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Significant figures (the important digits!)…
So how do we know how many digits to report for our measurements? Typically the tool used to make the measurement will have markings (like on a thermometer or graduated cylinder) that tell us all of the known digits to record and allow us to include one additional estimated digit. Balances and other electronic devices used for making measurements often report all of the significant digits (all known digits plus the estimated one) on the display. The key is to understand that there is always some uncertainty in the last reported digit from a measurement.
Although each instrument and tool is different, the principles of making a good measurement are the same.
As an example, consider the graduated cylinder below.
The device used to make the measurement will always show you all of digits that are known with certainty. You must always make sure that you are reading the tool correctly (for volume measurement with a graduated cylinder we want to read the volume at the bottom of the meniscus). Here, we see that the graduated cylinder is marked at every 0.1 mL. We know that the volume of water that it contains is 7.6_ mL with certainty. Lastly, we need to find the estimated digit (represented by "_" in 7.6_ mL). The bottom of the meniscus is closer to the 0.7 mL mark than it is to the 0.6 mL mark. So you could estimate that the final volume would be 7.67 mL (or any reasonable value for the hundredths place since it is an estimate).
So how do you figure out how many significant figures are in a measured quantity (i.e. a number provided to you from a measurement)? There are some simple rules to follow that will allow you to properly state the number of significant figures. This will be exceptionally important when we discuss the proper way to perform calculations with measured values.
1. All non-zero digits are significant
2. Zeros between non-zero digits are significant
3. Zeros at the beginning of a number (“leading” zeros) are never significant
4. Zeros at the end of a number (“trailing” zeros) are significant only if there is a decimal point
Zeros at the end of the number can be tricky, but it is usually assumed that if there is no decimal point then they are not significant. Using scientific notation can help with this, as any “trailing” zeros will be significant due to the decimal point. For example, 2.100 x 105 would have four significant digits (the 105 does not add any significant digits).
Let’s work a few examples.
State the correct number of significant figures in the following values.
A) 0.00785
B) 1.0003
C) 7000
D) 7000.
E) 1.350 x 105
Part A is an example of “leading” zeros. They are never significant so the answer to part A would be 3 significant figures. In part B, the zeros are sandwiched between two non-zero digits as mentioned in rule 2. This would lead to a total of 5 significant digits. For part C, we have trailing zeros. They are only significant when there is a decimal point. Therefore, the correct answer here is one significant figure. Likewise, part D shows a similar value with a decimal point this time. This would make the correct answer four significant figures for this one. Lastly, we have a number in scientific notation. It has one trailing zero, but scientific notation makes all of the digits significant (with the exception of x 105). Thus, the correct answer for part E is four significant figures.
To continue our discussion of numbers in chemistry, check out the next post which will cover significant figures in calculations.