## Chemistry for Dummies

**Appendix D. Significant Figures and Rounding Off**

Significant figures (no, I’m not talking about some supermodel) are the number of digits that you report in the final answer of the mathematical problem you are calculating. If I told you that one student determined the density of an object to be 2.3 g/mL and another student figured the density of the same object to be 2.272589 g/mL, I bet that you would naturally believe that the second figure was the result of a more accurate experiment. You might be right, but then again, you might be wrong. You have no way of knowing whether the second student’s experiment was more accurate unless both students obeyed the significant figure convention. The number of digits that a person reports in his or her final answer is going to give a reader some information about how accurately the measurements were made. The number of the significant figures is limited by the accuracy of the measurement. This appendix shows you how to determine the number of significant figures in a number, how to determine how many significant figures you need to report in your final answer, and how to round your answer off to the correct number of significant figures.

*Numbers: Exact and Counted Versus Measured*

If I ask you to count the number of automobiles that you and your family own, you can do it without any guesswork involved. Your answer might be 0, 1,2, or 10, but you would know exactly how many autos you have. Those are what are called counted numbers. If I ask you how many inches there are in a foot, your answer will be 12. That is an exact number. Another exact number is the number of centimeters per inch — 2.54. This number is exact by definition. In both exact and counted numbers, there is no doubt what the answer is. When you work with these types of numbers, you don’t have to worry about significant figures.

Now suppose that I ask you and four friends to individually measure the length of an object as accurately as you possibly can with a meter stick. You then report the results of your measurements: 2.67 meters, 2.65 meters, 2.68 meters, 2.61 meters, and 2.63 meters. Which of you is right? You are all within experimental error. These measurements are measured numbers, and measured values always have some error associated with them. You determine the number of significant figures in your answer by your least reliable measured number.

*Determining the Number of Significant Figures in a Measured Number*

Here are the rules you need to determine the number of significant figures, or sig. figs., in a measured number.

ü Rule 1: All nonzero digits are significant. All numbers, one through nine, are significant, so 676 contains three sig. figs., 5.3 x 10^{5} contains two, and 0.2456 contains four. The zeroes are the only numbers that you have to worry about.

ü Rule 2: All of the zeroes between nonzero digits are significant. For example, 303 contains 3 sig. figs., 425003704 contains nine, and 2.037 x 10^{-6} contains four.

ü Rule 3: All zeros to the left of the first nonzero digit are not significant. For example, 0.0023 contains two sig. figs, and 0.0000050023 contains five (expressed in scientific notation it would be 5.0023 x 10^{-6}).

ü Rule 4: Zeroes to the right of the last nonzero digit are significant if there is a decimal point present. For example, 3030.0 contains five sig. figs., 0.000230340 contains six, and 6.30300 x 10^{7}also contains six sig. figs.

ü Rule 5: Zeroes to the right of the last nonzero digit are not significant if there is not a decimal point present. (Actually, a more correct statement is that I really don’t know about those zeroes if there is not a decimal point. I would have to know something about how the value was measured. But most scientists use the convention that if there is no decimal point present, the zeroes to the right of the last nonzero digit are not significant.) For example, 72000 would contain two sig. figs and 50500 would contain three.

*Reporting the Correct Number of Significant Figures*

In general, the number of significant figures that you will report in your calculation will be determined by the least precise measured value. What values qualify as the least precise measurement will vary depending on the mathematical operations involved.

*Addition and subtraction*

In addition and subtraction, your answer should be reported to the number of decimal places used in the number that has the fewest decimal places. For example, suppose you’re adding the following amounts:

2.675 g + 3.25 g + 8.872 g + 4.5675 g

Your calculator will show 19.3645, but you are going to round off to the hundredths place based on the 3.25, because it has the fewest number of decimal places. You then round the figure off to 19.36.

*Multiplication and division*

In multiplication and division, you can report the answer to the same number of significant figures as the number that has the least significant figures. Remember that counted and exact numbers don’t count in the consideration of significant numbers. For example, suppose that you are calculating the density in grams per liter of an object that weighs 25.3573 (6 sig. figs.) grams and has a volume of 10.50 milliliters (4 sig. figs.). The setup looks like this:

(25.3573 grams/10.50 mL) x 1000 mL/L

Your calculator will read 2414.981000. You have six significant figures in the first number and four in the second number (the 1000 mL/L does not count because it is a exact conversion). You should have four significant figures in your final answer, so round the answer off to 2415 g/L. Only round off your final answer. Do not round off any intermediate values.

*Rounding Off Numbers*

When rounding off numbers, use the following rules:

ü Rule 1: Look at the first number to be dropped; if it is 5 or greater, drop it and all the numbers that follow it, and increase the last retained number by 1. For example, suppose that you want to round off 237.768 to four significant figures. You drop the 6 and the 8. The 6, the first dropped number, is greater than 5, so you increase the retained 7 to 8. Your final answer is 237.8.

ü Rule 2: If the first number to be dropped is less than 5, drop it and all the numbers that follow it, and leave the last retained number unchanged. If you’re rounding 2.35427 to three significant figures, you drop the 4, the 2, and the 7. The first number to be dropped is 4, which is less than 5. The 5, the last retained number, stays the same. So you report your answer as 2.35.