Remember: When you report a measurement, you should include digits only if you’re really confident about their values. Including a lot of digits in a measurement means something – it means that you really know what you’re talking about – so we call the included digits significant figures. The more significant figures (sig figs) in a measurement, the more accurate that measurement must be. The last significant figure in a measurement is the only figure that includes any uncertainty, because it’s an estimate. Here are the rules for deciding what is and what isn’t a significant figure:
✔ Any nonzero digit is significant. So 6.42 contains three significant figures.
✔ Zeros sandwiched between nonzero digits are significant. So 3.07 contains three significant figures.
✔ Zeros on the left side of the first nonzero digit are not significant. So 0.0642 and 0.00307 each contain three significant figures.
✔ One or more final zeros (zeros that end the measurement) used after the decimal point are significant. So 1.760 has four significant figures, and 1.7600 has five significant figures. The number 0.0001200 has only four significant figures because the first zeros are not final.
✔ When a number has no decimal point, any zeros after the last nonzero digit may or may not be significant. So in a measurement reported as 1,370, you can’t be certain whether the 0 is a certain value or is merely a placeholder.
Be a good chemist. Report your measurements in scientific notation to avoid such annoying ambiguities. (See the earlier section “Using Exponential and Scientific Notation to Report Measurements” for details on scientific notation.)
✔ If a number is already written in scientific notation, then all the digits in the coefficient are significant. So the number
✔ Numbers from counting (for example, 1 kangaroo, 2 kangaroos, 3 kangaroos) or from defined quantities (say, 60 seconds per 1 minute) are understood to have an unlimited number of significant figures. In other words, these values are completely certain.
Remember: The number of significant figures you use in a reported measurement should be consistent with your certainty about that measurement. If you know your speedometer is routinely off by 5 miles per hour, then you have no business protesting to a policeman that you were going only 63.2 mph in a 60 mph zone.
Example
Q. How many significant figures are in the following three measurements?
a.
b.
c.
A. a) Five, b) three, and c) four significant figures. In the first measurement, all digits are nonzero, except for a 0 that’s sandwiched between nonzero digits, which counts as significant. The coefficient in the second measurement contains only nonzero digits, so all three digits are significant. The coefficient in the third measurement contains a 0, but that 0 is the final digit and to the right of the decimal point, so it’s significant.
Practice Questions
1. Identify the number of significant figures in each measurement:
a.
b. 0.000769 meters
c. 769.3 meters
2. 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.