 All right, so if you have made it this far, there is a good chance that you just listened to me talk about significant digits for almost a half hour. So I don't know, good for you I guess, maybe you deserve a medal. Here's a medal for you. What we are going to talk about now is more about significant digits. However, what we're going to talk about is something a little bit more complicated. We are going to talk about how to use significant digits in your calculations. So there are times when we use measured numbers in our calculations. So as an example, if I say that I walked 31 miles and I measured it down to the ones place, you know maybe I went on a very long walk, and then I walked another 0.51 miles. But this time I measured it to the hundredths place. So the second measurement I knew down to the hundredth of a mile, the first measurement that I made, I only knew down to the ones place, to the single miles. You might think that if I asked what was the total distance that I traveled, that the answer should be 31.51 miles because, well, you just add them up. With significant digits, when you are using measured numbers, and you're doing some kind of calculation like we are here, we're adding numbers together, there are rules to prevent you from basically deluding yourself and deluding others into thinking you know a measurement more precisely than you actually know it. So the answer isn't going to be 31.51 miles, I'll explain what it is in a minute. The same holds true when you're multiplying or dividing measured numbers. If I say I have 32.3 grams, and I have to divide it by 0.8 milliliters for some reason, I'm going to get some number. You might think that the answer is going to be 40.375 grams per milliliter. What you're going to see, however, is there are rules for multiplying and dividing measured numbers and rounding whatever answer you get to the correct number of digits, and we'll talk about that in detail. So again, I'll try to explain how to do this, but the idea is that the rules for multiplying, dividing, adding, and subtracting measured numbers is to prevent you from overstating how precise your final measurement was. Okay, here we go. So this is a rule for multiplying or dividing measured numbers. What you do is you just multiply out, so suppose I had 32.3 grams, and I needed to divide it by 0.8 milliliters, and like I said on the previous slide, the calculator should spit out 40.375, and the units would be grams per milliliter because we're dividing grams by milliliters as well. The rule for significant digits when you multiply or divide is you round your answer so that it has the same number of significant digits as whichever starting number had the least number of significant digits. So we have to look at number one here and number two here, and we have to say how many significant digits did number one have? Well it had one, two, three, so this one had three significant digits. How many significant digits did the 0.8 have? It only had one significant digit. So what you have to say is, I want you to think about it this way. This is our weakest or least precise measurement, the one with one significant digit. And because of that, we have to round our answer. Even though our answer, if the calculator spits out all the numbers, all the digits, our answer is one, two, three, four, five significant digits. We shouldn't write five significant digits down because our weakest or least precise number only had one. And so the rule here is that you have to round your answer to one significant digit because that was our measurement. Our weakest measurement was one that had one significant digit. So in this case, we only round to one significant digit. So you have to look at 40.375 and say, well, what is it with one significant digit? Right now it has one, two, three, four, five significant digits. So we can't do that. We have to knock off the five. How many significant digits now? One, two, three, four. Can't use that either because we only want a measurement with one significant digit. We can knock off the seven and round this up to a four. 40.4 could be the proper answer. However, the four, the zero, and the four are all significant. So that answer has three significant digits. We're looking for one. We can knock off this four and I can write 40 with a decimal point. That, however, that ending zero is significant and so is the four. So if I wrote 40.40 with a decimal point, that's two significant digits. But we still have to round to only one significant digit. So the correct way to round our answer is to write 40 with no decimal point. This has one significant digit. So if I divided 32.3 grams by 0.8 milliliters, even though the calculator says that the answer is 40.375, we have to basically look at the beginning numbers and say which one was the crappiest measurement. Turns out that the 0.8 was the crappiest measurement, had one significant digit. So the rule is we are only allowed to present our answer with the same amount of precision as our crappiest measurement, which was one significant digit. So instead of writing 40.375, we write just 40. And so the proper, or the properly rounded answer is 40 grams per milliliter, not 40.375. The whole point, again, just to emphasize, is to prevent you from deluding yourself to thinking that you made a really precise measurement by writing all of the digits out, because you didn't. One of your measurements was pretty lousy. You only knew it to one significant digit. And so the idea is you're not allowed to add more precision than you actually have. So we're stuck at 40. So that's the rule for, whenever you multiply or divide measured numbers, the rule is the one that I just went over. And I'll do another practice one in a little bit. This is the rule for adding and subtracting. This one is a little bit more difficult to explain. But basically, you do the same thing as before. How you add or subtract your measured numbers. So I'm going to write 31 and 1.51, and we're going to add them together. And so it's going to be 5, 1, 2, 32.51 miles is what you would, if I said how far did you walk altogether, if you walked 31 miles and you walked another 1.51 miles, you might think that the correct answer is 32.51 miles. However, the rule for subtracting is a little bit different. You have to look at each number that you added and subtracted. Here we have only two. And you have to find the one that is known to the poorest position. So this second measurement, 1.51, is known to the hundredths place, or it's known to two decimal points. This first measurement is only known to the ones place. It's not even known. Our measurement is not known past the decimal point in this case. So in this case, the first number is quote unquote our crappiest, that's a technical term, measurement, because we only know it to the ones place. And so the rule for adding and subtracting is that you find your weakest measurement as far as positioning goes. You add and subtract, and then you round to whatever position that was that was your weakest measurement. So we have to round to the ones place in this case. So instead of saying 32.51 miles, I would say that I went 33 miles. And that's that. So I'm rounding up in case you notice. 32.51, since I'm a little bit over halfway to 33, I'm going to round to 33. So if I told you that I walked 31 miles, that basically tells you that my first measurement was known to the ones place. If I told you that I walked another 1.51 miles, that basically says that I used another measuring device that was better. It knew to the second decimal place. And so even though it seems like I walked 32.51 miles, this rule is preventing yourself from deluding yourself, from fooling yourself into thinking that you actually knew everything to the second decimal place. Because we didn't know the first measurement to the second decimal place. We only knew it to the ones place. So the rule says, look, you have to handicap yourself. You have to basically keep yourself honest and write your measurement only to the ones, your new answer only to the ones place. So in the end, we're going to just say that we walked about 33 miles. So those are the rules for adding and subtracting measured numbers. That's it. If you're in the online class, there is one lab where there's a lot of work on adding and subtracting measured numbers and rounding to the correct number of significant digits. After you pass that lab, I don't really care that you know these rules. There are some teachers who care very deeply and will demand that you round all of your answers to the correct number of significant digits. I am not one of them. Sometimes the teachers have very good reasons for demanding that you follow these rules. I'm going to ask that you follow them up until the end of that lab that requires it. After that, I'm not going to care. Primarily because I think these rules tend to freak students out. They tend to, and when they get freaked out about this, they sort of lose track of the more important concepts of being able to do the calculations that we're trying to do, and they worry too much about rounding. So I have to teach this, and here it is, and you're going to have to be able to use it for a little while, but after that, it's going to go away. So here's a little bit of practice. On the top, this calculation is basically adding three different numbers together, and I would like you to pause the video in a moment and add them up. They're going to add up to 130.9332 if I did that correctly, but I want you to be able to round them to the correct number of significant digits. It's sort of a test of whether you follow the rule correctly. The second calculation is multiplying three numbers together, same kind of thing. Pause the video in a moment and basically round everything and see if you get it correct. So pause the video now. When you're done, I will work through it. So here's the first problem. We're going to do 0.677 plus 48.1 plus 82.7655. So I just want to mark the position of all of the digits. And basically what you have to do is you have to look at number one, number two, and number three and figure out which one we know to the worst position. And it turns out that we know number two to the worst position. That's our weakest measurement. We only know number two to one decimal place. And what that means is our answer, which is up here, has to be rounded to one decimal place. And so instead of saying 130.9332, we have to knock off the last three decimals. So the properly rounded answer is going to be 130.9. And that's just how you should do it when adding or subtracting measured numbers. On the bottom one, when you are multiplying or dividing, you also have to find something slightly different. You don't have to find the number with the weakest position. You have to find the number with the smallest number of significant digits. So let's look at this number here. It has one, two, three significant digits. These zeros don't count because they come in the front. This number here has one, two, three, four significant digits because this zero comes at the end and somebody bothered to write a decimal point. And because of that, the middle zeros count as well. So four significant digits. And this one here also has four significant digits. So the first number, 0.0222, is our weakest number. We only know it to three significant digits. And what that means is whatever this answer is, whatever the calculator spits out, it has to be rounded to three significant digits. So here are the first three significant digits. We have to knock off all of the other answers. And so if you were going to write the answer quote, unquote, properly using these rules, the answer would be written as 0.135. So that's a little bit of practice using rounding and keeping track of significant digits. And I want to emphasize, though, that this is not the only way that people, this is not the only set of rules that people use in science for basically describing how precise their answer is. There are more sophisticated ways that require more math and are actually used more often. But this is probably the easiest one to present. And because of that, we present it to you. And that's sort of why it's shown to beginning students a lot of the time. So that's it for significant digits. Thank God, right?