In chemistry, the potential error associated with a measurement is often reported alongside the measurement, as in 
a.
b.
Practice Answers
1. The correct number of significant figures is as follows for each measurement: a) 5, b) 3, and c) 4.
2. The number of significant figures in a reported measurement should be consistent with your certainty about that measurement.
a. “
b. “
Doing Arithmetic with Significant Figures
Doing chemistry means making a lot of measurements. The point of spending a pile of money on cutting-edge instruments is to make really good, really precise measurements. After you’ve got yourself some measurements, you roll up your sleeves, hike up your pants, and do some math.
Remember: When doing math in chemistry, you need to follow some rules to make sure that your sums, differences, products, and quotients honestly reflect the amount of precision present in the original measurements. You can be honest (and avoid the skeptical jeers of surly chemists) by taking things one calculation at a time, following a few simple rules. One rule applies to addition and subtraction, and another rule applies to multiplication and division.
Addition and subtraction
In addition and subtraction, round the sum or difference to the same number of decimal places as the measurement with the fewest decimal places. For example, suppose you’re adding the following amounts:
Your calculator will show 19.3645, but you round off to the hundredths place based on the 3.25, which has the fewest number of decimal places. You round the figure off to 19.36. (See the later section “Rounding off numbers” for the rounding rules.)
Multiplication and division
In multiplication and division, you report the answer to the same number of significant figures as the number that has the fewest 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 (six sig figs) grams and has a volume of 10.50 milliliters (four sig figs). The setup looks like this:
Your calculator will read 2,414.981000. You have six significant figures in the first number and four in the second number (the 1,000 mL/L doesn’t count because it’s an exact conversion). You should have four significant figures in your final answer, so round the answer off to 2,415 g/L.
Notice the difference between the two rules. When you add or subtract, you assign significant figures in the answer based on the number of decimal places in each original measurement. When you multiply or divide, you assign significant figures in the answer based on the smallest number of significant figures from your original set of measurements.
Tip: Caught up in the breathless drama of arithmetic, you may sometimes perform multi-step calculations that include addition, subtraction, multiplication, and division, all in one go. No problem. Follow the normal order of operations, doing multiplication and division first, followed by addition and subtraction. At each step, follow the simple significant-figure rules, and then move on to the next step.
Rounding off numbers
Sometimes you have to round numbers at the end of a measurement to account for significant figures. Here are a couple of very simple rules to follow and remember:
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Rule 1: If the first number to be dropped 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.
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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.
Examples
Q. Express the following sum with the proper number of significant figures:
A. 671.1 miles. Adding the three values yields a raw sum of 671.05 miles. However, the 35.7 miles measurement extends only to the tenths place. Therefore, you round the answer to the tenths place, from 671.05 to 671.1 miles.
Q. Express the following product with the proper number of significant figures:
A.
Practice Questions
1. Express the answer to this calculation using the appropriate number of significant figures:
