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!
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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.