Multiplying and Dividing 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.
Here is an example that you might commonly encounter in a lab. You need an estimate of the mass of some glass beads that you will use in an experiment. If you take a single bead and place it on the balance, the mass is too small, and the balance still reads 0.00 g. So, you decide to weigh a bunch of them together to get an average mass. You carefully count out 17 beads and weigh this group of 17 beads together. The total mass is 1.35 g. To find the average mass of one bead, you whip out your handy dandy calculator and divide the total mass by 17, resulting in 0.079118... showing up in your calculator's display. What should you report for the average mass of one bead?
Rule for Carrying Significant Digits through Multiplication and Division
The rule for multiplication and division is to look at the total number of significant digits in all of the numbers involved in the calculation. Then, decide what is the smallest number of sig figs you are working with. Report your answer to this smallest number of sig figs.
In the example above, we were working with two numbers and division: 1.35/17
How many sig figs does each of these numbers have?
1.35 has three sig figs. The balance automatically rounds the mass to the nearest hundredth's place.
17 has an infinite number of sig figs. Why? Because we like you. Oops - showing my age here. Seriously, because it is an exact number. You can count that there are exactly 17 beads in the sample. There is absolutely no uncertainty in this measurement.
So, we compare infinity to 3, and determine that 3 is the smaller number. Then, we report the answer to 3 sig figs. The average mass of a glass bead is 0.0791 g.
Rule for Addition and Subtraction
The rule for carrying sig figs through addition and subtraction is a bit different. You should be aware that it is different since you will see it in CHEM 1411 and other science classes. But, we are not going to cover it here.
For this class, round your answers in all calculations to the same amount of sig figs as the number in the problem with the fewest sig figs.
Examples:
Perform each of the following operations and round to the appropriate number of significant figures.
a. (6.02 x 1023)(0.39) =
b.
|
(7.9) |
x |
(1) |
x |
(6.02 x 1023 ) |
|
(1) |
(12.01) |
(1) |
|
|
Solutions:
a. (6.02 x 1023)(0.39) = 2.3 x 1023
The answer is limited to the two sig figs of 0.39
b. Assume that the 1's are exact numbers.
|
(7.9) |
x |
(1) |
x |
(6.02 x 1023 ) |
= 4.0 x 1023 |
|
(1) |
(12.01) |
(1) |
|
|
|
The answer is limited by the two sig figs of 7.9