In this video, we're going to talk about sections 1.5 and 1.6 is going to be significant figures and scientific notation. As as always, feel free to skip through, fast forward, slow down, rewind, whatever topics that you may need. Significant figures. Okay, so significant figures is not necessarily um something that is intuitive to everybody. So if you're having trouble understanding um definitely talk to me um and I can help try to reexlain a little bit of things. I can also recommend some great YouTube videos from creators that are I trust to give you correct information. So basically significant figures come to us in in most important aspect is measurement. So in the lab, so if you're in the lab for this class, you will probably understand significant figures a little bit better because you're actually making measurements. And so we kind of have two different numbers, uh, two different types of numbers. Either exact numbers. Exact numbers come from counting or part of a definition. So like if I'm counting my fingers and my toes, it's exactly 10, right? It's not 10.5. I mean 10.2, 2 10.1794 like it is an exact number. So when you're counting people, you're counting thing, any kind of counting number is going to be an exact and we we think about it as exactly as kind of infinite digits. So um an exact number if I'm counting pens or something like that. So like 10 pens would be exactly like 10.0000 0 0 0 to infinity to infinity. It's exactly 10, right? Cuz we don't really have fractions of a thing we're counting necessarily. I mean, yes, technically, I know you're like, "Hey, well, that could be like half a sandwich or something." Sure, I get it. But like where when you're counting stuff, um, it's exact. If it is like half a sandwich, right? If you have 10.5 sandwiches and it's still exactly 10.5. I know you can probably get into more fractions and y'all can sit and argue with me about it, but if you're thinking about how do I answer a question on a test like is this a counting number or it's part of a definition? It's exact. For definitions, any of your SI prefixes, centi, mi, millie, kilo, mega, giga, whatever, those are definitions. Um, so it's exactly 100 centimeters in a meter. It's exactly a thousand grams in a kilogram. Um, it's exactly, you know, 24 hours in a day. I know it's not really exactly, but when we're talking about definitions and things that we just say, hey, these things are equal to each other. These are just definitions. They're not measured. Yes, a lot of these things come from a root thing that is measured. For example, a kilogram is based off of an actual piece of metal that's like in a museum that we said, "Hey, that's going to be one kilogram. We base everything off of it." But like when we're thinking about prefixes and definitions and like those kind of things, we have to let some things be exact because not everything is measured um with any uncertainty. An inexact number is anything that I measure or I observe and there is some uncertainty. Like think about when you measure with a ruler. You can't just measure to the infinity decimal place, right? You're limited by the markings on the ruler and to what place value they are. So like 15.3 cm. Basically, this last number, we're guessing that three. What What kind of ruler does this come from? This comes from a ruler that has a marking at 15 and has a marking at 16. Doesn't have any other markings. And maybe what I'm measuring is like right here. And so I'm saying, hey, I think that's 15. And then it's between 15 to 16, a little bit closer to 15. So I'll say 15 whatever, right? 15.3. Maybe you look at it and you say, that's 15.2. Um, but we can't really say any number past that because it's marked between 15 and 16. It's marked every one. So, we can read it to every 0.1, but we can't really read it past that. So, the marking is to the one's place. So, the marking is to the one's place. We read it one place after that, which would be the 10th's place. So brush up on your place values like do you know what tens, ones, tenths, hundredths, thousandth like uh place value measurements is going to be helpful because I'll I'll use those words like tenths place um etc. So brush up on those. If it's marked to a certain place, you can read it one decimal place after but not more than that because it's not the ruler is not precise enough to give us that. Um, so if you're measuring something, if you're observing something, that's an inexact number and that last digit is really uncertain. We're kind of guessing that one or the machine is guessing it like some it's guessing something. Something is being guessed in there on that last number. So, usually you'll see things like plus or minus one or plus or minus.1 or plus or - 01, right? That's kind of telling you how good it is at reading up to that decimal place. Significant figures are all the digits in a measured number including one uncertain or one estimated digit. So, um all the numbers we know for sure plus that one we're kind of guessing on, we call those numbers significant. Um nonzero numbers are always significant. So if I measure something and I just said, "Hey, how many significant figures are there?" If there's no zero in the value that I give you, all of those numbers are significant. So this would be three significant figures and this one would be six significant figures. I'm going to um write significant figures as sigfigs. I'll say sigfigs or I'll write it as SF because I'm not trying to write significant figures. I'm not even trying to say the word significant a lot of times. So I'll say sig figs or sf. So first rule of significant figures of trying to figure out how many there are. If it's not a zero and it's there, it's significant. It's the zeros that get give a little bit of trouble here. So rules for zeros. Uh a zero counts if it's between two nonzeros. So I like to call these sandwiches. So sandwiches are significant. So if you if you are like you like sandwiches, then hopefully that'll help you memorize that. But sandwiches are significant. If there's a zero sandwich between two non-zero digits, it doesn't even matter if it's a big sandwich. Like if it's a stack sandwich, you've got lots of zeros. As long as you have nonzero numbers on the outside of it could be 20 zeros and a one on either side of them, those zeros are significant now. They're part of the measurement and they're part of that numbers that I know for sure and that one number that I'm guessing on. So remember, all significant figures are basically what numbers do I know for sure and one number that I guessed on when I was measuring. So when you get a number in a problem, imagine that somebody measured it and guessed on that last number and they're trying to tell you all these numbers are significant here and you kind of need to know how to interpret that. So zeros that are sandwiched, they're meant to be there. You can't just get rid of them. Um and they are they're telling you, "Hey, we know that's a zero there. We know that's a zero there. Um we know that's an eight. We kind of are guessing on the seven. We know that's a two. We know that's a nine. We know that's a zero. And we're kind of guessing on the five. It's the last number we're guessing on, but others we know." So if the zero is in between two non-zero numbers, we know that is a zero in that place. If the zero comes at the end of a number with a decimal. So if a number has a decimal and the zero comes after nonzero numbers. So after nonzero numbers. So after nonzero digits and it's got a decimal in it, then that zero counts. So, if I had 3.75, but I then gave you these zeros on the end, these don't actually change the value, right? I could put in my calculator 3.75 and it'd be the same as 3.75 0. So, I'm telling you, I know the three, I know the seven, I know the five, I know the first zero, and I'm guessing on that last zero. That would be five significant figures. One interesting thing we can do is we can put a decimal at the end of a number. And now whatever zeros are in front of it, as long as they're after non-zero numbers, count. So let's say I know the six, I know the two. I'm guessing it's a zero there. If I leave it as 620, and I don't put the decimal, what I'm saying is I know the six, I'm guessing on the two. But if I want to if I want to tell you, no, no, I'm not guessing on the two. I know the two. I'm guessing on the zero. You have to give me a decimal there. So this would only be two significant figures because that zero does not have a decimal after it. If you'd said 620.0, now both of those zeros count and that would be four significant figures. So there's a difference. Like sometimes you want to put a decimal without anything after it cuz you want to tell me, hey, I guessed on that zero right there. Um but I know the two. And that's a way to which you can do that. You can also put things in scientific notation to help you with significant figures, but we can talk about that a little bit later once we talk about scientific notation. And then a zero does not count if it's at the beginning of a number or at the end of a number that doesn't have a decimal. Basically, these numbers, these zeros are placeholders. They're telling me the number doesn't start here. It doesn't start here. It doesn't start here. It starts three digits after the decimal. That's when the number starts. The number doesn't start here. It doesn't start here. It starts here. Right? So, these zeros are placeholding. They're telling me, hey, not the ones place, not the tenth's place, not the hundredth's place, but the thousandth's place is where we actually want that number to be. This is not the same as 2.45, right? So, you can't just take those zeros out because they are important because they're holding the place, but they're not really part of the measurement as far as the numbers that I know and the numbers that I'm guessing. Um, if the zeros come at the end, but with no decimal, they're kind of the same thing. They're like, well, it's not really part of my measurement, but my measurement starts at 2,00 not 200. So this is not the same as 257, right? Um this is not the same as 1 2 4 55. So if I just get rid of those zeros, the number actually changes in the value. Those zeros are maybe placeholders. They're telling you something about where the actual measurement is. The measurement is not in the ones place. I'm not measuring the ones place. I'm measuring the thousand's place, the hundred's place, and the 10's place. But I'm not really measuring the one's place. But I don't have anything to go there because the seven was the last number I guessed. I can't say more numbers after that because I don't know. So I'm just putting a zero there to placehold it. If I wanted that zero to be significant, I just stick a decimal on the end of it. And now that zero would be significant. So if I said 2570 point, that would be four significant figures. Now I'm telling you, hey, I'm guessing that zero here. I'm saying, "Hey, I'm guessing the seven is in the 10's place, not the seven is in the onees place. The seven is in the 10's place. I have no idea what's in the onees place." So, I just put a zero there because I cannot measure that. Maybe because my instrument is not calibrated far enough for that. Hope that makes sense. And this is a very simple significant figure rule this textbook does. It's actually the way I teach it if other if other textbooks are trying to be more complicated. Um, and I think that is kind of simple. So, zeros count only if they're sandwiches or if they're after a nonzero with a decimal. If they're after a nonzero without a decimal or if they're at the beginning of a number, even with a decimal, they don't count. They're just they're just placeholder. They're just telling you where you did not try to measure. So, we have significant figure rules. Um, you may have had this problem before in math. You're like, "Hey, I'm doing this math. How many numbers do I put?" My calculator gave me 15 numbers and what do I put? This will help you answer that question. So, if you did this math where you're taking this miles per hour, right? This many miles, that many hours, and you want to figure out your average speed, miles hour, and you get this in the calculator. Well, do you write all those numbers down? No. Let's look at what you measured. You measured four significant figures and you measured two significant figures. When we do multiplication and division, it's the fewest number of significant figures that we go with. We're limited by the number of significant figures in our measurement. So, if we want to use more digits in our answer, we have to do measuring with more decimal places, right? So, I have four significant figures in the miles and I have two significant figures in the hours. So I have to go with the smallest amount of significant figures in my answer. So my answer must have two significant figures. So how would I round that? So I would say um I like to underline. So I want this number. I want the 10's place and I want the onees place. So I want two significant figures. I got to drop it here. So wherever I'm going to drop it off or cut it off, we always look to the next digit, the one that's about to be dropped. If it's four or fewer, you just drop it and all remaining numbers. If it's five or greater, we're going to round the previous number up. So, for example, this 8 is five greater than five or five or greater. So, we're going to say not 63, but 64. If this were a four, we just leave it at 63. So, we're going to round based on the number of significant figures we need based on the math rule that we're doing. So, we round that to 64 miles hour. Um, here's an example of rounding um that you can look through to kind of give you an idea maybe to refresh your memory on rounding, but pause and take a look at that if you need to. For addition, subtraction, most people get the addition multiplic or multiplication division rule pretty well. Addition, subtraction is a little bit harder to think about. However, we use it less. So it can be important because we do a lot of this work in like temperature conversions and sometimes in lab. Um but it is not really the primary method of calculation we do in the class. So don't let it break your back if you're like hey this is like the worst thing ever. Um you know just kind of do your best with it. With addition subtraction we're not looking at number of significant figures. not number of significant figures. We actually don't care. We're looking for the decimal places or the least precise place value. I don't like to say decimal places sometimes because sometimes a number doesn't have a decimal in it and you're like, well, what's the decimal place? It's the place value. The least precise place value is the where you kind of stop with your answer. So this number, this measurement goes to two digits after the decimal, right? Two decimal places or you can say to the hundredth's place. This number goes to the 10th's place. Which is least precise? The one that is farther to the left is less precise. So if we think about our place values uh we have like if I say 1 2 3 4.5 6 7 you've got least precise moving that way more precise moving this way. So the seven is the most precise decimal place here. That's the thousandth's place with a th. Then you've got the hundredth's place, the tenth's place, the ones place, the 10's place, the hundred's place, the thousand's place. So if this one goes to two digits after the decimal, this one goes to one digit after the decimal, only one digit after the decimal is less precise. So my answer can only go to the least, you're only as strong as your weakest link. So your answer can only go to that less precise place. So if I get 6.51, I have to cut it off at one digit after the decimal. So I'm going to round to 6.5. It doesn't matter that this was two sigfigs and this is four sigfigs. We're not looking at number of sigfigs. We're looking at the precision of the decimal places. Scientific notation is um important to know kind of what it means um because we're going to be using it a lot because chemistry any number that is really small or really big we're going to use scientific notation for and because we talk about tiny little particles all the time there's a lot of them. Um but basically scientific notation has this setup. A coefficient is a number between 1 and 10. It cannot be 10. it it's it's um uh it should be uh between one and 10. So one through nine is what you can put there, right? You cannot put 10 there, but you can put one through nine there. Um it could be point something. It could be 1 point something 9 something. It can go all the way to 9.99999 to infinity. That's fine, but you can't put two digits to the left of the decimal. So, one digit to the left of the decimal and then we have time 10 to an exponent. The exponent is a positive or negative whole number if the exponent is positive. So, if your exponent is positive, uh let's just do x is positive. x is positive the value of the number is greater than one. If x is negative, the value is less than one. You don't have to remember if it's negative, move your decimal to the right or move it to the left because it changes if you're going into or out of scientific notation. So just remember if you see a positive exponent that means that number is bigger than one. So I like to think about it as would you like to get a paycheck of this thing? If it's if it's more than times 10 the 1, yeah, give me a paycheck. I'll take it. I'll take $10. I'll take anything greater than a dollar. Like I'll take it. If the paycheck is 10 to the negative something, it's like 0.0 something or 0.00 something. Like don't waste the paper and like writing a paycheck for me. I don't you know unless you're getting a lot of those little numbers that it's not really going to mean anything, right? So positive value if positive exponent the value is bigger than one. Negative exponent the value is less than one. That way that'll help you that will help you figure out how do I move my decimal to go in and out of scientific notation. make it a big number or make it a small number. So to convert um you would take your number and you would move your decimal to give a number between 1 and 10 or 1.9 one and 9.999999. Right? And then your x is the number of places the decimal was moved. Right? Don't again if you want to memorize this move left move right thing you can. Just remember x is positive. That's a number greater than one as the overall value. Is 2500 greater than one? Yes. So the exponent will be positive. The number of times you have to move the decimal is the number of places or that's your exponent. If your x is negative, if the number is less than one, do you want a paycheck of that? No. Don't like save your paper. Don't give me that. Right? I do want this money. Don't worry about this money. Right? This is a small number. So less than one. So, it gets a negative exponent. We move the decimal two times. So, you don't have to memorize left and right, all that kind of stuff. It changes depending on the type of question you're being asked. Um, and I know that some people have issues with left and right. I get really confused when I'm doing it, trying to memorize, move it left, move it right. So, I just say big number greater than one, small number less than one, call it a day, and don't really think about um think about it past that. To me, that's work smarter, not harder. So here are some other examples. If you want to pause and just kind of take a look at this, you can. Here's how we can kind of move them around. So if you have the exponent is positive, we got to make it a big number. So move the decimal so that it makes it big. If the exponent is negative, move the decimal so it makes it small. So if you memorize right and left, it's it's opposite. Big versus small is opposite if you're going into or out of scientific notation, right? So for me, I'm just like, hey, big number, positive, small number, negative. Make it a small number, move the decimal that way. Make it a big number, move the decimal that way. That is a lot easier to me. If you got questions, um, let me know in class today. We'll be doing some practice this next class. Uh, and yeah, always stop by office hours if you want to go over any of this. Uh, and I'll see you in class.