How to Identify Significant Figures
How many significant figures does the number 1.25 have? What about 1.255? Or 1.250? What’s the quick and dirty method that you can use to find out? Keep on reading for Math Dude’s significant tips!
Jason Marshall, PhD
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How to Identify Significant Figures
As we learned in the last episode, What Are Significant Figures?, some numbers are exact and some are not. In particular, any number that is the result of a measurement always has some uncertainty about its value and is therefore not an exact number.
Since all measurements have limited precision—simply because we and the tools we use are never absolutely perfect and are therefore prone to errors and uncertainties—we also learned that a number obtained from a measurement has only a limited number of digits (or figures) that hold any significance.
But how can you tell how many of these significant figures a number has? In other words, how can you identify which figures in a number are significant? And more importantly, is there a quick and dirty method that you can use to figure this out? Those are exactly the question we’ll be answering today!
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Accuracy, Precision, and Significant Figures
Before we start talking about identifying significant figures, let’s quickly talk about last week’s brain teaser about whether significant figures have more to do with accuracy or precision? What do you think? To answer this question, we need to remember two things:
- Accuracy tells you how close a series of measurements are (on average) to the true value.
- Precision tells you how close a series of measurements are to each other.
Significant figures are about precision not accuracy.
In the analogy we talked about before, an archer can be accurate but imprecise if she sprays arrows all around the target but far away from one another. She can also be precise but innacurate if her arrows all land close together but far away from the bullseye. Of course, she could be both precise and accurate if all of her arrows land near the bullseye. And she could even be both imprecise and innacurate if her arrows go every which way—in which case, watch out!
With all of this in mind, do you think significant figures are about accuracy or precision? If you think about it, you’ll see that the significant figures in a measurement tell us how precisely we have determined the value. They tell us nothing about the accuracy of that measurement. Which means that significant figures are about precision not accuracy.
Rules for Identifying Significant Figures
With that out of the way, we’re now ready to answer today’s big question: If somebody gives you a number—any number—how do you identify which of its digits are significant? As luck would have it, the rules are pretty simple. Just remember that all digits are significant except:
- Leading zeros in all numbers (more on what those are in a second), and
- Trailing zeros in numbers without a decimal point
This means that every single non-zero digit—and all zero digits stuck between non-zero digits—are significant…always. So the number 15 has exactly two significant figures, the number 3.14 has exactly three significant figures, the number 135.9 has exactly four significant figures, the number 2,509.1 has exactly five significant figures, and so on. These are the easy cases. Things get trickier when dealing with leading and trailing zeros.
Leading and Trailing Zeros
Leading zeros—which include any 0 that comes before a non-zero digit—are never significant. Why? Because those zeros aren’t something you measure. Another way to say this is that those zeros have no uncertainties associated with them. They’re only present to set the scale of the digits that you do actually measure. For example, the zero in 0.1 is not significant since it comes before a non-zero digit. Similarly, neither zero in 0.05 are significant. Both of these numbers have only one significant figure.
Leading zeros are never significant.
Trailing zeros in numbers without a decimal point are not significant either…at least not usually. For example, most of the time the pair of zeros in 1,300 are not significant. Writing a number in this way implies that the number is only known to a precision in the hundreds. But notice that I said “usually.” And that’s because it is possible for a number like 1,300 to have four significant figures if both zeros in the tens and ones place were actually measured. In cases like this, it’s helpful to write the number with an ending decimal point—as in 1,300.—to indicate the significance of those zeros.
Whenever one or more trailing zeros show up in a number that has a decimal point like this, they are always significant. This means that not only does the number 1,300. have four significant figures, but a number like 92.30 has four significant figures, too. Why is this trailing zero significant? Because it shows that the value of that digit was actually measured to be zero.
Significant Figures and Scientific Notation
Although these rules for identifying significant figures aren’t too hard to follow, they nonetheless can sometimes be a little confusing to remember. Thankfully, you don’t have to because there is an easier way to identify the number of significant figures in a number. The trick is to look at the number written using scientific notation.
As you’ll recall, the idea behind scientific notation is to write all numbers as a decimal number times some multiple of 10． For example, the number 1,234.5 is written 1.2345 x 10^3 and the number 0.00678 is written 6.78 x 10^-3 in scientific notation. The beauty of scientific notation is that the number of digits in the decimal portion of the number is always the same as the number of significant figures. In other words, all of the numbers in the decimal portion are significant.
This is especially helpful when it comes to a number like 1,300. When you really mean 1,300—with only two significant figures—you write it 1.3 x 10^3 to show that the trailing zeros are not significant. On the other hand, if you really mean 1,300.—with four significant figures—you write it 1.300 x 10^3 to show that those two zeros really do mean something.
Wrap Up
Okay, that’s all the math we have time for today. Be sure to check out my mental math audiobook The Math Dude’s 5 Tips to Mastering Mental Math. And for even more math goodness, check out my book The Math Dude’s Quick and Dirty Guide to Algebra.
Remember to become a fan of the Math Dude on Facebook where you’ll find lots of great math posted throughout the week. If you’re on Twitter, please follow me there, too. Finally, please send your math questions my way via Facebook, Twitter, or email at mathdude@quickanddirtytips.com.
Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading, math fans!
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About the Author
Jason Marshall is the author of The Math Dude’s Quick and Dirty Guide to Algebra. He provides clear explanations of math terms and principles, and his simple tricks for solving basic algebra problems will have even the most math-phobic person looking forward to working out whatever math problem comes their way.