Richard C. PinkeHonl and Chester E. Gleit North Carolina State University Raleigh, 27607
The Significance of Significant Figures
I n our world, a person learns to make numerical ca.lculations at an early stage in his education. Only at some higher level in his scientific training does he hump up against the notion of significant figures. He soon learns that his data always have some uncertainty, and this must be taken into account in deriving information from quantitative observations. It is strange that one can find little mention of significant figures in the literature of mathematics. I n fact, we have not been able to find any. The concept is well presented in certain texts in the applied science^.^ I n other texts the idea is ignored completely or treated
' The concepts presented in this paper were developed by the late Professor Pinkerton. The assistance of H. Z. Dokueogus is gmtefully appreciated. For example, SKOOG, D., AND WEST,D. M., "Fundamentds of Analytical Chemistry,'' Holt, Rinehart & Winston, Ine., New York, 1963, or SALZBERG, H. W., MORROW, J. I., AND COHEN, 5. R.. "Laboratorv Course in Physical Chemistrv." " . Academic ~ r e s s ' h c . New , ~ & k 1966. , $ I n the binary number system there are only two digits. Here, lower ease "0" and "in are used to symbolize the zero and unit digits. As in the decimal system, the position of the first non-zero digit, indicates the order of magnitude of the number. "his also constitutes what the chemist usually thinks of ss a qualitative analysis. Any process of analysis has an inherent sensitivity, and the answer to the question, "Is such-and-such an element present?" depends entirely on this sensitivity. The answer may be "no" for one method and "yes" for a more sensitive one. The popular idea that s. qualitative analysis tells us what is present, while s quantitative one tells us how much, is misleading. The two ideas are not that distinct. This is even further emphasized if the logarithm of the fraction is reported, rather than the fraction itself, as will be shown.
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superficially. Convenient rules-of-thumb are often stated and then outrageously violated. Answers to exercises are given to more significant figures than are possible. Few elementary texts in the experimental sciences mention the importance of finding the piece of data which limits the calculation and controls the precision of the results. Hours are wasted in computation. This paper is an attempt to clarify some of our ideas about numerical data. I t discusses the following questions: What is the relationship between precision and the digits we call significant figures? When arithmetic is applied to experimental measurements, is there such a thing as a number with an infinite number of significant digits? Would such a concept not imply that we had an infinite amount of information concerning the value of this measurement? To begin with, a somewhat different view of the idea of precision will he explored. The treatment usually given by the theory of errors has its application but is helpless in the common situation in which only one numerical result is available. Here, the concepts of significant figures, precision, resolution, and inherent informational content of anumber will be given more extensive definitions. The expression of the chemical content of a substance will serve as an example. The concentration of a certain element will be expressed as a fraction. The binary number system will he used.a This will allow us to state the numerical result as yes-or-no answers to a series of question^.^ Also, the connection with information theory will he made clearer. The first ques-
tion will then be whether the sample contains more than 50% of the given element. If the answer is "no," we record 0.0. We next ask whether the sample contains more than 25%. If the answer is again "no," the fractional composition will be o.oo. However, if the element is present to more than 12.5% (or the composition will be set down as o.ooi. The answer will be said to be correct to one significant figure in the binary system. The next step in the process would be to establish the value of the fraction in a more quantitative fashion; in other words, to provide more significant figures. The question will be, "Is the fraction o.ooii or o.ooio?" As this process is continued, more significant figures will bc accumulated. I n theory this may be repeated until as much information as desired is obtained. I n practice some limitation in the physical method will be reached. Much will depend upon the price in time and equipment we wish to pay for the information. At this point, we are always left with some residual uncertainty. If the errors are randomly distributed and if sufficient observations are made, a certain interval may be established in terms of the standard deviation, and confidence limits calculated. A comprehensive statistical treatment may bc employed. A thorough knowledge of statistics is essential to the scientist working in fields such as biology and sociology, where the collection of a large body of data is necessary. Its use becomes more questionable for the analytical chemist who is running three, or perhaps only two, samples. Finally, on more occasions than we realize, only one reading of a particular kind will be made. How often do we bother to replicate simple measnrements of length or take a second glance at the illnminated scale of a modern single-pan balance? One might argue that for single measurements the precision of a particular instrument is known. This is misleading. As we now realize, the observer, the object observed, and the measuring instrument interact, even if only to a small degree. Further, when only one numerical result can be obtained, as is often the case in destructive tests and small sample analysis, how should the values be regarded? It is at this point that we must form a clear idea of the meaning of a significant figure, for the statistician can no longer help us. If we are given a number such as 2.017, we generally assume that the last figure is in doubt, and that the number falls between 2.016 or 2.018. Let us use a somewhat narrower interpretation, for reasons which will become apparent. Assume that the number has been rounded off by some agent, human, machine or natural, and that the value given lies within 2.0170 + 0.0005. This makes the range of error exactlyone unit in the last digit given. Now let us return to a binary example. The number o.ooio will be assumed to be greater than o.oooi but less than o.ooii. The range will be o.nooi; and the relative range, ANIN, will be o.i (or in decimal units). This may be used as an index of precision. A somewhat more useful concept is that of resolution, which is simply the reciprocal of the relative precision. Resolution has the advantage that its numerical value increases as the precision increases, which is somewhat more logical. I n this case, its value is io. (or 2). Unless stated otherwise, every number sequence has a certain
inherent resolution. For example, the number ioo. (or 4) has a resolution of ioo. The number o.ooioo has the same resolution, defined as N / A N . We also note that there are exactly three binary digits in the resolution, or three significant figures. If we allow that this is an inherent quality of the number, it may be expressed unambiguously as follows: F, = 1 log,R, where F2is the number of significant figures and R the resolution, as defined above.
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Now let us examine two numbers in the binary system. The number iooooo.o lies very close to iiiii.i. I n the first case the number has 7 significant figures. The second number, according to the usual convention, has only 6. Yet each has close to the same inherent resolution. If we had obtained these two numbers in a series of measurements from some instrument, we would consider the result as satisfactory. We would not hold the idea that some drastic change in the resolution or the sensitivity of the instrument had occured. If the above formula is used to define significant figures in a continuous and rational way, the number of significant digits in iiiii.i is not 6, but 6.98. These same notions are directly applicable to the familiar decimal system. The number of significant figures, FI0,will be given by the expression: Flo = 1 10gmR. The number 1.0 contains, as always, two significant figures; the number 10.0, three. However, the number 5.0 we would now say contains 2.7 rather than two significant figures. This rational approach eliminates the dilemma of the student who discoven that he has "lost" an order-of-magnitude in precision in going from a 10.0 gram sample to one of 9.9 grams. Further, suppose that we should choose to change the scale of an instrument. For example, without changing the marked lines or divisions on the scale, we might substitute a two for a one, a four for a two, etc. We would not have changed its inherent precision in any way. Neither would the resolution have been changed. Yet if the idea of significant figures is to be associated with resolution, it must also be formulated so that it is independent of the numbering system used. (This assumes, of course, that the base of the number system itself is not changed. A significant figure in the decimal system is worth about 3.32 bits, or significant figures, in the binary system.)
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The foregoing may seem like an over-sophisticated treatment. With the present organization of teaching in the sciences, offering such ideas to the student may seem to place an additional burden on him with little justification. There will develop certain practical benefits. To cite an example, consider the use of significant figures in the operation of multiplication. When multiplying 4 X 4, although each number has one significant figure, the result need not be rounded off to 2 X lo1. Using the logarithmic concept, each number has 1.6 significant figures. The product should be given as 16 + 4, an answer which retains 1.6 significant figures. When multiplying or dividing, the logarithmic nature of resolution gives a much better estimate of the true range to be expected in the result. I n fact, it can be shown that for large resolutions the expected range possible in a calculation approaches exactly that predicted on a logarithmic model for significant figures. The estimation of uncertainty is Volume 44, Number 4, April 1967
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simplified, and the relationship between the result of a calculation and the least precise term is more easily seen. Specifically, we suggest that the student first be introduced to the concept of resolution, or relative precision. After practice in locating the least precise term in an equation, the relationship between this term and the result of the calculation is demonstrated. If desired, the concept of significant figures, conventional rules-of-thumb and the limitations of these rules can then be presented. We are led by these ideas to suspect that there is a difference between the idea of number given to us in mathematics and the use of number in applied work. Many of us, especially those not brought up on the "Xew Math," will be surprised to learn that the definition of number as now accepted by the mathemai tician, due to Frege, was only formulated in 1884.5 Of course, methods of practical calculation have changed as well, but not always in the parallel way. At some stage in our mathematical education we carry with us the concept of an ideal integer such as two. By this we really mean 2.000. . . . Similarly, the fraction is, at times, 0.33333. . . Numbers like a, e, and 2'Ia also possess an infinite number of significant figures, although we can never know all of them. In dealing with measured quantities, however, there are no such numbers, except those arbitrarily defined. For example, there is for us only one such object as a mass of exactly one gram (having associated with it an infinite number of significant figures), and this resides at some Bureau of Standards. Any other object possesses a weight, which is the ratio of the mass of the particular object t o the mass of the standard. Weight is really a dimensionless ratio, and our information about a praticular weight is carried in a number. This number will have a limited or finite number of significant digits in it. This is not to say that there is any real disparity between the correct definition of num-
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ber according to the mathematician and commonly used figures. If we wished to report the amount of an element in a sample, we could, in imagination, give an integer answer if we could only state the exact number of atoms of that element present. The absurdity of this proposition is obvious. Even neglecting the time factor, there are physical laws which limit the precision of any analytical instrument. For the chemist, there is the basic assumption that the carbon isotope 12 has an atomic mass of 12.000. . . with an infinite number of significant digits. This is an arbitrary assignment subject to change, as we well recognize. For all other elemental quantities, including sub-nuclear particles, it should not be considered old-fashioned to refer to atomic weights (not masses). After all, they are ratios, and numerically cannot be determined beyond a certain number of significant figures. In this presentation, two ideas have been emphasized. First of all, a single numerical datum contains an inherent, finite amount of information. As developed in the binary number system, natural for machine and computer use, the information is expressed in terms of bits. But it is also connected with the older concept of significant figures. Second, no matter what number system we are working with, every number may also be said to have certain inherent resolution, or precision. This is an assumption which must be accepted whether or not a statistical approach to an experiment is possible. Just as this resolution is a continuous concept, growing larger as the number of digits obtainable increases, so is the number of significant figures. The idea of significant figures may be related logarithmically to the resolution. If it becomes really necessary to use significant figures, one should take the true nature of this concept into account. 6 RUSSELL,B., ''Intr~duetion t o Mathematical Philosophy," The Maehlillsn Co., New York, 1920, p. 11.
