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Rules of significant figures

Learn the "rules of thumb" for significant figures. Created by Sal Khan.
Video transcript
Based on the examples in the last video, let's see if we can come up with some rules of thumb for figuring out how many significant figures or how many significant digits there are in a number or a measurement. So the first thing that is pretty obvious is that any non-zero digit and any of the zero digits in between are significant. Clearly, the 7 and the 5 here are significant. And the 0 in between them, it's also going to be significant. So let's write this over here. So any non-zero digits and zeros in between are going to be significant. That's pretty straightforward. Now, the zeros that are not in between nonzero digits, these become a little bit more confusing. So let's just make sure we can rule out some of them. So you can always rule out when you're thinking about significant figures, the leading 0's. And when I'm talking about leading zeros, I'm talking about the zeros that come before your non-zero digit. So these are leading zeroes here. These are leading zeroes. There is no leading zeroes here. No leading zeroes in this one, this one, and this one. But in any situation, the leading zeros are not significant. So leading zeros not significant, I'll write it over here. Leading zeros not significant. And so the last question, all you have left, I mean you only have non-zero digits and zeros in between. You could have some leading zeros, which you've already said are not significant. And so the only thing left that you have to figure out is what do you do with the trailing zeroes, the zeroes behind the last non-zero, or to the right of the last non-zero digit. So these trailing zeroes here. There's actually two trailing zeroes over here. And then there's three trailing zeroes over here. So let me make a little-- so trailing zeroes. Trailing zeroes, what do we do with them? So the easy way to think about is if you have a decimal, if there's a decimal anywhere in your number, count them. If you have a decimal, count them. Count them as significant. They are significant, count them as significant. If there's no decimal anywhere in the number, then it's kind of ambiguous. You're kind of not sure and this is a situation. So clearly over here, there's a decimal in the number, so you count the trailing zeroes. These are adding to the precision. Over here there's a decimal, so you count the trailing zero. There's a decimal here, so you count the trailing zeroes. There are no trailing zeroes here. And over here-- well, the way I later put a decimal here. Here you would count it. So if you have the decimal there, you would count all five. If you didn't have the decimal, if you just had 37,000 like that, it's ambiguous. And if someone doesn't give you more information, your best assumption is probably that they just measured to the nearest thousand. That they didn't measure exactly the one and just happened to get exactly on 37,000. So if there's no decimal, let me write it this way-- it's ambiguous, which means that you're not sure what it means, it's not clear what it means. And you're probably safer assuming to not count it. If someone really does measure, if you were to really measure something to the exact one, then you should put a decimal at the end like that. And there is a notation for specifying. Let's say you do measure-- and let me do a different number. Let's say you do measure 56,000. And there is a notation for specifying that 6 definitely is the last significant digit. And sometimes you'll see a bar put over the 6, sometimes you'll see a bar put under the 6. And that could be useful because maybe your last significant digit is this zero over here. Maybe you were able to measure to the hundreds with a reasonable level of precision. And so then you would write something like 56,000, but then you would put the bar above that zero, or the bar below that zero to say that that was the last significant digit. So if you saw something like this, you would say three significant digits. This isn't used so frequently. A better way to show that you've measured to three significant digits would be to write it in scientific notation. There's a whole video on that. But to write this in scientific notation, you could write this as 5.60 times 10 to the fourth power. Because if you multiply this times to the fourth, you would move this decimal over four spaces and get us to 56,000. So 5.60 times 10 to the fourth. And if this confuses you, watch the video on scientific notation. It will hopefully clarify things a little bit. But when you write a number in scientific notation, it makes it very clear about your precision and how many significant digits you're dealing with. So instead of doing this notation that's a little bit outdated-- I haven't seen it used much with these bars below or above the high significant digit, instead you could represent it with a decimal in scientific notation. And there it's very clear that you have three significant digits. So hopefully that helps you out. In the next couple of videos, we'll explore a little bit more why significant digits are important, especially when you do calculations with multiple measurements.