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Barbara Anderlik
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The Leelanau School Glen Arbor, MI
Whatever became of Significant Figures? The trend toward numerical illiteracy
Watching the evolution of calculations for the last 25 years, from pencil and uaper throuph slide rule, and on to calculators--suddenly Ifeel things are getting out of hand. For some time, the slide rule held the student to a modest number of digits, hut now the calculator has opened a pandora's box, and the number of digits a student can produce from a problem is limited only hy the width of his paper. The Chem Study course was my first real exposure to the derivation and use of uncertainties in measurements and the subsequent use of significant digits in lab experiments. I can recall the frustration of the chemistry teachers themselves a t Dr. Lippincott's Ohio State Summer Institute when we were required to work out the results from the early labs using the maximum-minimum error and the percentage methods. The derivation of significant figures proved to he too much for most of my chemistry students, so after a couple of years, I settled on a compromise. I require uncertainties on all measurements and on~ suhsrauent or subtractions. ~ . -~ -~ -. ~ ~~~additions ~ In multiplication and division, I recommend they use one more digit -. hut the final " than is significant when cornoutins. value must he reported using the correct number of significant figures-no more, no less. For many years I taught slide rule a t the heginning of my chemistry course, and most students could be convinced that the slide rule did not rob them of valid numbers in lab calculations. But things have changed. For example, in arithmetic class the product of 532 X 621 is 330,372, and most calculators will light up to give a t least that many digits. To the student, that 372 is sacred. How does one persuade students, when working with physical measurements, to part with these excess numbers and become numerically literate? My attack, the first week of school, begins with a Measurements Lab (given in Appendix I). Although this lab is designed to acquaint the student with metric measurements and lab eauioment. it is alsoan excellent . . device for the introdrwtion ot r~nwrtainriesin me:isurementi and significant 1'1rurcs.The~mly insrrwriun before the li~hin " " this regard is to read every measurement to the smallest calibration and olace this value after the measurement as a f : 1.34 f 0.01 g'or 10.2 i 0.2 ml, etc. No propagation of these uncertainties is mentioned. After the labs are turned in, a sample set of data for the metal har in Exercise 1is placed on the hoard and calculated using significant figures correctly. ~~~~~~
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The following item from the Chem Ed Compacts section of this Journal IFeh. 19771 is a prime examnle of simificant figure abuse: The circumference of a basketball is 30 in. or 76.2 cm (National Federation of State High School Associations Basketball Rulebook, 1975-76 edition: Rule 1-12. o.7.)
How did an increase of three digits from the data to the molar volume occur? Are significant figures important to anyone hut me? There are no questions on the ACS-NSTA High School standardized exam concerned with uncertainties or significant figures. I've never seen them mentioned in your high school pages. Is it going to be worthwhile in college for the high school graduate to have mastered this skill?
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Appendix I Experiment 1: Measurements Purpose: T o get acquainted with laboratory equipment and metric measurements, and an introduction to uncertainty in measurements. General instructions: Four Exercises are set up in the Chemistry labs. Data, the measurements taken in the laburatmy, should be written directly in your lab notebook. (The organization for several are suggested below in the Exercise instructions.) All measurements and calculated values should be identified as to Exercise number and have a unit attached. The Exercises may be done in any order. Label all measurements with an uncertainty which corresponds to the smallest calibration. Ex. 13.2 f 0.1 ml Exercise I Materials: wooden blocks, metal bars, ruler, balance. (a) Find density of any one wooden block. (h) Find density of any one metal bar. Volume = w X I X h might Density = volume
Wood block #
Height
1.9 f 0.1 cm Width of bar Length 15.1 1 0 . 1 cm 0.6 i 0.1 cm (limits final value to 1 sig. fig.) Thickness 17 cm" (sensible to use an extra digit here) Volume Weight of bar 145.16 i 0.01 g Density of bar 8 gicm"
Examples of densities read off student papers illustrate the instinctive misuse of significant numbers of digits. Since the textbook I am currently using (Silver, Burdett), has no section on significant figures, I pass out a sheet with that information (given in Appendix 11). From then on, a guaranteed part of each quantitative lab score is based on the correct use of significant figures. The lesson is not an easy one and student resistance is great and prolon~edin many cases. It seems to he completely a ~ a i n sthe t instin& except f ~ iar small percent of the better students who grasp the common sense of the whole procedure.
Metal bar #
Width Length
Volume Weight DenSiN
E w r ~ . i s I1 e Materials: Burets, eudiometers (gas measuring tubes), thermumeters (O.lO,0.2'. l o Celsius), aneroid barometer, mercury barometer, mercury in small graduate. Take readings at all set-ups on all tables. Carefully identify all values with units. ./ ~ Y V ? " ~ < OI l l hlntrrmli: 3 l r r a k r r j wirh hquidi. graduated eyl~ndrri, extra h a ken, Im1;me. ! 1 ' 1 LKC ~ grdu:+t? ~ ~ .4 fur liquid A, H for 13, rt?., Find t w d m s ~ t wC U I t h e r h r Iiquds ~ A, I\, and (.'. ~
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Volume 57. Number 8, August 1980 / 591
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Suggested data table: Liquid
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Volume of liquid used Weight of liquid and container Weight of empty container
Weight of liquid Densit" of ii""id Exercise IV Materials: 5-ml graduated pipet, 50-ml Erlenmeyer flask, 10-ml graduate, 100-mI graduate, 1/2-I graduate, water. Fill the Erlenmeyer flask using successive fills of the graduated pipet to determine its total capacity (filled to the top). Repeat the measurement using the 10-ml graduate. Next fill the flask and measure its volume by pouring the water into the 100-ml graduate. Repeat with the 112-1 graduate. Be sure to record each meaaurement. Erercise V (Optional) Take one of the rectangles of aluminum foil. Using the density of aluminum as 2.70 g/ml, and only the equipment available on the table (balance and ruler), determine the thickness of the foil. (Show calculations.)
Appendix II Significant Figures A platform balance is a common laboratory instrument used to measure mass, and it is usually precise to 0.1 g. Let us sssume that the mass of an object weighed on a platform balance is 56.3 g. Since the balance is precise to i O . l g, the last digit of 56.3 is uncertain. Significant digits in a number are all the digits in that number known with certainty in addition to the first digit which is uncertain. A zero, however, is significant only under certain conditions. If it is used to locate a decimal point, it is not a significant figure. The
592 1 Journal of Chemical Education
measurement 0.0065 g, for example, has only two significant figures. A zero to the left of an understood decimal point and t o the right of a nonzero digit is not simificant unless a bar is ulaced above it. 6500 km has two s7gnificant f;gures, however, 6580 km has three significant firmres. If a zero a o o e m between nonzero dipits.. the -~ - zero ~ ~is simificnnt. ~-Thus the medsurrmmt 106 g has three siynii~cnntf i g u r r . 11 a w r o nppear., after a nunaen, d l r which ~ t d l o r - n dermal p h i , thr rcro is iigniticdnt. O.flP111.1has fiwr iiirnifkmr figurer, ihr ,crj n, rha r r ~ h t of the decimal is not significant
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Arithmetic and Significant Figures The result of any computation can be no more precise than the least precise figure used. Addition and Subtraction. In the addition of the fallowing measurements: 18.033 + 0.001 g 0.1 g 6.7 0.01 g 4.62 29.353 0.111 g
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The sum of the three numbers can contain no more significant figures than the number with the fewest significant digits, which in this case is 6.7. Thus, the sum can contain only three significant figures, and must he rounded off to 29.4 g. The added uncertainties in the above problem also clearly indicate the number of significant figures in the sum, or 29.4 f 0.1 g. Examples for rounding off: 29.45 to 29.4 (last sig. digit even, drop 5) 29.35 to 29.4 (last sig. digit odd, round up) Multiplication and Diuision. Aproduct ( a X b) or quotient (alb) has the same number of significant figures as the least precise component (a or b). Thus, if the dimensions of arectangle are 10.3 cm and 1.2 em, the area of the rectanele. as calculated bv lone hand. is 12.36 cm2. However, the product e& only contain two significant figures, and must be rounded off to 12 cm2.
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