implied by the number of significant
digits to which that result is written. Our work shows that the standard
rule is too conservative, leading to a loss of precision over 50% of the
time.
While a simple, accurate, and safe rounding rule
has been called for [1], this work proves that there is no a priori
rule that can accurately predict the number of significant digits in all
cases. With no perfect rounding rule possible, the best rounding rule is
the simplest rule that is relatively accurate and safe. Because the alternate
rule is simple, more accurate than the standard rule, and never leads to
a loss of precision, it is far superior to the standard rule and should
be adopted as the new standard.
Appendix: Generalization to a series of multiplications and divisions
The results of this work can easily be extended to
calculations involving an arbitrary number of variables, m, involved
in a series of multiplications and/or divisions. To apply the alternate
rule to such a series, each step in the calculation must be considered
as a simple multiplication/division and evaluated in turn. The application
of the alternate rounding rule upon this series must, by its nature, preserve
precision because precision is preserved at each step in the calculation.
Consider x = (1.2 / 3.45) · 6.000 as an example. Using
the alternate rule for each step leads to the result x = 4.14 ·
6.000 = 24.84, which has four significant figures (Nx = 4). Now
consider the same example written as x = (1.2 · 6.000) /
3.45 and x = (6.000 / 3.45) · 1.2. The application
of the alternate rule at each step will now lead to x = 24.84 (Nx
= 4) and x = 24.8 (Nx = 3), respectively. Thus, the order
in which the operations are performed may affect the predicted number of
significant figures even though the result of the calculation is unaffected.
Because any prediction made by the application (or series of applications)
of the alternate rule must preserve precision, the smallest predicted number
of significant figures is the minimum number required to preserve precision.
In the preceding example, this number was found to be Nx = N
+ 1, where N is the smallest number of significant figures used
in the calculation (and the number predicted by the standard rule).
The general case of m variables involved
in a series of multiplications and/or divisions can now be considered.
Applying the alternate rounding rule leads to m! / 2 predictions
for Nx, where m! / 2 is the number of unique orderings
of simple multiplications and/or divisions that can reproduce any series.
Extensive numerical testing shows that the alternate rule can yield no
more than m 1 unique values of Nx and that these values
lie between N + 1 and N + (m 1), where N
is the smallest number of significant figures in the m variables.
In the special case where all m variables have the same number of
significant figures, all possible values of Nx are equal to the
minimum value of N + 1. Thus, according to the alternate rounding
rule, the minimum number of significant figures needed to preserve precision
in a series of multiplications and/or divisions is N + 1. The fact
that the alternate rule, in exactly the same form, applies to simple and
multiple operations of multiplication and division makes its application
as simple and straightforward as that of the standard rounding rule. In
practice, the results of a series of multiplications and/or divisions should
be carried out using full calculator results and then rounded to N
+ 1 significant figures.
It is also interesting to consider the special case
investigated by Schwartz [2], x = y^{n}. By generalizing
Eq. (4) to include n terms and substituting Eq. (2) into it, it
can easily be shown that Nx = Ny Int[log(n) + log(Cy
/ Cx)]. This implies that the number of significant figures needed
in the result (x) decreases with increasing number of multiplications,
n. This is exactly the overall behavior that Schwartz [2] observed.
The somewhat erratic deviations observed by Schwartz are due to the log(Cy
/ Cx) term which is very sensitive to the result (x) and can
fluctuate between 1 and +1. As Schwartz also discovered, the operation
x = y^{n} can yield results where the true uncertainty
is an order of magnitude larger than the result written with just a single
significant digit. In the current notation, this occurs when Nx
£ 0 and the precision of the result degrades
to a value that surpasses 100%. Thus, the results of Schwartz can be completely
described and explained by the formalism developed in this paper.
References
[1] R. H. Good, Phys. Teach. 34, 192 (1996).
[2] L. M. Schwartz, J. Chem. Educ. 62, 693
(1985).
[3] B. L. Earl, J. Chem. Educ. 65, 186 (1988).
[4] S. Stieg, J. Chem. Educ. 64, 471 (1987).
[5] J. R. Taylor, An Introduction to Error Analysis:
The Study of Uncertainties in Physical
Measurements, 2nd
ed. (University Science Books, Sausalito, C.A., 1997), pp. 3031.
[6] B. M. Shchigolev, Mathematical Analysis of
Observations (London Iliffe Books, London,
1965), p. 22.
[7] P. R. Bevington and D. K. Robinson, Data Reduction
and Error Analysis for the Physical
Sciences (McGrawHill,
New York, 1992), p. 5.
[8] Park and Miller, Comm. of the ACM 31,
1192 (1988).
[9] See, for example, C. E. Swartz, Used Math
for the First Two Years of College Science,
2nd ed. (American Association
of Physics Teachers, College Park, M.D., 1993), pp. 615.