# How to Calculate with Significant Figures

How do addition, subtraction, multiplication, and division work when you're dealing with significant figures? Keep on reading to find out!

So far we've talked about the big ideas behind significant figures (things like where they come from, why they matter, and what they mean), and we've also talked about how to identify significant figures in a number. But we haven't yet talked about how to actually make calculations using significant figures.

In other words, we haven't actually talked about how to work with significant figures in the real world! So, how do real world significant figures work? That's exactly the question we'll be talking about today.

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## Calculations with Significant Figures

What do significant figures have to do with real world calculations? Everything! First of all, as we've talked about before, significant figures are one of the ways we can quantify the precision of real world measurements. Which means that every time we make a calculation about something we measure in the real world, we need to worry about significant figures.

We need to understand how to add, subtract, multiply, and divide with significant figures.

In the example of estimating the speed that an American football player is running down the field that we talked about before, the question might be: How do you divide a distance measured to the nearest tenth of a yard by a time measured to the nearest few tenths of a second to come up with an estimate of a player's speed to an appropriate precision? What if you then wanted to add a bunch of these values together to come up with an estimate of the average speed of several players?

In order to perform any of these real world calculations—and any of the other much more important real world calculations that scientists, engineers, business and financial professionals, and countless other people make every day—we need to understand how to add, subtract, multiply, and divide with significant figures.

## Addition, Subtraction, and Significant Figures

Let's start by talking about how addition and subtraction work when dealing with significant figures. Here's the rule: the result of an addition or subtraction problem should always contain the same number of decimal places as the input value with the fewest number of decimal places.

What if one or more of the numbers is an integer and therefore doesn't have any decimal places? In this case, the more general rule to remember is that the result of an addition or subtraction problem should always be rounded to the same place as the largest input digit whose value is in doubt. What does that mean?

Here are a few examples to clarify:

- 0.31 + 0.1 = 0.4 — Why? Because, according to our rule, the result should have the same number of decimal places as the input with the fewest number of decimal places. Yes, 0.31 + 0.1 = 0.41, but it's equal to 0.4 when rounded to the tenths place as required.
- 0.35 - 0.1 = 0.3 — Why? Not worrying about significant figures, the result of 0.35 - 0.1 = 0.25. But since we do need to worry about significant figures, the result of our calculation should be rounded to the tenths place—so, 0.35 - 0.1 = 0.3.
- 9 - 2.4 = 7 — Really? Yes! Why? I know it seems kind of weird, but the number 9 in this calculation has an uncertainty in the tens place (since it doesn't have a decimal component). According to the general rule, the result should be rounded to the same place as the largest input digit whose value is in doubt. So 9 - 2.4 = 6.6—which, when rounded to the tens digit, gives 7.

## Multiplication, Division, and Significant Figures

The rule for determining the number of significant figures in the output of a multiplication or division problem is much simpler. You just have to remember that the result of a multiplication or division problem always has the same number of significant figures as the input value with the least number of significant figures. For example:

- 14 x 8 = 100 — Why? Again, I know this looks kind of strange, but the point is that we only know the input value 8 to one significant figure, which means that it's impossible to know the result of multiplying some other number by 8 to more than that one significant figure. So although 14 x 8 = 112, the fact that we're taking significant figures into account means that we need to round this to one significant figure. The answer is therefore 100.
- 8 x 7 = 60 — The reasoning here is the same as in the previous problem. Do you understand why?
- 5.5 / 1.1 = 5.0 — Why 5.0 and not just 5? Because both inputs have two significant figures, which means that we know the result to two significant figures as well.

## How to Calculate with Significant Figures

There's one small but important detail that we should talk about briefly before wrapping up. And that is that whenever you're doing a calculation involving significant figures, you should always round to the appropriate number of significant figures at the very end of the calculation. In other words, use all available digits when doing the actual calculation and only round to the appropriate number of digits at the end. Doing so will ensure that you don't end up with the wrong answer due to accumulated rounding errors.

Follow this advice and you should always arrive at the correct answer. But just in case you want to double-check that your answer is indeed that correct answer, here's a cool significant figures calculator I found that should help. Enjoy!

## Wrap Up

Okay, that's all the math we have time for today. It's nearing the end of the month, which means that the next installment of our new "Frequently Asked Questions" series is just around the corner. So be sure to send your math questions to mathdude@quickanddirtytips.com.

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