welcome to the second video for chapter one section five uh in this video we'll be discussing sig figs in a little more detail and we'll also be talking about accuracy and precision so for this video the learning objectives are to define accuracy and precision to correctly represent uncertainty in quantities using significant figures and to apply proper rounding rules to computed quantities which again is going to be a key step in representing values correctly using their sig figs okay so uh the first thing we're gonna talk about is how to count sig figs uh or significant figures in a given number uh real quick before we jump in i'm gonna be using the sig figs a lot and you're gonna hear this all the time it's just a lot more convenient to say than significant figures but one of the reasons that we uh need to know about sig figs we need to we we need to know how many sig figs a number r uh has in it um pretty regularly we're going to talk about uh why when we talk about how to um use sig figs in in computations so uh for example if we're wondering how many sig figs this number right that we talked about earlier had we're gonna say we need to know how many we want to count how many significant figures or how many significant digits this guy has and that might be different than say 21.64 milliliters if we had had a better uh graduated cylinder or different measuring tool to measure that that um the number of sig figs here is different and that tells us something about the um precision that we will and we'll define that term a little bit later but it tells us something about how uncertain our measurement is and it's going to be important for us as we do calculations later so uh counting sig figs is a crucial skill and we're gonna i'm gonna dive right in okay so the first thing that we're gonna do is say that all non-zero digits are significant so these guys are actually really easy because they don't have any zeros in them right so all non-zero digits are significant so this number has three significant figures two one six three three sig figs okay and this guy has four two one six four so there's four sig figs in this guy um so that's really easy if there's no zeros if there are zeros life gets harder um because zeros can be placeholders but they can also be measured values so we need to know what the zero was doing in our number in order to understand how to treat it with uh in the in the realm of sig figs so essentially if it's a placeholder it's not going to be a significant figure so that's a placeholder is going to be non-sig i'm going to say non-sig um a measured value is going to be um is going to be significant or sig right so if it's just a placeholder it's not significant it's just to denote that the number wasn't 10 times larger than than we should have written down um and if it is a measured value as in our scale said no no you actually have you know like there's a zero in there because that's the correct digit that's a significant figure because it's acting as a digit not a placeholder so yeah so we just need to determine how our zeros are acting in order to understand how many significant figures we have so we're going to define our zeros in three types of zeros we have leading zeros we have captive zeros and we have trailing zeros and if we consider them in this sort of context of if it's a placeholder or a measured value it becomes a little bit easier so a leaping zero is a zero that shows up before any non-zero digits um so right so if i were to write um 21.6 i could write 0 to 1.6 mils that's the same number it's really weird we don't usually do that but that's the same thing as it's a it's a leading zero um or here we've got a leading zero that just holds some hold some decimal places empty before so that's a leading zero and this is totally a placeholder right a leading zero is absolutely a placeholder you can see it really clearly it's just a placeholder um and so that's going to be non-significant then you have captive zeros captive zeros are guys that get stuck in the middle right they get stuck in the middle of between digits that are non-zero right so this zero is between a three and a nine this zero is between an eight and a two they are captive between non-zero digits those guys are always significant those are always significant they represent measured values right they can't be placeholders because they're between things that are acting as measured values they are also measured values so those are always significant and then we have trailing right and then we have trailing zeros and they can be either they can be either a placeholder like here right this guy is it is a placeholder um it's before a decimal it's just hanging out to say that this is not 309 it's three thousand and ninety right it's just moving everybody up a tens place um this this guy though is after decimal there's no reason to report that right there's no reason that we would have to report that zero the number doesn't change if we write the zero or not so that indicates that this is a measured value this is a real digit um so it depends right this is this is this depends it's significant if it's a measure value and it is non-significant if it's a placeholder and uh so essentially that's if it's if it's uh where it where it falls in terms of a decimal so if it falls after the decimal then we then it's going to be significant if it falls before the decimal then it's going to be non-significant so there is a solution to all of this messiness right this is this is unfortunately complicated and the solution is when in doubt write it in scientific notation right write all the digits you need to that um that are that that you would write down if you were going to write this guy in scientific notation so let's look at 3090 first off so 3090 um if we were going to write this in scientific notation we would consider well the decimal would be here so um scientific notation is always one digit with a decimal after it um so we're gonna go ahead and figure out how many uh we'll just you know we'll write the rest of the digits and then we're going to multiply by 10 to the however many times we had to move the decimal over so we need to move the decimal over one two three times so this will be ten to the third the question is do i need to report the zero or not and here i actually don't i don't need to report this zero um so i'm going to go ahead and erase it it doesn't do anything right there's no indication that i needed to report okay eraser let's please keep the eraser there's no need for me to report that right there's no requirement um that that zero is here to uh keep the name number the same um as what i i had it this is this guy was being a placeholder right um i don't there's no reason that i need to keep it i didn't measure it necessarily there are some ways that you can indicate that that was a real value that you um that you did report for reasons um including putting a decimal at the end of your number um to indicate that that was a real number but here there's no decimal so it's not it wasn't a real measured value so we're gonna leave it out this guy we're gonna change right so zero point zero zero eight zero two zero we're gonna move um this into scientific notation and here we're gonna write all of the digits that we would have had to write down to keep this number identical and this one we do write the zero because it's being a measured value right there's no reason that i would have had to write this down if i didn't know it here i wrote this number down just because i had to otherwise the value of the number would change and so it doesn't we don't keep it in scientific notation this guy we do um keep it because there's no reason to write that down it doesn't change the value of the number so i am going to keep that guy and then i'm going to write times 10 to the and then i had to make this number smaller so 1 2 3 so it's gonna be negative three because the number has to get smaller um okay so right now i this is easy right so i can i can right away figure out how many uh how many digits i've got so this guy i have three sig figs here i've got four sig figs and this is the same answer that i'm going to get if i use these complicated values right so if i count here there aren't any leading values in this number there's one captive zero in this number and then there's one trailing zero which is a placeholder so it doesn't count so i count my significant digits one two three and hey three sig figs over here same thing i don't count my leading zeros they are never significant i do count my captive zero because it's always significant and then here the trailing zero is a measured value um it's not acting as a placeholder so it also counts so i count up my digits one two three four four sig figs hey look i got the same numbers so when in doubt write it inside uh scientific notation it's just it's gonna it's gonna make your life a lot easier okay now we are going to deal with sig figs in calculations and how we deal with that so um the basically why we have to deal with this is that if you use any measurements in a calculation the uncertainty of your measurement is going to define the uncertainty of the result so in other words results calculated from a measurement are at least as uncertain as the measurement itself sometimes they are more uncertain because of a different measurement and uh to report our result we need to round appropriately to um to display our result with the proper sig figs so this is where we need to know the number of sig figs uh or the the correct sig fig we need to represent our our measurements correctly in order to um report our results of any calculations correctly okay so we're going to talk about the rules for significant figures how we deal with uncertainty for addition and subtraction because they are different than multiplication and division so we're going to talk about these two separately so when you're dealing with addition or subtraction we're going to use the decimal places as the limitation of um of our our our uncertainty and we always use the fewest sig figs or the fewest the most uncertain measurement is going to be the limit right because the other guy if it was more certain but then we do something with a less certain measurement we don't really know as much as we we could have if we had more certainty so we're always going to use the least certain measurement as our limitation so let's say we measured a couple of things so let's say first of all we measured um something that was 4.7 grams and we then also measured a different one that was uh 1.13 grams hey we had a better scale this time we had a better balance we had more information for this guy this this guy again was less certain and it's going to limit our our certainty that we can report in our final answer so i'm going to go ahead and show you how we're going to deal with the sig figs here so we're going to what we're going to do is we're going to line them up with the decimal places in line and these guys both have units on them which is important okay so i'm going to add these together and notice i have lined up my my decimal places so i'm going to go ahead and line them up and i'm going to wind up that my answer is 5.83 grams but my question is how many sig figs am i allowed to report and i have to cut it off here this is my limit the fewest number of decimal places is going to be my limitation and same thing if you have tens places or something it would it would be the same way um you line it up with your decimal places and then the guy with the fewest um places the fewest the lowest certainty is your limitation so i'm not allowed to report this last digit in my answer i have to round it so this is actually going to round to 5.8 grams so i can only report two sig figs because i am limited by my fewest number of decimals okay um and this works the same way as with subtraction all right with multiplication and division it's similar it's limited by the least certain value but here that's going to be the actual number of sig figs because you're multiplying um you can move decimals get moved around quite a lot so we're going to deal with the actual number of sig figs not the actual place of the decimal so let's say that i have 1.1 milliliters and then i'm going to multiply by a density to try to figure out a mass of something um so and then my density was given with more significant figures i had 120 or 1.21 grams per mil so the first thing that i need to do when i'm working with multiplication or division is count the sig figs in each in each value um so these ones are easy they don't have any zeros here i've got two sig figs and here i have three sig figs okay so again i'm going to be limited by the least certain which means the fewest number of sig figs here so i'm going to go ahead and do out this calculation and it turns out this is equal to 1.331 grams um and the question is how many sig figs do i get to report right and um so when you do multiplication and division you can you can wind up with a ridiculous number of sig figs and then it becomes really really crucial to um to truncate or well you don't ever truncate you round but it becomes really important to round off appropriately when you do these these um calculations okay so i'm going to be limited by the fewest number of sig figs so that's this guy right two sig figs is my limit so i need to round this guy to two sig figs so here i'm gonna round to two sig figs and i round off to 1.3 grams so um that's how you report each of these numbers with the correct number of sig figs all right um so these two rounding examples hopefully were pretty straightforward but we are going to talk about the exact rounding rules to use just in case you need a refresher or you aren't familiar with them and that is the last thing that we're gonna do in this video okay so the rounding rules are is if the digit to be dropped so we're gonna look at the digit that we need to drop um and then we're gonna make some decisions based on the value of that digit so for example um if i have let's say 1.12 and i need to round to two sig figs and we're gonna look at all these guys with two sig figs um so we're just gonna do two sig figs okay so if i need to round this guy off to two sig figs so i'm gonna look at the digit that i need to drop so i need two sig figs okay this two is the guy that needs to be dropped it is less than five so i'm going to round the digit uh to the left down which means this guy is going to stay as 1.1 okay all right in a different case let's do a rounding example where the digit to be dropped is greater than 5 right so again two sig figs that's this 4.7 portion but i don't know what this last number that i want to write down is i'm going to look at this last digit that i'm dropping and that's an 8. it's greater than 5 so i need to round up so this guy's going to round that seven up and i will report this as four point eight okay so you see the digit to be dropped was greater than five we round up okay and then this last case when it's exactly five uh this is when things get a little bit tricky because it's exactly halfway between so i d the idea is that you're rounding to the best possible representation of what that number is if you didn't have that last digit um to sort of help you understand so if it's less than five it's smaller it's less than half then we round down if it's greater than five it's greater than half we round up and the problem is if it's exactly five which way do you go and there's a whole bunch of different rules for this depending on what you want but basically the idea is that you want this to be statistically equal you want to round up half the time and you want to round down half the time and the easiest way to do that is just try to get an even number so let's look at two different cases 2.35 and 2.45 so in both cases the digit that we're dropping is exactly five and we don't know where to go to that so we're gonna just we're gonna just try to get to an even number right so we wanna write down our last digit as an even number this means this guy i'm gonna round up to 2.4 and this guy i'm gonna round down to 2.4 because it's the even number um and the way that this works is that half of numbers are even and half of numbers are odd so statistically um it should work out ish more or less that you round up half the time and down half the time and that over time with your calculations that'll work out to be about right and that's that's the rationale behind it so that's what we do when the number that you are dropping is exactly five so it's a little bit tough okay all right so the last topic that we're going to cover in this video uh for this section is accuracy versus precision so these are two words that you're going to need to be familiar with in chemistry because they have very specific meanings um and these are not the same meaning as like in in sort of modern language right when you talk about accuracy or precision um they may not mean different things to you uh in just sort of normal english usage right but they do mean very specific very different things in science language and that's really important that we are using proper science language when we're talking about science so um essentially they're defined slightly differently so one of them is that they yield a result close to the true or accepted value and that guy is accuracy so if you have a really accurate result that means you're really close to something that we know is true or something that is accepted like the defined published value for say the density of water um alternatively the other guy is precision and a precise accuracy sorry a precise measurement is a measurement that when you repeat it a whole bunch of times you get similar values so a way that we often talk about this is with this uh target practice right so imagine that you're shooting arrows at a target so you can have uh i don't know if you've ever done this or with a rifle um but you often aren't great at shooting arrows when you first start out so you almost never get this bullseye answer right away right um usually when you start out shooting arrows or rifles at targets you're pretty bad at it right you you miss the bullseye and also you shoot all over the target like you all over the place um and essentially that is when you are not accurate you miss the bullseye and you were not precise you were not shooting um in a way that you could repeat right it's all over the place it's just it's just you're not good at it right um this is a very sad place to be in the science land if you are not accurate and non-precise um then you really need to work on your experimental protocol to improve one or both of those factors uh the next thing that happens is as you get better at shooting things you tend to become more repeatable but you're still missing the bullseye right so this is um this is something that happens or or often if really good marks people are shooting with a new weapon right a new bow or a new rifle um and they're shooting super well they're just not hitting the bullseye right so this is this is um it's a pretty close what we call close grouping in archery um but it's not accurate you're missing the bullseye right you did a really good job of shooting the same way every time but you missed so that is precise but not accurate um and so that's you know well kind of okay you're getting there right you're getting there precise precision is important um so this is if you measure the density of water you get 1.25 every single time even though that's not the accurate that's not the true accurate value you're getting the same number every time that's that's you're moving in the right direction and then the best thing um when we're really happy in lab is when we are both accurate and precise so that is if you are getting your result is close to the true value um and all your measurements are really close together so that's accurate and precise this is awesome there is also one last option where you are accurate but not precise and that's also a sort of halfway face so this is like if you so here's the bullseye sorry so this is like if you miss but like every time you miss but all of your values average out to the true value so your average value is pretty close to the true value but all of your measurements are really far apart this is accurate but not precise this is uh not a great place to be this is this is probably a frowny slightly frowning place this is this is pretty bad you might as well just lucked out like it's pretty bad um so anyway so this is accuracy and precision and these are the terms that we often think about we will think about uncertainty in terms of in science