 So in the previous video we were talking about the difference between measured numbers and exact numbers and one of the things that I mentioned is that measured numbers are not perfectly precise and I want to sort of go over that in slightly more detail using an example of a scale or something that's going to measure how much you weigh, again pretend in some fantasy world that I actually weigh somewhere in the neighborhood of 128 pounds. So let's say that this is, we're talking about a scale called Scale A and let's pretend that this is Scale A and Scale A says that you weigh, that I weigh 128 pounds. So probably I don't weigh exactly 128 pounds, I might weigh 128.2, I might weigh 127.9 But this scale is only good down to the ones place as far as its ability to measure my weight. What this scale is basically saying is I know that you weigh more than 127 and I know that you weigh less than 129. So I'm going to call it 128 but where you are with respect to 128 I'm not exactly sure. You could have a different scale, you could have a more precise scale called Scale B, so here we're going to have Scale B and in this case maybe it says that I weigh 128.2 pounds. Even Scale B is not perfectly precise, Scale B is basically saying look I know that you weigh more than 128.1 and I know that you weigh less than 128.3. So I'm going to call it 128.2 but where exactly you are with respect to 128.2 I'm not exactly sure, that's as good as I know. I know down to the tenths place, so this is the tenths place, excuse me, and then pretend you have a much more precise scale, Scale C and it knows how much I weigh down to many many digits past the decimal point. Even this scale is not perfectly precise, it just knows down to one two three four five six seven seven digits past the decimal place how much I weigh but it is conceivable that there are other digits after that last two and that's why I'm putting those question marks there. There are other there are possibly other digits that contribute to my exact weight that even Scale C isn't capable of figuring out so again whenever you make a measurement um a lot of times the the well almost all of the time the measurement is not perfectly precise this is true even if I have zeros at the end. Scale A pretend I have uh I'm weighing someone else and that person weighs 135 pounds, Scale A only measures down to the ones place so Scale A is saying I know you weigh more than 134 less than 136. In Scale B you might think that it's uh measuring with the same precision as Scale A but it isn't it has a point zero at the end and in math class what they tell you is 135 and 135.0 are the same thing. What I'm going to tell you is that that's not necessarily true when you're making a measurement. All you really know with 135.0 is that you weigh more than 134.9 and you weigh less than 135.1 and so the scale is calling it 135.0 but there may be other numbers other digits hidden after the zero that your scale just is not good enough to detect and it just so happened that it got lucky that it landed on the zero so it looks like it's exactly the same as 135 but Scale B is measuring more precisely but it's still not measuring perfectly precisely and then Scale C um again it is saying it's saying I know how much you weigh down to the second uh digit past the decimal place but after that I don't know so it could be that even with Scale C that you weigh 135.00 uh three pounds something like that but the scale was not good enough to figure out the third digit past the decimal place so you have to realize that even when you see numbers uh with uh 0.0 or 0.00 um and their measured numbers those those measurements are still not perfectly precise. All right um there is a way of describing how precise a measured number is it is one this is one way of describing I should say and this uh this way that we're going to talk about in a fair amount of detail is called significant digits I want to mention that there are other ways students typically hate the significant digit section and I kind of hate it too um there are other ways that are used at least in beginning chemistry courses a lot of times the the course makes you feel as though significant digits is used constantly and and all the time and that that's not really true but unfortunately I have to teach you something and this is kind of the standard thing that gets taught so settle in um get some popcorn uh I don't know maybe in some beer so here we go uh significant digits are a way of describing how precise a measurement is so the way that you do it and this isn't going to make complete sense until we get to the end is you take all of the digits in your number and you break them into two groups you break them into group into digits that are called significant and the other group is the digits that are not significant and then what you do is you count the number of digits that are in the significant group and the idea is that if you have a lot of uh if your measurement has a lot of digits that are significant that it means that your measurement was very precise compared to another measurement that has less or fewer significant digits and that's basically the punchline I I imagine that this doesn't make much sense um at the moment but uh with some examples it probably will let's see so this is how to count how many significant digits you have in your number so you look at each digit in a measured number this is a measured number it's 309 meters that 309 is the number meters is the unit we had to use some kind of ruler to measure 309 meters so we have to look at each digit we have to look at the three the zero the nine and the idea is a digit here here come the rules the digit is significant if it's not a zero in other words if it's one two three four etc up to nine so we can already look at 309 is the three significant um you should all be screaming yes because it's not a zero is the nine significant you should all be screaming yes because it is also not a zero then the question is is the zero significant if you're saying no um at least at the moment what I will tell you is that we don't know I didn't say I said if it's not a zero it's definitely significant if it is a zero that here's where the problem is sometimes it might be sometimes it might not be we have to wait for more rules so this is a question mark so right now 309 meters has at least two significant digits maybe three because we don't know if the zero is considered significant yet um here's another rule a digit is significant if it's a zero and it's in between digits that are already significant now we can answer the question about whether this zero is significant is it significant uh you should all be screaming yes because it's a zero and it's sandwiched in between a three and a nine which are already considered significant so that a zero in 309 meters is significant so we now know that 309 meters has not maybe definitely three significant digits so there's two of the rules if you're not a if you're a digit that's not a zero you're significant if you're a digit that is a zero and you're in between digits that are already significant you're all right you're all also significant here's another rule about the zeros um if a zero comes at the end of a number and I'll explain what I mean by end in a minute and the number contains a decimal what I mean by that is somebody actually bothered to write the decimal point into the number then the zero at the end is considered significant so here the number 300 I'm going to write it up here 300 it has a zero at the end there it is um you have to ask yourself the second question did somebody actually bother to write a decimal point um somewhere in that number the answer is no there's a decimal point implied to be somewhere but it's not written and so this is basically code for saying this zero right here not significant um and so if we go through this number um basically the three is significant and these two zeros are not so if I write the number 300 without writing a decimal uh decimal point then this number has one significant digit and what that is telling you that is basically a code that tells people that if I measured something and it came out 300 without a decimal point that my measurement was kind of crappy suppose that I measured something and I said it was 300 meters maybe from my uh from where I'm sitting to my car is 300 meters but I didn't write the decimal point what that is telling people is that the ruler was so crappy that was used to make that measurement that all it really knows is that my uh my car is more than 200 meters away but less than 400 meters away in other words what it's saying is I only know down to the hundreds place um that's only how that's how good my measurement is I don't know the tens place and I don't know the ones place now if you look at the second number it looks almost identical except it says 300 with a decimal point at the end so here I would say that the zero comes at the end or maybe a better way of saying that is it's the zero is the last digit and did somebody bother to write a decimal point um excuse me you bet your ass they did I'm sitting right there at the end and so because of that this uh this last zero is significant or it's considered significant and so the three is significant the last zero is significant and according to the previous rule if there's a zero in between digits that are already considered significant then it's significant as well so if I write 300 with a decimal point at the end my measurement has three significant digits which means that it is more precise than if I just write 300 without the decimal point what the decimal point is telling people is that I know my measurement was good enough um uh to be uh precise down to the ones place or what it's saying is my measurement was so good that it knows that the distance to my car is more than 299 meters and less than 301 meters so a lot of times um students are looking for a little bit extra meaning in these rules the way that I would think about these rules is they are like a code for people who have been taught the rules if I have if I write 300 without a decimal point that is a code to people who have been taught the rules that my measurement is only good to the hundreds place if I write 300 with a decimal point that is also a code it's a tip off to people in the know that my measurement knows the hundreds place it knows the tens place and it also knows the ones place this is just like a little clue to people um again a couple more examples 300.00 now again in math class when they're uh talking about very abstract things uh a lot of times your math teacher will tell you that 300 and 300.00 are the same thing when you're making a measurement that's not necessarily true 300.00 is the three significant yes um here's a zero that is the last digit there it is did somebody bother to write a decimal point somewhere in this number yes there it is because there's a zero that is the last digit and there's a decimal point written this last zero is also significant and by the second rule by this rule um the other zeros that are in between they are also considered significant because the three is considered significant and the last zero is considered significant so if I write 300.00 as part of my measurement that measurement has five significant digits what that means is it's much more precise than just writing 300 with no decimal point 300.00 says that I know that whatever it is that I'm measuring is more than 299.99 and it's less than 300.01 so the the idea here is that five significant digits for 300.00 compared to one significant digit for just plain old 300 with no decimal point you can see just by counting the significant digits which one is more precise which one is a more precise measurement the one with more significant digits same thing here 309.5900 this zero is the last digit and somebody bothered to write a decimal point so 309.5900 has one two three four five six seven significant digits so those are some of the rules for uh determining whether a digit in a measurement is significant all right so on the previous slide we were talking about rules uh for figuring out when a digit was significant here are some rules for figuring out when a digit is not significant so not um this is only going to deal with zeros because uh if you're not if you're a digit that's not a zero you're definitely going to be significant um here's the first rule if you have a zero that's part of the beginning of a decimal number then the the zero is not significant so as an example if I make a measurement and I say that I measured something and it's 0.0 uh three nine three meters long um these zeros that come at the beginning or the front not significant um this three definitely is this nine definitely is and this other three definitely is so if I make a measurement and it's 0.0393 meters long something that I measured um this measurement only has three significant digits um however just a slight variation on this if I write and I measure something and the measurement comes out to be 10.00393 meters then these zeros no longer come at the beginning they don't come at the front of the number anymore because the one is coming at the front and so now this measurement 10.0393 has one two three four five six significant digits so I want you to realize that the uh the slight difference now every once in a while a student sort of and and this I understand the confusion here they sit there and they say well look those zeros come at the front um and it sure seems like they ought to be significant or at least the one after the decimal place seems like it ought to be significant so right 0.0393 meters certainly this guy um the second zero here why isn't it significant I need it there because if it wasn't there all the three the nine and the three would slip over to the left and I would get the wrong answer so right isn't it significant it's not considered significant um the reason it's not considered significant is that all I have to do is change the units and I can make those zeros disappear so instead of saying 0.0393 meters what if I converted this uh number to centimeters if I did that because there are 100 centimeters in a meter um and I rewrote the measurement just I didn't change anything all I changed was the units I didn't change the measurement at all um and if I change the units to centimeters my measurement would be 3.93 centimeters all of a sudden you realize that those zeros are no longer there um because they're no longer there these zeros are have like a little bit of a fancy name they're called placeholders they're holding the place of the three the nine and the three just because of the unit that we're using basically you can think of it as we're not quite using the most appropriate unit it would be better if we switched units to centimeters and when we do those front zeros disappear and because they're just placeholders they are considered to be not contributing to describing the precision of the number and those zeros are considered to be not significant so when I say 3.93 uh these are the only three digits there and I have three significant digits so this is a long-winded way of saying that the front zeros in a number in a measured number are uh if they come at the very front they're considered not significant over here with 10.0393 meters there is no way that I can switch the units and get rid of my zeros I can't go to centimeters or I could go to centimeters but then that would be uh 1003.93 centimeters zeros are still there I'm stuck with them can't get rid of them I can go in the other direction I'm still stuck with the zeros because those zeros are in the middle now I can switch the units all I want but they're never going to go away and I need them to uh to basically tell me the complete precision of my measurements so in that case they're significant but if you can just switch the units and make the zeros go away and that will only happen if they come in the front then your uh your front zeros are considered not significant and last but not least the last rule for uh zeros not being significant basically if the zero is the last digit and you did not bother to write a decimal and the example that I gave in the previous slide was 300 without a decimal um here it is basically 500 without a decimal that is say that number has only one one significant digit sometimes people call significant digits significant figures so they're kind of used interchangeably so if you see that that's what they mean and again 500 no decimal point uh one significant digit and what it is telling you it is a big tip off that the ruler that you used was only good to the hundreds place if you measured something that was 500 meters long and you wrote no decimal point you're basically saying my ruler was so crappy that it could only tell that my distance was more than 400 meters and less than 600 so I'm going to call it 500 but I don't really know that for sure it could be 525 for all I know so uh what's the purpose of significant digits it is a way it is not the only way of describing how precise a measurement is in fact as I mentioned earlier um there are other ways that are actually more commonly used in the lab but they are a little bit uh more math intensive and there there's a little bit more theory behind them so for whatever reason um this one is the one that gets taught to beginning chemistry students over and over again and it causes a certain amount of pain let's see so it's a way of describing how precise a measurement is imagine that you have two different devices two rulers to measure the length of your computer screen so here's my computer screen I've got one ruler and I'm measuring the diagonal and ruler number one says that uh the diagonal on my computer screen is 34 centimeters ruler number two says it's 33.8 I know that you can just look at this and tell intuitively which one is more precise it's this one but if you want to do it in the sense of um counting significant digits ruler the measurement that came from ruler number one has only two significant digits and the measurement that came from ruler number two has three significant digits excuse me and so ruler number two gives a more precise measurement there's one more uh rule about significant digits or counting uh significant digits in a measured number that I want to talk about before we move on to a little bit more detailed information about significant digits and here's the setup for explaining the rule I told you earlier that if I write the number 300 if I say I took a measurement and something was 300 meters long and I actually bothered to put the decimal point in I said that that number with the decimal point has three significant digits and the reason was that the rule was if your last digit is a zero and it is and somebody bothered to write the decimal point then the last digit that zero is significant and the three is significant because it's not a zero and then this middle zero is also significant because it's in between digits uh that are already considered significant so that measurement has three significant digits um so what this is telling you is whatever ruler you used it is good to the excuse me to the ones place that's how that's how well it measures um what I want to ask now uh to to introduce the rule is what happens when you actually have a measured number that doesn't uh measure let's say to the ones place maybe it's only good enough to measure to the tens place in other words uh and I happen to get a measurement that ends with two zeros so the three is definitely significant um but this only the second zero only the one that I highlighted in red here um is significant as well because this ruler is a little bit worse than the the one I was mentioning up above this ruler over here that gave us uh that also measured 300 meters um it's only good to the tens place so that's as as good as it can measure it can measure to the tens of meters and it just so happens that when we measured some distance um the the tens place ended up being a zero but this zero actually we want to tell people that it is significant it's uh it's part of the precision of the ruler this one the third or the third digit or the second zero third digit is not significant so in this case this number we we want to tell people that it only has two significant digits the hundreds place and the tens place and the ones place isn't uh is not significant so um how do we actually do that uh one way would actually be to just color um the digits that are significant and that that would be a reasonable tip off the way that it's typically done um unfortunately because coloring is a little bit problematic um is people use scientific notation and if you wanted to write this number and tell people that it only has two significant digits you would write it like this you would write three point zero and if you'll notice the coefficient can i spell coefficient maybe um the coefficient has two digits and we wrote three point zero times ten to the two meters this is scientific notation way of saying 300 meters but for people who have been taught this rule what it is also saying is that um whatever the measurement is it's known to two significant digits so when a number is written the punch line here is that when a number is written in scientific notation the the number of digits that are written in the coefficient equals the number of significant digits so uh if i write three times ten to the two meters this is a way of saying 300 meters but it's also a way of saying only the first digit is significant because i only have one digit here if i write three point zero times ten to the two meters that is a way of also saying 300 meters but what it's saying is whatever measurement was used to to make the second measurement it has two significant digits it knows the hundreds place and it knows the tens place i i suppose i could do this as well i could write three point zero zero times ten to the two meters that is another way of saying 300 meters as well but what this one uh does is it there are three digits written in the coefficient so what it's telling you is all three digits in 300 meters are significant so one significant digit two significant digits three significant digits let's go back so again just to emphasize if you see a number written in scientific notation and you want to count the number of significant digits what you do is you look at the coefficient and you count the number of of digits shown there the number of digits in the coefficient will equal the number of significant digits in that measurement