If I give you a numerical value, but you have no idea how precise it is, then the information is useless to you. We have to have a prior agreement (perhaps an implicit one) that allows you to know how much you can "trust" my information. When using the English language (or any other spoken language), we have ways of indicating precision. If I tell you that I'm "about 29 years old" (Ha!), then you know it's approximate. If I tell you that I am "78 years, 3 months, 2 days, 8 hours, 47 minutes, and 32 seconds old", that is less approximate (and more precise). Neither age is accurate, by the way. Unfortunately, when simply reporting a numerical quantity (with its unit, of course!) there are no cues concerning precision. "29 years" might mean anything from 28.50000000000000000001 years to 29.49999999999999999999 years. You wouldn't know what "29 years" means without further information. Similarly, if I didn't know the time of day of my birth, then I would have no right (i.e., I'd be lying) to say that I am "78 years, 3 months, 2 days, 8 hours, 47 minutes, and 32 seconds old". In this course, you and I will trade a lot of numerical data. We had better discuss how to interpret this information so that we can communicate.

  What we will do is agree to report measured data in the following fashion: report all digits that you are sure about, plus one more digit that you have estimated. For example, if you tell me that you are 170.18 cm tall, I will understand that to mean that you have measured your height and your measurement device convinced you that you are between 170.1 cm and 170.2 cm (the device must have been clearly marked off to the tenth of a cm). Further, it looks to you as though you are considerably closer to 170.2 cm, so that you estimate that you are 0.08 cm taller than 170.1 cm (i.e., you believe that you are 170.18 cm tall). To give me fewer digits would mean that you are "holding back" information that you do have. To give me more digits would be lying (or, at least, an exaggeration). Our agreement on how many digits should be reported allows you to send information about precision along with your numerical value and it allows me to receive that information.

  Now if I use your information in a calculation, I have to regard my result as being limited in its precision by the data (your data) upon which it is based. That way, if I pass my result on to someone else (or even back to you) the recipient gets the appropriate precision information. Therefore, we have to establish another agreement about how to determine the appropriate number of significant figures to include in a calculated result. You will find our agreement (I am confident that you will agree!) below.

  What you will find below is simply a set of time saving rules. By obeying these rules, your answers to my questions will agree (in terms of precision) with my answers to my questions. These are not the only rules that exist, nor are they the "best". Some sets of rules do a better job than others. By that, I mean that "better" ones more closely duplicate the results of a complete and laborious error analysis. The most accurate sets of rules tend to be difficult to use. When they are used, there seem to be many "borderline" cases where more than one rule *might* apply. Since we are using the rules to save time, so that we can devote more attention to other important matters (such as chemistry), I have adopted a particularly simple procedure for the General Chemistry courses. It is not the best, but it's not too bad. It has so few ambiguities that we can expect every student in the class to be able to arrive at the same answer (provided that they don't make a mistake). At the same time, I point out and hope that students note, that these simple rules are APPROXIMATE and that if they ever do a calculation for which the precision of the result is of great importance, then they should NOT use the rules, but should sit down and do a careful data analysis. If lives will depend upon your results (e.g., you design bridges for a living) please don't use my rules after you've completed General Chemistry I and II!!!

If you solve a problem that involves ONLY addition and/or subtraction (well, possibly mult/div by a number of "counted objects" or by "defined quantities", neither of which is considered to be subject to the "random" error of measurements), then use the ADDITION/SUBTRACTION RULE.

THE ADDITION/SUBTRACTION RULE: round your answer off to the same DECIMAL PLACE as the least precise piece of data that was involved in the calculation (including conversion factors that depend upon measurements rather than definitions and those that have been rounded off, e.g., 1 mile ≈ 1.6X10³ m).

If you solve a problem that involves ONLY multiplication and/or division, use then use the MULTIPLICATION/DIVISION RULE.

THE MULTIPLICATION/DIVISION RULE: round your answer until it contains the SAME NUMBER OF SIGNIFICANT DIGITS as the least precise piece of data that was involved in the calculation (again, including "measured" conversion factors), regardless of the position of the decimal point.

If you solve a problem that involves both addition (or subtraction) AND multiplication (or division), use the multiplication/division rule, EXCEPT every time you perform an addition or a subtraction, assess how many sig figs would be used in its answer (via the addition/subtraction rule) and then regard that additive/subtractive result as you would an item of data (i.e., additions might increase and subtractions might decrease the number of sig figs in the overall answer).

ONLY ROUND OFF WHEN REPORTING AN ANSWER. If you know (from looking at the data) that you'll eventually be rounding the answer off to a particular number of significant figures (or to a particular decimal place) then keep AT LEAST TWO (2) EXTRA DIGITS (preferably more) throughout your calculation in order to avoid contributing "round off" error to the result.

If you need to use a reported result for further calculation, use its value from "before rounding".
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  Now, if you disagree with these rules, that's fine. They're not perfect. This procedure does not duplicate the outcome of a careful error analysis 100% of the time. But it does a pretty good job and involves little ambiguity. For a course of nearly 700 students, it's workable. If I taught bridge design, we wouldn't use it. However, in order to draw SOME ATTENTION to questions of precision, without going overboard with issues of applied mathematics, the procedure described above is a good compromise.