*Posted 11/22/21*

As we’ve seen, significant figures aren’t just for measuring. I mean, sure, they’re something we take account for when doing measurements – however, it’s usually not the case that data is collected and then just looked at. Instead, we usually do calculations using them, and we have to continue using significant figures to ensure that people know how good our data are. For those of you who dislike math, I’m sorry to be the one to give you this bad news.

In any case, I’ll try to make this as painless as possible. Let’s see how it goes.

**Addition and subtraction:**

When doing significant figure calculations involving addition and subtraction, the important thing to determine is the decimal place of the rightmost significant figure in the number. For example, in the number 130, the rightmost significant figure is the 3, which is in the tens place. Likewise, in the number 34.0, the last zero is significant, which is in the tenths place.

Now that you know which are the last significant digits, we’re ready to go. Let’s say that we want to do the following calculation:

**130 grams – 54.0 grams = ?**

Here’s how to solve it. Keep in mind, this process only works for addition and subtraction, not for multiplication and division. We’ll get to those later.

**Step 1**: Do the calculation on your calculator: In this case, when you plug it into the calculator you find that the answer is “76 grams.”

**Step 2**: Figure out what decimal place the rightmost significant figure is for each number. We’ve already figured out that the 3 in 130 is in the tens place, meaning that the number is precise to the nearest 10 grams. Likewise, the rightmost significant digit in 54.0 grams is the zero in the tenths place, which means that our number is precise to the nearest 0.1 grams.

**Step 3**: Determine which number is least precise. Clearly, having a measurement precise to the nearest 10 grams is less precise than having one rounded to the nearest tenth of a gram.

**Step 4**: Round the answer you got in step 1 to the decimal place of the least precise number you determined in step 3. Since the least precise number is rounded to the nearest ten grams, round the answer from step 1 to the nearest ten grams.

**76 grams rounded to the tens place is 80 grams! Which is your answer! Yeeha!**

Likewise, let’s say that you’ll add 4.50 cm to 3.3 cm. When you add them together with your calculator, you get the answer 7.8 cm. Since 4.50 cm is precise to the nearest hundredth of a gram and 3.3 cm is precise to the nearest tenth of a gram, we need to round our answer to the nearest gram. Since the number 76 grams doesn’t need any rounding at all to be shown to the nearest gram, that’s our answer!

Easy peasy somethingsomething

**Multiplication and division**

Multiplication and division are even simpler than addition and subtraction. All you need to do is to figure out how many significant figures are present in each of the numbers you’re multiplying or dividing and then round your answer to whichever is smallest.

Let’s look at this problem:

**450 cm x 21.2 cm = ?**

450 cm has two significant figures. 21.2 has three significant figures. Since 2 is less than 3, all we need to do is to perform the calculation and round our answer to two significant figures.

The unrounded answer: 450 cm x 21.2 cm = 9540 cm^{2}

**This answer rounded to two significant figures: 450 cm x 21.2 cm = 9500 cm ^{2}**

To do another example, let’s divide 45 grams by 3 mL to find the density of something that has this density. 45 grams has two significant figures while 3 mL has one significant figure. As a result, our number will have to be rounded to one significant figure. This gives us an answer of 20 g/mL, rounded from 15 g/mL.

And that’s how you do calculations with significant figures. Yeeeha!

**Photo credits:**

**Cinnamon photo**: It’s in the public domain, downloaded from Wikimedia Commons. You can access it here: https://commons.wikimedia.org/wiki/File:Cinnamon_(Alabama_Extension).jpg

**Lemon photo**: It’s in the public domain, downloaded from Wikimedia Commons. You can access it here: https://commons.wikimedia.org/wiki/File:Whole-Lemon.jpg