If you missed our second post on the basics of numbers in chemistry, please click here to check it out. This post will focus on significant figures in calculations.
Now that we have learned how to identify the digits that are significant in a measured value, we need to broaden our discussion to include performing calculations and reporting the answers to the correct number of significant figures.
As we have noted before, there is always uncertainty in measured values. We now need to consider this when performing calculations with measured values so we do not report our answers incorrectly (i.e. we do not want to over- or underrepresent our precision!). There is a basic principle that can help guide us to correctly perform any calculations that we may encounter in chemistry. This principle is such that the measurement with the least amount of certainty will dictate the number of significant digits that we report in our final answer for any calculation. This may seem complicated, but as you will see we can avoid over- or underrepresenting the number of significant figures in our answer by following two simple rules (one for addition/subtraction and one for multiplication/division). This will allow us to perform calculations and report our answers with only one uncertain digit.
1. For addition and subtraction, the resulting answer will have the same number of
decimal places as the measurement with the fewest decimal places.
Let’s consider an example.
As you can see, the final result is limited by the number “3” in 2023 which is highlighted green to help us see the least significant digit. Our answer reported to the correct number of significant digits will be 4046. We cannot forget the rules for rounding. The calculator will display 4045.5, and we must round by considering what follows the uncertain digit in our answer.
When rounding you always look to the leftmost digit that is not significant and if it is > 5 you round up, and if it is < 5 you leave the digit to be rounded unchanged. As we see highlighted in red, the “.5” tells us that we need to round our uncertain digit up to 4046 meters.
Take a look at another example (and a different method).
Sometimes it is easier to see the cutoff for what is significant and what is not by drawing a line just to the right of the number with the fewest decimal places (the green dashed line in the example above). Our answer can have no more precision than is allowed by 0.02 as it has the fewest digits past the decimal in this example. Now we can report the correct answer by considering all of the significant digits and performing rounding where necessary. Here the correct answer is 1.25 since the “4” to the right does not require us to round up to 1.26 mL.
Another example.
When scientific notation is involved, it can get confusing even when performing addition and subtraction. Try to remember that the rule still applies. The digit with the fewest decimal places will limit the significant figures in the final answer. So, if you have a hard time seeing that 1.17 x 10^−3 has the fewest digits past the decimal, then you may want to convert the numbers to standard notation and stack them like the method we examined above.
Now we can see that the final answer reported to the correct number of significant digits (when we round correctly) is going to be 0.00117 or 1.17 x 10^−3 cm.
1. For multiplication and division, the final answer will contain the same number of significant figures as the measured value with the fewest significant figures. Just like addition and subtraction, if the answer contains more digits than allowed by the rule we will round appropriately.
Let’s see this in practice.
Here, we are tasked with computing the volume of a rectangular prismatic object and are provided with the dimensions. Volume = length x width x height (provided above, respectively), so all we need to do is multiply to obtain the volume. The measurement with the fewest significant figures is highlighted in green. Our answer reported to the correct number of significant figures should have three digits. When we use our calculator, it displays 302.48 cm^3 (note that when you multiply, you multiply the numbers and the units!; cm x cm x cm = cm^3). As before, we have noted the limits of significant digits with green while the red digit is noted for rounding purposes. This would make the correct answer 302 cm^3.
Let's try another example.
Exponents/scientific notation was troublesome when performing addition and subtraction, but here it is quite simple. The digits to the left of the “x” are all significant. The value with the fewest significant figures is highlighted green. This will limit our final answer (an area in this example since we are multiplying two lengths in units of centimeters) to two significant figures. When we plug this into our calculator and compute, the display reads 1.404 x 10^−9 cm^2. The green digit notes our limit on significant figures, and the red digit helps us with rounding. As you can see, the final answer reported correctly is 1.4 x 10^−9 cm^2 (a very tiny area!).
To conclude, let's discuss performing calculations with multiple operations. To do this correctly we will want to follow the order of operations. PEMDAS is the acronym used to remember the correct order. P = parentheses, E = exponents, M = multiplication, D = division, A = addition, and S = subtraction. Be sure to assess the problem and perform the calculations in the correct order! This means handling anything in parentheses first, followed by exponents, etc.
Let’s do an example to see this in practice.
In this example, if we follow the order of operations we need to compute a difference of two lengths in parentheses, and then divide by another length. If we look closely at the subtraction portion, we need that 1.570 x 10^3 is the same as 1570. The “0” highlighted in green represents the limitations of the significant figures in this portion of the calculation (four significant figures).
If we complete the subtraction on our calculator, the display reads 1.568795 x 10^3 or 1568.795 mm^2. It is extremely important to note that when there are multiple operations, you don’t want to round anything until the final answer is to be written. Therefore we will just note the number of significant figures that would result if we stopped here (in green) and use the unrounded number (or at least keep a couple of extra digits) in the next operation.
If we divide the unrounded value that we just obtained by 12.7 mm, the calculator displays 1.235271654 x 10^2 mm. Note that the final unit is “mm” and not “mm^2” since diving by a common unit cancels out one of the “mm” (see the orange lines that cancel one mm/mm in the image above). Do not get bogged down with this as we will discuss it in more detail in a later post!).
Now we need to apply the rules for significant figures in division operations to report the answer correctly. 12.7 has three significant figures, and our subtraction step was limited to four significant figures (note the green above). The final answer should contain three significant figures (1.235271654 x 10^2 mm) which leads us to the correct value of 1.24 x 10^2 mm after rounding correctly (see the green and red digits to note the significant digits and what was used to round to the final answer).
If you found this helpful, then be sure to check out our next post covering dimensional analysis!