Adding and Subtracting with Significant
Digits
Dr. MJ Patterson
This lesson focuses on correctly carrying significant digits through a calculation. Now that we can identify which digits are significant in a reported measurement, we need to know how to best represent any calculations made from that data.
The main idea behind calculations involving experimentally measured numbers is that the calculated value cannot be any more precise than the experimental values. The experimental numbers limit the precision with which we can report a calculated value.
For addition and subtraction, the rules focus on the number of decimal places, instead of the total number of significant figures.
Rule for Carrying Significant Digits through Addition and Subtraction
The rule for addition and subtraction is to look at the number of decimal places in all of the numbers involved in the calculation. Then, decide what is the smallest number of decimal places you are working with. Report your answer to this smallest number of decimal places.
In the lab, you might need to take the difference of two masses. If you used different balances to weigh the two objects, you would have two different numbers of decimal places. For example, if you weighed a beaker containing a lot of mercury, it could weigh 524.9 g, which would require you to weigh it on one of our rough balances (the total mass is too high for our analytical balances). However, the analytical balances are more convenient to use, and so when you weighed the empty beaker, you might have found a mass of 122.687 g. To find the mass of the mercury, you need to subtract the two measurements:
524.9 g - 122.687 g = 402.213 g = ?
How many decimal places should we report the answer to? The first number has only 1 decimal place, while the second number has 3 decimal places. We are limited by the smaller of these, and so we should report the answer to 1 decimal place.
524.9 g - 122.687 g = 402.213 g = 402.2 g
Examples:
Perform each of the following operations and round to the appropriate number of significant figures.
a. 45.94 - 44.74 =
b. 45.687 + 121.4684 =
Solutions:
a. Both numbers have 2 decimal places, so we need to report our answer to 2 decimal places.
45.94 - 44.74 = 1.20
You must explicitly write the terminal zero.
b. The first number has 3 decimal places, while the second has 4 decimal places. The answer should be reported to 3 decimal places.
45.687 + 121.4684 = 167.1554 = 167.155
To recap the rules for carrying significant digits through calculations:
Multiplication and Division
Look at the total number of sig figs in the problem. Report your answer
to the smallest number of total sig figs.
Addition and Subtraction
Look only at the decimal places. Report your answer to the smallest
number of decimal places.