hi and welcome to the section of the chemistry tutor and in this section we're going to cover the very important topic of significant figures or they may be called significant digits in your book basically we're going to be doing a lot of calculations in chemistry and when you're doing calculations with more than one number it's really important to look at how accurately each of those numbers in your problem are measured because your final answer really should be how many digits you put in your final answer should really even be dependent upon what you started with and so I think the easiest way to really illustrate that and why that's important is from a real-world example not a chemistry example but just a real-world example that we all have some experience with so we'll do that really quickly and then I'll give you the rules of significant figures and then we'll go ahead and produce some practice so that you can get the hang of it it's one of those things really that you can memorize rules all day in the book but until you sit down and really practice it and count significant figures and get some practice with doing some calculations you really won't get it until you do that so I want to definitely spend some time at the end giving you that practice so this gets some motivation for why this is important let's say you're going down the road and you're traveling at some speed right what speed is that let's just take a example from real life let's say your speed in a car or a train or something like that is ten point eight two meters per second this means meters per second so this means that every single second I am traveling 10.8 two meters so really need to wrap that in your brain so another second goes by another second goes by another second goes by every time a second goes by I have traveled ten point eight two meters so the fact that I actually have a decimal point here in other words I didn't measure this as 10 meters I didn't measure this is ten point nine meters I didn't measure this as you know ten point eight two three six seven nine three two meters but I measured it as ten point eight two and what that means is that I have sort of an implied accuracy to the speed that I'm really traveling so the train or the car is really going at some speed that's truth but when I measure it you know I have to get a camera out or I have to get a yardstick and I have to do my best to judge it and I only measure it to two digits after the decimal point that's important so hold that in the back your mind now let's say that I travel this speed for a time a time let's say and the time that I'm traveling if the speed is two point six five four to seven seconds all right so I'm going down down ten point eight two meters per second and I'm traveling for two point six five four to seven seconds now the first thing you should notice is that the the accuracy with which I measured my time is much greater that means that whatever device I used to measure the time I was able to put more decimals after the decimal point because it was it was a more accurate representation of the true time so I'm going down the road I go in a certain speed I measure the best I can and this is the best I can measure that guy with the measurement tool I have I click a stopwatch I see I've gone down the road this many seconds and I have more decimals because the way that I'm measuring time maybe they have a really good clock clock or something like that and so I'm able to put more decimals after the decimal point so let me now ask you a question if I'm traveling at the speed and I'm traveling for this many seconds how do I calculate how far I have traveled well if you think about it change the numbers to something easier if I'm going one meter per second for five seconds how far have I gone well I just multiply those two numbers one meter every single second but I'm going for five whole seconds so I multiply those two numbers to give me the distance how far have I traveled over that time period so the distance there's an example of a calculation that you might actually have to do in real life when we change colors here the distance that I'm going to travel is the velocity times the time V times T all right so if I were going to do that I would go in my calculator and put ten point eight two and I would multiply it by two point six five four to seven because this is the distance I'm sorry this is the velocity m/s this is the time in seconds I multiply it now when I do this what do I get now in the calculator the calculator will give me the following thing the calculator will give me two eight point seven one nine two zero one four meters I'm gonna put in a bracket so you can sort of see it stands apart from the number twenty eight point seven one nine two zero one for most of us before we take a chemistry class or a physics class that would be a perfectly good answer and we would round the number maybe we would round it to two or three decimal places or whatever we've just picked a number out of our out of thin air to be able to round that guy but you should be able to convince yourself that it doesn't really make sense that this is really the true answer because for the distance that I've traveled I mean it's it's accurate in the sense that the calculator gave us this number but your answer is only as good as the information you put in and this guy was only able to be measured to really to two decimal places after the decimal point right so it doesn't really make sense to include one two three four five six seven decimals after the answer there just doesn't really it doesn't pass the smell test because if I measure if I'm doing a calculation and I measure one quantity really really accurately million decimal places and then I measure another quantity for the calculation to like two decimal places and I add them or subtract them or divide them or multiply them does it really make sense even though the calculator gives you all those decimals the calculator just going to give you numbers it's just doing math but to interpret the answer it doesn't really make sense to take this as a true answer because when you give this many decimals after the answer it implies that you really are confident in these numbers that they're correct and in fact they really probably aren't correct because our velocity was only measured very sort of inaccurately compared to to the time so if you were gonna round it just you know without using any real rules of significant digits but in your head what would you pick you might pick I don't know let's just do it 28 point what makes sense what's round 2 7 2 7 2 meters now I haven't given you any rules of significant digits yet but this looks like it might make a little bit of sense because one of our original numbers has four digits in it really two digits after the decimal the other guy has a lot more but when we do the math between them it makes sense that the final answer should be something with fewer digits in the end there because it's more representative of what the information we have begin with this so this is the concept of significant digits and I'll just give you one little example this guy has four significant digits sig digs you might you know your teacher might call them but really it's significant digits so one two three four the decimal point has nothing to do with it really you just count the digits and we're going to get into a lot of examples it gets a little bit more complicated than that but that's the basic idea you're really you're really in chemistry need to be careful because everything is being measured when you measure five milliliters of a liquid when you measure 28.2 grams 28.2 grams is measured but if I instead measure 28 point two seven nine three nine to seven grams then I'm definitely confident in my measurement in the second in the second hand way and the second example they're more than just measuring it to one decimal place so when I do a lot of math involving numbers I really need to be careful that my final answer only has the right number of digits representative of what I put into my problem that's what significant digits is all about so keep that in your mind it wouldn't make sense to round this to 30 meters it would make sense to round it to 29 meters wouldn't make sense to Ram it to 28 point no seven to nine meters in this case it's going to make sense to round it to this and it's sort of an accepted way to do it that you're gonna use all throughout your class so you need to get comfortable with significant digits you're going to use it in every calculation there and it's really intended so that the students and not only students but in real life when you're doing real experiments that your answers kind of make sense based on your on your in put information alright so let's talk about how to count significant digits how do we know how many significant digits do we have in a number so let me know in title the significant digits all right now the strategy here be honest with you is we need to outline what we're doing and I'm going to put it on the board so that you can see how to count them and then really what we're gonna do is learn how to use this when we do some problems so you can try to memorize them and hopefully get some information about what I'm gonna put on the board here and use it and file it away in your head but really and truly you're not gonna really nail it home until we do some problems so the first thing you need to remember when you're doing significant digits the first thing the first thing that you should really be doing is you count the digits it's pretty simple count the digits yeah the number all right so let me give you an example of that let's say we had the number twenty three point four five six it does not matter that there is a decimal there all right so this is one two three four five and go ahead and put it like this just to make it absolutely clear this is five significant digits one two three four five so congratulations you've already successfully you know counted significant digits in a real number so it's not that difficult what you're really gonna be doing most of the time when you're doing this significant digit stuff is just counting the digits in your numbers the thing you got to be careful about is zeros that's really the only thing you have to really worry about when you have digits like this with no zeros in it it's very simple you just count the digits that's the number of significant digits that you have but when you have a number that has a zero limit either before a decimal or after decimal or what we'll talk about it here in a minute but zeros in the problem then you need to be a little bit more careful so let's look at that down okay let's go and say the next little rule a zero before a decimal point is not counted and let me give you an example of that zero point one two four at first glance you might be tempted to count this and say there's four significant digits but there actually isn't there's actually three significant digits in this guy now you might say why well I'm giving you the rule it's an accepting it's an accepted standard when there was a zero in front of a decimal point you don't count it the reason you don't count it is because when you think about it it doesn't really add any accuracy to this number to put a zero here in fact I could actually cover the zero up and just put point one two four and it would convey the exact same information so digits are really only counted in terms of significant digits when they add accuracy measurement accuracy to a number in this case it doesn't because you know that you can actually cover it up and if I write down point one to four you're gonna know exactly how accurate I was able to measure that guy as if I had just put the zero there the zero is cosmetic in other words it's implied by the decimal point that there's a zero there when there's no other number here it's implied it's always there right so when Z and that's why zeros are a little tricky sometimes when you put a zero in a number it's actually adding to the accuracy of your measurement we'll talk about when in a minute sometimes when you have a zero in a number it like right here it's not really it's there but it's not really telling you that the number is any more accurate because in fact you can eliminate it and it would still mean the same thing so if there's a zero and only a zero before a decimal like this it's three significant figures I should say it's not counted so in this case it's three significant digits if I had another digit here it would be four significant digits and so on all right let me go ahead and give you another little rule leading zeros to the right of the decimal point I not counted and again it's dealing with zero so I told you zeros are a little bit special let me give you an example of that what if I have zero point zero zero zero two two three four how many significant digits is this guy well you should know from this rule that this one doesn't count at all but when I'm telling you with this rule leading zeros to the right of a decimal see these are what we call leading zeros they're zeros that are before the main digits but in fact this guy has four significant digits one two three four the reason you do not count these is exactly the same reason as before because they don't really lend any additional accuracy I mean they're they're basically stuck there to position and tell you how small this number really is they're placeholders in other words they're stuck in there yes they are part of the number yes they have to be there but they're not really adding to the accuracy the digits here at the end this is what the scale or the flask or whatever it is you have they're telling you how accurate you're able to measure this number these zeros are just put here because my number is very small they're placeholders in other words hopefully that makes sense to you if it doesn't really should just memorize it and don't fret too much about it because you're gonna use this so much in chemistry that really in just need to memorize these things trying to give you a couple of reasons so that hopefully they'll sink in a little bit but basically leading zeros before a decimal don't count them leading zeros after a decimal don't count them so far it's pretty easy leading zeros before the decimal leading zeroes after nothing well don't count only count the digits all right so that's sort of another rule now let me give you one more trailing zeros after a decimal point are counted all right are counted let me give you an example of that one point two three zero zero how many significant figures does this number have well based on this rule we have one two three four five significant digits now you can just memorize the rule trailing zeros these are called trailing zeros after a decimal point right these are counted the reason they are counted is because there's some value in having these guys here in other words I don't really have to put these zeros here I can cover them up and just put one point one point two three but if I actually go through the trouble of writing these extra zeros here it means that my scale or my flask or whatever a measurement tool I'm using has that much additional accuracy to measure it that much more accurately to stick these two zeros at the end it's kind of hard to describe other than just telling you that but these guys are merely placeholders because the number I'm measuring is very small so these are just used to shift the numbers down over here these guys are placed here because I'm actually measuring this guy with much greater accuracy what if I put 10 zeros there one point two three zero zero zero zero zero zero zero zero well I don't have to put those zeros there but the fact that I do put them there tells me that whatever I'm using to measure this this mass or whatever has a precision and the accuracy to be able to pull it off so they are counted so let's review the meaning a little bit here and you count your not your digits okay no problem if you have a zero before a decimal you don't count it because it's cosmetic if you have leading zeros after a decimal you don't count them because they're cosmetic if you have trailing zeros after a decimal you do count them all right and let me go ahead and tell you here let's see here just to make it clear since we're talking about zeros zeros between other significant digits are counted alright let me give you an example of this what about 702 alright 702 this is a significant digit this is a significant digit this is not a cosmetic number this is actually a part of the number it's telling me the accuracy with which I've measured this guy is at least to 702 so when I have a zero sandwiched between other numbers it is counted so this guy would be three significant digits okay let me give you a couple more rules and then we will review them and I know I'm giving you a lot of things to look at but really it's better to just build it out and that way you can memorize them and use them rather than guessing for scientific notation which you're going to use a lot in this class only count the digits prior to the power of 10 you know in significant figures I should say scientific notation is always going to have that so as an example what if you have 2.4 3 times 10 to the 6 this is an example of scientific notation right 2.4 3 times 10 to the 6 we talked about that before how many significant figures does this number have well you don't count the 10 and the 6 is what we're really trying to say you count these numbers before so it has 3 significant figures 3 significant digits okay 3 significant digits so when you're doing scientific notation just look at the guys before their and count those guys that's all you need to do alright one more thing I want to talk about is the following significant digits only apply to measured quantities because that's the whole point right talking about measurement accuracy then this is a very important sentence so open your ears here counted quantities are perfect in other words significant figures don't even apply to things that we count because the whole point of significant figures is based on measurement you don't want to use too many or too few decimal points in your answer because you're measuring something with a certain accuracy right but if you're counting golf balls or you're counting tires or you're counting marbles then significant figures don't really matter because that's an exact number that you've really picked so you don't even really even have to deal with that so a good example would be two balls four significant digits not applicable you don't even you don't even need to worry about them I guess what trying to say what about a hundred coins not applicable alright because I've counted out two balls I have two exact balls it's totally exact there's no point to even talk about significant digits because I know what the exact answer is a hundred coins I've counted a hundred discrete little coins I know what the answer is exactly so that doesn't even matter and that brings up one final point I'm gonna leave you before we work some problems here this is a number that's an important number when I have there's no decimal here but when I have a significant digit and then zeros that fall after it these zeros in this particular case because I'm counting coins I told you significant digits don't matter but what if I what if they're not related to coins what if I just have a hundred of something 100 grams of something these two digits are they may be significant and they may not be significant and I know that that even that even gets uglier and it's not very pretty to say but that's the truth they may or may not be significant because when you put two zeros after and there's no decimal point here implying accuracy you don't really know if you measured exactly a hundred grams like exactly because if I put a hundred point zero then I would know I would know with with total certainty that how accurately because I have a decimal place how accurately I measured but if I measured a hundred grams I mean who knows maybe my scale is really not capable of giving me anything other than you know whole numbers of grams so when you have hunting ulam 100 like this with zeros like that it gets a little bit ambiguous so to remove any ambiguity in your answer sometimes it might be better to represent this in scientific notation maybe one point zero zero times ten to the two if I were talking about grams of something right because if I do that then going by the rule above this has three significant figures and what we're gonna use this for later is when we do calculations it's important to know how many significant figures we have at a number because when we do a calculation we're going to use that information to choose how many digits are in our final answer so that's why it's important so if you have a hundred grams of something and you look at this and you're not sure if it's one significant digit you don't count these guys or if it's three significant digits you're not really sure it can really affect your final answer because you might truncate decimals right off the end of your answer without knowing that so if you're not sure or if you have any power over it then you would just write it in terms of scientific notation and that would tell whoever is looking at this that there's definitely three significant figures here when you look at this guy since there's a two you would be shifting this decimal to so these two these two numbers are exactly the same thing 100 coins or 100 grams of something and 1.00 times 10 to the 2 represents the same thing it's just that here you're totally clear that it's three significant figures and over here it's really not so clear but certainly if you're counting things or if in a book it's like telling you that there are you know three flasks on a table you're not gonna even knew significant digits there because when you're counting objects those numbers are exactly perfect all right what I want to do here is go over these really quickly just to kind of drill them in a little bit but like I said you can study rules all day long but you won't really get comfortable until we actually use them in some calculations so we're gonna do that next significant figures here is a laundry list of rules bottom line is you want to count your digits don't care about decimals they don't even play here so just count your digits 1 2 3 4 5 when they're non-zero you just count them five significant digits if you have a decimal what you're going to have a lot of in this class cuz you're measuring things if it's a leading zero before a decimal it doesn't count because it's cosmetic you only count the digits after right now on the digits afterwards if you have leading zeros before your significant digits here you do not count these because again they are cosmetic they're just placeholders to push your number and show you how small of a number you really have so really none of these digits count when we count our four significant digits here if we have a decimal with trailing zeros behind significant digits they do count because it's implying that my scale or my you know my balance or whatever has the required accuracy so that I can measure this out this bar so I do count them so five significant digits here now when I have a zero sandwiched between two other numbers it always counts it's something you should just remember it always counts because it's it's part of the magnitude of the number which means I'm able to measure it to at least 702 you know accuracy so this guy is definitely going to count when you have a zero sandwich between numbers like this when you're using significant digits which a lot of times in your problems you will you might have 3.29 times 10 to the minus 2 grams they might give it to you like that well the significant digits are only counted before the scientific notation party only this guy so in this case three significant digits all of the other rules of zeros and everything applied to this but you only look before this guy finally when you're counting objects maybe in some certain problems you may have a something that's counted an exact number that means it's totally exact you haven't measured the number of of balls with with any ambiguity you know how many balls you have you know how many coins you have so you don't even care or factor in the significant digits in your calculation if you've actually counted something like that but to remove any ambiguity if you have a number like 100 with some trailing zeros and no decimal point here maybe have 300 grams of something maybe I have 20 grams of something that zero there we're not sure if it's significant we're not sure if it is significant so it's better to write it in scientific notation if you have to we're gonna do an example to show you that in a minute now this is the these are the rules that we use to to determine how many digits significant digits a number has now we're gonna learn some rules to do calculations and they're not going to be very hard but we're gonna learn them right now and then we're gonna do a lot of examples so let's go and figure this out now if you were multiplying and dividing numbers multiply and divide significant digits what do you do alright basically the answer has same number of significant digits as the number in your problem with fewer or fewest significant digits that's a lot of words what is it basically saying to give you a concrete example is if I'm taking the number three point four how many significant digits do I have two significant digits and I'm multiplying it by something like one point nine eight seven six right if I do that in my calculator what I'm going to get is six point seven five seven eight four this is the calculator number and I know that I want to keep all those digits because one of my numbers I used to begin with it wasn't even measured very accurately this guy was measured a whole lot more accurately so how many digits do I keep basically my answer that given by the calculator is given by this guy and I'm gonna keep the same number of significant digits as the number and my problem with the fewest significant digits in other words this number has two significant digits therefore my answer must have two significant digits so the way I'm going to write that is six point eight I'm gonna round it here because I have a five here this is gonna round up to eight and I'm gonna truncate everything right there two significant digits two significant digits that makes more sense because whatever I'm multiplying here I was not able to measure this guy nearly as well as I was able to measure this it doesn't matter how many decimals are after the decimal point it just matters the total significant digits two significant digits here two significant digits here all right let's go and list and that's for multiplying and dividing let's go and look at the equivalent rule for adding and subtracting so to add and subtract significant digits alright the answer has the same number of digits after the decimal point as the number in the problem with the least number of digits after the decimal with least number numbers after the decimal so this is a little bit different of a rule all right so let me give you an example and then we'll do a ton of examples on the next board here what if I have twenty three point nine nine seven and I'm going to subtract eighteen point one okay so I put that in my calculator to three point nine nine seven minus 18 point one what I'm going to get out of my calculator is five point eight nine seven this is from the calculator all right but it doesn't really make sense to keep that as my final answer because this guy I was not able to measure nearly as well as this guy so I need to figure out what to do here when I'm adding or subtracting it really matters more what's on the right-hand side of the decimal point and there's theory behind that and all that but really the bottom line is it's accepted standard so you know really you should just memorize it so here this has three significant figures after the decimal this has one significant figure after the decimal so the answer that I'm gonna actually keep is five point nine I'm gonna round this up right here five point nine with significant digits because this rounds up and I'm going to only keep one digit after the decimal notice that this number actually has five significant digits this number has three significant digits but in my answer I only have two significant digits because this is different than multiplying and dividing I don't really care about the total number of significant digits I care about what the significant figures are behind the decimal point that's the difference so really quick summary multiplying and dividing I'm concerned with the total significant figures in both numbers the answer is going to have the number of significant figures as the smaller number in total it doesn't matter what's behind the decimal or in front it's the total number of significant figures driving the answer this one is different it only matters how many digits the significant figures I have behind the decimal here I have one digit behind the decimal so my answer is going to have one digit behind the decimal I know it's a lot to soak in you had to learn what significant figures are why do we care about them how do you count them all of the rules and then when you do calculations involving them how do you count them in the final answer what we're gonna do now is erase the board and give you several problems really quickly here to give you a good practice and good footing so that you'll get really comfortable with significant figures and then the good news is or however you want to look at it good news bad news whatever when you're doing every problem from here on out in chemistry class you're gonna be doing the significant figure so much that you're not really gonna even think about it too much once you learn these rules it's gonna be common sense to you really to look if you have two digits here and four digits here and you're multiplying them what do you do so let's erase the board work some problems and give you some practice okay for our first problem what we want to do is count how many significant figures there are in the number that's all we're going to do what about the number 8008 how many significant figures do we have and that number well obviously this is a significant figure this is a significant figure we have these two zeros but they are sandwiched between significant figures so they count one two three four this has four significant digits okay what about zero point zero zero zero seven five how many significant figures do we have here well obviously these counts so there's two leading zeros before a decimal do not count leading zeros that come after a decimal but before my other ditches these don't count either so this is all it counts only has two significant digits in this number okay what about zero point zero four nine three zero zero zero point zero four nine three zero zero how many significant figures do I have here well leading zeros do not count leading zeros after a decimal do not but trailing zeros after a decimal do count and of course these counts so I have one two three four five significant digits five significant digits all right what if I have 6.02 times 10 to the 5 right this could be a measurement in grams could be a measurement of a volume could be anything what do you do here well do you count any of this here course not you don't count the exponent you don't count the X of course you have 1 2 3 3 significant figures notice that in this in this case this zero does count the reason that that zero does count is because it's sandwiched between two other significant figures like that alright so it does count so 3 so you have to configure okay II what about four point two zero zero four point two zero zero times 10 to the 5 what do I do here well this counts this counts these are trailing zeros so they also count one two three four significant figures okay what about F zero point one zero five zero well this does not count because it's a leading zero this is a significant figure this is a significant figure the trailing zero is significant this zero is sandwiched between other significant figures so it also counts so we have a total of four one two three four four significant digits all right and what about G what if I have five point zero zero times 10 to the nine how many do we have there same sort of deal so we have this counts of course these these don't count after the decimal they have two zeros here but they they do count because they're sort of like trailing zeros if you can think of that way so they have three significant digits here one two three three significant digits so after the decimal the the only reason to even put these zeros here is to show additional accuracy because really the number can be expressed without them the fact that we put the decimal and left them in there gives us the additional accuracy is telling me that my measurement device is accurate accurately capable of displaying that so you count them as three significant digits let's do an easier one 451 how many significant figures do we have 451 grams of something like one two three three significant digits and what about 0.01 six nine again this does not count this does not count because it's a leading zero on the right hand side these trailing three do counts so we have three significant digits so just wanted to give you a good overview of some of some example problems that you could see the smattering of what you might see on your exam and so or on your test or in your homework so those are just simple significant figure problems and I'm gonna I'm not gonna lie to you it does take a little bit of practice to get hang of those zeros when do they count when do they don't count and so just do the best you can it's really designed to try to prevent you from using way too many digits in an answer a lot of students when they first start they'll put every digit the calculator gives them one of those digits at the end don't mean anything but if your input data only had two digits to begin with that's what we're trying to do here give you a good feel for what makes sense okay now we're going to do a few actual of arithmetic problems let's say we're doing the following calculation thirty-six point five minus two point one six plus three point four five two now if you actually put this in a calculator what you're going to get is thirty seven point seven nine two that's from the calculator what would be the answer that we would actually circle or write down on our homework it's not going to be thirty seven point seven nine two that's just too many especially since a lot of these numbers didn't have that accuracy to begin with so what do we do we're adding and subtracting here right so when we add or subtract significant digits like this we really only care what is after the decimal the smallest number of significant digits after the decimal this has three significance after the decimal this has two significance after the decimal this has one significant after the decimal so the final answer is only gonna have one significant figure after the decimal thirty-seven point I'll round this up to eight this is the number that we're gonna pick with our significant digits thirty seven point eight the reason we pick it is not because this is three significant figures in the answer it's because we're rounding our answer to only have one digit after the decimal because this is this this is a smallest number of digits after any of these decimals in our problem and that's what we do when we add or subtract what about one hundred and fifty-one plus four point one six minus zero point zero two two zero what do we do with that well we're adding and subtracting so if we were going to actually put this in our calculator the calculator would spit out one hundred fifty five point one three eight and that's what the calculator would give us but that's not going to work out but let's look at what we have here we have one two three significant figures after the decimal in this number two significant figures after the decimal this number and here we actually don't have a decimal at all so we have basically no significant figures after a decimal so in that case we're gonna round this guy to 155 and just leave it like that that's going to be the number that we're going to pick because we don't want to have any significant figures after a decimal point here because this is the least number of significant figures after a decimal again this one have three after the decimal this one had to enter the decimal this one had none after the decimal so our final answer because we're adding and subtracting we want the zero significant figures no significant figures after the decimal so we just round it accordingly okay I know it may not make sense to you to do this calculation and not put any decimals afterwards but it really makes sense because the the weakest link so to speak is this one we couldn't measure him as accurately for whatever reason so our final answer yeah we have these digits here and it's pretty to write them all down yeah but they don't really give you any confidence that there recked because this guy didn't have any digits at all after the decimal all right now let's do some multiplication let's do two hundred and sixty five point zero two times zero point zero zero zero five eight one times 12 point one eight now if we put these all in a calculator and multiply them out what we will get is one point eight seven five four three five and some other digits and we're gonna get this in our calculator like that but I think you figured out by now we're not gonna actually keep all those digits but here we're not adding a subtractive we're multiplying so what we want to do now is we're not as concerned with what's after the decimal we're going to look at the total number of significant digits in each one of these guys this has one two three four five this does count because he's sandwiched between two significant figures so one two three four five five significant figures okay this guy only has three significant figures because these zeros here they're they're bleeding zeroes behind the decimal so they don't count so we have three significant figures here and here we have four we just count the digits so again five significant figures three significant figures total and four significant figures total we're not looking at what's after the decimal or before we're counting total significant figures so the smallest of these is the three significant figures here so we want to round to three significant figures 1.88 this is a 5 so we round this up to eight we have three significant figures in our final answer and that is the answer okay let's do one more problem let's do 33 point five eight multiplied by one point zero zero seven and then you take the answer that you get here and you divide it by zero point zero zero 705 okay so you multiply those guys on top divided by the number on the bottom and what you will get is four seven nine six point four six two four and then some other digits after that and this is what the calculator will give you but you're not going to keep all of those what we need to do here because we're multiplying dividing we want to look at the total number of significant digits in each one of these numbers not what's after the decimal or what's before if like for addition and subtraction but the total number of significant digits here we easily have four we can count the four here here we also have four believe it or not these zeros do count because they're sandwiched between two significant figures one two three four all right and this guy is three significant figures this doesn't count because it's leading zero this doesn't count because these two zeros are leading on the other side here this zero does count because it's sandwiched between two significant figures so one two three three significant figures so what we want to try to do is round this guy to three significant figures so let's try to do that this is the third significant digit unfortunately it's a nine and this is a six so if we try to round up we're gonna get something like four eight zero zero or something because it's kind of a ripple effect this is a six if we try to round this up to a ten then this is going to go to an eight because we're going to round from 79 to 80 basically so four eight zero zero so you could circle that and that's sort of the best you can do but it's a little bit ambiguous how many significant figures are in this number this is significant this is significant but these two here I mean we're doing the problem so I know we're trying to do here we're trying to say that at least this guy's significant but if you just write 4,800 down they're not going to know did you count 4,800 items is a significant at all or did you really measure 4,800 in which case they're all significant in other words this number is really difficult to to write in terms of three significant figures because of the fact that the zeros popped in and zeros are kind of ambiguous when they're left behind a whole number like that something you kind of have to get used to I know it's a little bit confusing but the way you get around it to write your answer down in a bullet proof way that will never get you the long answer is write it in scientific notation instead of writing it this way write it as like this four point eight zero times ten to the three and when you think about it this is what you're really trying to say because if you move the decimal three spots one two and then one more it really is 4,800 it's expressing the same number but because I write it in scientific notation this is significant this is significant trailing zeros are significant also after a decimal so because I write it in terms of scientific notation all three of these are significant I have three significant digits in our answer and then there's no ambiguity here when I try to write an answer now like that so if you ever get into a situation where you're trying to round something up and you really can't round it to the exact number of significant digits that you want primarily that's going to pop up when you get zeros here because you're not going to be sure if those are significant or not then just turn around and write in scientific notation so it's a good idea for you to be comfortable with scientific notation in general but it's very very important for that too all right I think we're done with this section we've done a quite a bit it's important and I could work 25 more problems but really what we're going to do is get practice doing all this as we go and work problems in the next section and the one after that and the one after that because we're constantly going to be counting significant figures and writing down the right number of of significant figures the basic idea is you want to count the significant figures in your in your input numbers so that when you get an answer you know how many digits to keep when you're multiplying and dividing significant figures then or numbers that you're trying to count these significant figures are you're going to be more concerned with the total number of significant figures in the input numbers whichever one has the smallest number of significant figures like this guy this had the smallest number of significant figures so we wrote our answer to three decimals and not to three decimals to three significant figures total but when you're adding and subtracting you're more concerned with what lies beyond the decimal point so this guy has three significant figures this guy has three significant figures this guy has three significant figures total but after this guy after this implied decimal I have zero significant figures this guy has two significant after the decimal and this guy has three significant after the decimal so since this is the smallest number I want to keep zero significant figures after that decimal point it's a lot to absorb but as you go through chemistry you'll find that it gets easier and easier as you do more more calculations review this section as many times as you need to get the hang of it plow through the rest of your sections I promise it will come easier with that you'll get your right answers on your test and it's really important to get the hang of this because if you if you start guessing a lot and getting the wrong number of significant figures then your teacher is probably going to take off a few points we're not including the right number of digits in the answer it's worth your while to hang on to those points and get the right answer by understanding these rules here