Rules of significant figures

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 is significant and the 0 is in between them it's also going to be significant so let's write this over here so any nonzero digits non-zero digits and zeros in between zeros in between are going to be significant are going to be significant that's pretty straightforward now the the zeros that are not in between nonzero digits these become a little bit more confusing so let's let's just make sure we can rule out some of them so you can always leat rule out when you're thinking about significant figures the leading zeros I'm going to talk about leading zeros I'm talking about the zeros that come before your non-zero digit so these are leading zeros here these are leading zeros there's no leading zeros here no leading zeros in this one this one and this one but in any situation the leading zeros are not significant so leading zeros not significant all right over here leading leading zeros not not significant and so the last question all you have left I mean you only have nonzero digits and zeros in between you could have some leading zeros which we'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 zeros the zeros the zeros behind the last non-zero or to the right of the last non-zero digit so these trailing zeros here there's a trailing there's actually two trailing zeros over here and then there's three trailing zeros over here so let me make a little so trailing zeros trailing trailing zeros what do we do with them so the easy way to think about it is if you have a decimal if there's a decimal anywhere in your number count them if you have a decimal count them them count them as significant they are significant than they are count them as significant if there's no decimal no decimal anywhere in the number then it's kind of ambiguous you kind of not sure and this is a situation so the so clearly over here there's a decimal in the number so you count the trailing zeros 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 zeros there are no trailing zeros 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 to the one and just happen to get exactly on 37,000 so if there's no decimal let me write it this way there's it's ambiguous ambiguous which means that you're you really don't you're not sure what it means it's not clear what it means and you're probably safer assuming to not count it so not not count not count the trailing the trailing zeroes 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 have you do measure and let me do a different number let's say you do measure 55 56,000 and there is a notation for specifying that six definitely is the last significant digit and sometimes you'll see a bar put over the six sometimes you'll see a bar put under the six and that could be useful because maybe 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 have to you would write something like five hunt or there's to still write 56,000 but then you would put the bar above that zero or the bar below that zero to say that if that was the significant digit so if you saw something like this you would say three three significant digits this isn't used so frequently a better way to show that you've measured two three significant digits would be to write it in scientific notation and there's a whole video on that but to write this in scientific notation you could write this as five point six zero times 10 to the fourth power right because you have you'll have to if you multiply this times 10 to the fourth you would move this decimal over four spaces and get us to 56,000 so five point six zero times ten to the fourth and if this confuses you watch the video on scientific notation 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 you instead you can represent it with a decimal in in scientific notation there it's very clear 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