 Hey everybody, welcome to Tutor Terrific. In this video, we're going to look at numbers in physics, specifically how we deal with uncertainty, accuracy, precision, and numbers that are very large or very small. So it's a real lesson on measurement in physics. So measurements, numbers are understood in physics as measurements of all different types, comparing an unknown amount of a quantity to a given standard unit. That's what we're doing. And it's very nice to do so because it allows us to communicate amounts of different measurements, different numbers. We have lots of measuring tools that we use in physics or outside of physics to measure and create numbers, numerical data for certain types of quantities of objects. You see lots of tools that we use around the house over here on the left and lots of more specific tools, depending on what field you're in for measurement as well. So measurements are how we understand numbers in physics. There is no number that is not a measurement in physics. Now, measurements must have units to make sense. I cannot tell you that the length of my finger is 7. That means nothing. It means absolutely nothing to you. And it shouldn't mean anything to you because I need to know the units. I need to know if we're in inches or meters or feet or miles, hopefully not miles. And all units, all measurements must be together. Measurements must have units. The system we use for measuring numbers and the units we use could vary. In the beginnings of your physics understanding, what we are going to use is called the SI system. SI is French acronym for System International System. So the international system picks a particular unit for each measurement type. So the first measurement type here that I have here is called length. We need to measure length all the time. There are many symbols in physics used for length, such as L, H, if it's a height, or R, if it's a radius. And the SI unit for length is the meter. Now, you know we can use inches or centimeters or feet or miles or kilometers to measure length. But the international system uses specifically the meter. The meter. OK. Next, mass. Mass needs a special unit, an SI unit. We can measure mass in grams, nanograms, kilograms, other measurement units. We use kilograms in the SI system. Kilograms, kilograms. Time. Another important measurement, fundamental unit in physics. We use the second. The second. Of course, I could use hours or minutes or years or days or nanoseconds. But we default to seconds for time. That's the SI system. Now, there are many more units. There are many more types of measurements. But for the beginnings of physics, we focus on length, mass, and time specifically. Now, our measurements have a certain degree of precision and a certain degree of accuracy if we're looking at multiple measurements. Precision is the study of how close measurements are to each other. So if you take repeated measurements of the same object, if they are close together, you have high precision. If they are not close together, you have low precision. Accuracy, this word here, is defined as the measurement of how close your measurement is to the actual accepted value for the measurement. So we're comparing one of your measurements to the actual measurement that's accepted and might have been calculated by the most fine-tuned machines that can be used to measure that quantity, whatever it is. We could combine these two ideas when we look at measurements to grade how good they are. And the dartboard is often used to describe this. So when we look at this dartboard here, this first one, the bullseye represents the accurate, accurate measurement right there, the center of the bullseye. If we have a good, accurate measurement, our dart will hit the bullseye. Our darts are our measurements, OK? A low-accuracy and low-precision set of measurements would be a bunch of darts all over the board, not centered really around the accurate spot of the bullseye. Technically, what I do is I put these darts so that the center or the average position of the darts is not the center of the bullseye. So they almost look like they are here. You'd want this dart maybe over here. None of them are close to each other. None of them are close or centered around the absolute value of accuracy, the bullseye. Now look at B right here. We have here a situation in which all the darts are basically hitting the same point, or very near close to it, in the same section, at least, of the dartboard. This person, who's run this dart, their measuring device, analogy to that, would have high precision. All the measurements are close to each other. But their average position is very far from the bullseye. So that would be low accuracy. High accuracy and low precision darts would be all centered around the bullseye, and fairly close to it, but not close to each other. Low precision, not close to each other. And finally, the measurements we want, when we take repeated measurements, are highly accurate and highly precise. All those darts are right near the bullseye, and all the darts are right near each other. So that's precision and accuracy. Another very important measurement in physics is uncertainty. Let me explain. We've got this meter stick here, and it's kind of old. It's got some problems on the edge. It's been ground down over time. It's hard to read some of the tick marks. There's even a tick mark missing. If I were to measure this purple rectangle here, you'd see that this device could be used to measure the length of that rectangle. No problem. But what if I asked you to measure the diameter of this white circle? Good luck using this device to do just that. It's not going to go well for you. If you line that circle up with, maybe I'm lining it up with this here edge, and it's this long, but this is not the right device for measuring the diameter of this little white circle at all. How could I do it? Well, first, you need to understand that no measuring device is infinitely accurate. You cannot get infinitely close to the actual value for any measurement you're taking or what it's supposed to be, nor can you be infinitely precise, meaning infinitely precise. So each number is exactly the same as every other number you use to measure something. No measuring device allows you to do that. They all have limitations. And this is an example of that with this meter stick. You are only able, when speaking about accuracy specifically, to be as accurate as the smallest division of your measuring device. How do I explain divisions? Well, let's look specifically at this meter stick that we're zooming in on. The way this meter stick works, if this is inches, which it is, on one side of your meter sticks you have inches on the other side, you have meters centimeters and millimeters. This is the inches side. What we see here is that we can, of course, measure whole inches. Here's inch one, and here's inch two. It also splits that in half. So the longest tick mark represents the half of the inch. So right here, this would be 1 and 1 1⁄2 inches, if measured from the edge. And we have half of that again. We get 1⁄4 inch, this is a little bit shorter line. And half of that again, we get 1⁄8 of an inch. If you were to go between tick marks, that would mean that represents an 1⁄8 of an inch. We can only be, with this measuring device, accurate to the nearest eighth inch. I can't say, oh, this here ball's diameter is 0.12683 inches. I can't do that with this device. It doesn't allow me to get that accurate. So let's try to measure this particular purple bar using this device, okay? Here's what you do. You find the smallest division on the measuring device, and we already did. 1⁄8 of an inch, each tick mark, consecutive tick marks are separated by 1⁄8 of an inch. Number two, round your measurement to the nearest value of that smallest division. Well, if we line this purple rectangle up so that it zeroes on the edge of our meter stick, which represents zero, it comes closest to the 1-inch, whole 1-inch mark, okay? I know it's a little bit bigger than that, but it's not as big as the next eighth. It's closer to this line for the 1-inch mark. So we are rounding, in other words, estimating or rounding our value of the length of this to 1-inch. Now, to say and explain to those who are with us or who are reading about our measurement or who are, we are communicating our measurement to that we have some uncertainty and we're not pretending that we don't because that's not good science. We will state the rounded value that we got from step two here plus or minus the smallest division value. So one plus or minus an eighth inch. This means that it could be 1⁄8 inch bigger or smaller, but I rounded to the 1-inch specifically. And this is my smallest division of my value, one plus or minus 1⁄8 inch. So it's between 1⁄8 inch and 7⁄8 of an inch. So that's basically as accurate as I could be and I'm being more honest about my limitations of my measuring devices. So that's uncertainty in measurements. Now I wanna look at numbers in general. Significant figures is something, if you've taken chemistry, what you should have, you've had to deal with, okay? This is a measurement in which we show the limits. This is a method by which we show limits in the accuracy possibilities of our measurements. So for example, 4.5 versus 4.51230. Which one is more accurate? Which one has more digits? Which one measures to a smaller division in the measurement device? Well, that would be the second number, 4.51230. That's quite accurate if it's close to the actual value, but is it possible? Can we accurately state that? Are we allowed to state that? We might not be. Also look over here, 50,000 without a decimal versus 50,000 with a decimal. What is that decibel saying? Is it saying something about all these zeros past the five versus it not being there? Yes, inside of the discipline of physics, it is and chemistry as well. Three rules for looking at significant figures in measurements, okay? Not deriving them, but looking at measurements that have already been found and determining how many significant figures they have. That's what we're doing right now. There are three rules for doing that. Number one, all non-zero digits are significant figures. Significant digits in other words. So for example, 4.5123, those non-zero digits are always significant. So if you've included them in your final measurement that you're publishing, they're always considered significant. Number two, let's talk about the zeros. There's multiple types of zeros. All zeros to the right of the first non-zero digit. For example, this zero here in 4.51230, it's to the right of the four, the first non-zero digit. It's significant if we have a decimal written. These are called trailing zeros also, trailing zeros. They are significant if a decimal is written with the number. If not, only zeros between non-zero digits are significant. So here we have an example of rule number two, where we have a decimal written versus a decimal not written. For this particular number, 50,000, only the five is significant because these trailing zeros are not followed by a decimal. So the 50,000 without a decimal has one significant figure. If I write the decimal, all the zeros trailing the first non-zero digit are actually significant now. So now I have five significant figures. One, two, three, four, five. Next, we're gonna talk about leading zeros. These are zeros to the left of the first non-zero digit. For example, 0.00326. 0.00326. This number has three zeros to the left of the first non-zero digit. Those, no matter what, are insignificant. They are insignificant. They are just there to hold the place, to show where the decimal exists. And those are used generally for numbers that are less than one. So those are our significant figures rules for measurements that already exist. Now, let's say we have to do a calculation or two to determine significant figure amounts, okay? We assume that the numbers we are already working with are properly rounded to the proper number of significant figures. When we are adding or subtracting with our operation, we have to look at significant figures a specific way. The answer must have its rightmost significant digit in the same place as the number in the calculation with its rightmost significant digit farthest to the left. That's certainly a tongue twister. This rule is best shown visually. So let's say we're adding these three masses, 23.4567 grams plus 2.2 grams plus 12.112 grams. All these decimal type numbers end with a different number of decimal places. 2.2 stops first. In other words, it stops farthest to the left when we're looking at all these numbers with their decimal lined up. This position, the tenths place, is as far as I can go in my final answer when I am adding or subtracting. You get a raw answer in the calculator, 37.7687 grams when you compute this in your calculator. This adds like you would in grade school, whenever you learn to add a decimal. So a third or fourth grade. Seven plus zero, seven, six plus two is eight. Five plus one is six. You get all these numbers here. But when we're talking about accuracy, you can only be as accurate as your number that has the least amount of decimal places when it comes to adding or subtracting. So we have to round this 37.7687 to 37.8. So we're rounding to the tenths place because the least accurate number stops at the tenths place. Okay, and remember rounding rules. You're rounding to a certain place. You look at the next digit. If it's five or larger, we round up. This is classic rounding. When I teach physics, that's what I teach my students to do. Classic style rounding, okay. Multiplying and dividing, the rules are different. You do not now look at where the digits, the significant digits are with respect to the decimal. Now you look at how many there are, regardless of the position of the decimal. So here's the rule. Answer must have as many significant figures as the number in the calculation with the least number of significant figures. So let's look at a multiplication example. 3.69 times 2.3059. This first number has three significant figures, all non-zero digits. This next number has five. Remember that zeros in between two non-zero digits are always significant called sandwiched zeros. When I multiply these two together, the raw answer in the calculator is 8.5088. Now I'm looking back at my original numbers and determining which one has the least amount of sig figs, not how many. So this one has three and this one has five. So my final answer must be rounded to three digits. I don't care where they are. I have to start at the left, go to the right. When I get to the third one, I have to stop. And round there. So I will round this zero up because the next digit is an eight in the thousandths place. So I have to round to the hundredths place. It turns out for this one to get three sig figs, I rounds up to 8.51. So these two rules matter. And you will see that in physics, most of our clasions have multiplying and dividing in it. The general understanding is to, when you have any multiplication or division in physics you use the multiplying and dividing rule as a general rule. The last thing I wanna show you guys is scientific notation. Scientific notation is a way to handle very large or very small numbers, which is quite common in physics as you will see throughout this course. So if you have a really small measurement like .000000000623, or you have a giant number like .00000000043, we have to deal with, that's actually .00000043, quadrillion. These are giant numbers. We don't like to write them out with all those zeros that are just place-holding. Scientific notation is a way that allows us just to see without any place-holding zeros which digits are significant and how many we have. Let me explain. You're gonna move the decimal if it shows up or if it's not actually visible because it can't be used due to the significant figure's rules. You're gonna move it till it's to the right of the first significant digit, usually a non-zero number. And you're gonna explain then with a certain notation afterwards how many places you had to move to get it there. So for example, this first number is .000000623. I'm going to move that decimal one, two, three, four, five, six, seven, eight, nine, 10, 11 spots to the right to get it to the spot I have to get it to according to this rule above. 11 spots to the right. Since the number was very small, we're gonna use a negative exponent to describe the powers of 10 we would multiply it by to get it there. So 6.23 times 10 to the minus 11. That shows me that I am moving it 11 spots to the right to get it where it is after the six, okay? So when you're moving to the right from standard to scientific notation, you decrease the power of 10 by one for each place you move. So even just one spot to the right means 10 to the minus one. And if you move another one, that's 10 to the minus two, okay? That's how it works for small numbers. For very large numbers, you have to move the decimal in standard notation to the left to get it into scientific notation. One, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13 spots. I had to move it to the left to put it from where it is understood to be after this last zero to between the four and the three. When you move to the left, you have to increase the power of 10 by one for each place you move. I had to move 13 places to the left. So I write 4.3 times 10 to the 13. When I actually compute this in the calculator, it will produce me the standard number. So it's another way to write the same number. But look at how compact these are compared to the standard notation. Many less typing keystrokes if you're typing or many less zeros to write by hand. You get a good understanding of the size of the number. If you have tons of zeros, you can almost get lost inside them. But if you have scientific notation, you just have a number representing how many zeros there were. Now, when you write in scientific notation, you can only include significant digits. You may not put any zeros that are not significant. That's the whole point. That's the whole point. So if you are converting to scientific notation, do not add any extra zeros that you're not allowed to have, either before or after. You're not allowed to do that. It must have every scientific notation style number, the correct number of significant figures. All right. Now, let's discuss arithmetic operations with scientific notation. Keep in track also of our significant figures rules from previous slides. Sometimes you have to add, subtract, multiply, or divide numbers that are all in scientific notation. What are the rules? Adding or subtracting is where you have to be really careful. When you are adding or subtracting numbers in scientific notation, you can only do so if you have the same power of 10, okay? You must move the decimal for one or more of the numbers so that the exponent is the same, the power of 10, for all the numbers involved. So that means sometimes you might have to move a number out of standard scientific notation, the real scientific notation in order four, those exponents to be the same. Let me show you what I mean. 512 times 10 to the 5, plus 5.3 times 10 to the eighth. We've got a problem here. These exponents don't match. Now, eight is greater than five. And so what it says is to change the other one so that the 10 to the five becomes 10 to the eighth. What would I have to do to make this exponent 10 to the eighth? Well, I'd have to move the decimal from its normal spot three places to the left to make 0.512 times 10 to the eighth. Now that I have my decimal, I'm sorry, this decimal in these numbers is a little high up, but pretend it's the decimal. 0.512 times 10 to the eighth is what I rewrote this first number, 512 times 10 to the five as. 0.512 times 10 to the eighth can now be added to 5.3 times 10 to the eighth. Now you have to keep track of the rules. I can't have all of these digits. This is adding. So what I have to do is I have to stop at the nearest the tenths place because my number that stops farthest to the left stops at the tenths place. So I rounded the tenths place, okay? 5.8 times 10 to the eighth. I know I'm technically multiplying when I have these scientific notation, but we apply the addition rule when we're adding numbers in scientific notation. Okay, now multiplication and division is slightly different. We do not care if the original numbers in scientific notation have powers of 10 that are different. It's okay. What you must do is you must multiply the decimals, but you must add the exponents then rewrite improper scientific notation. So let's look at an example of that. For multiplication, it would look something like this. We have 2.5 times 10 to the 17 times five times 10 to the 14. You will multiply the two decimals together. 2.5 times 5.0 is 12.5. The two exponents you have, you will add together, which would give you 31. 17 plus 14 is 31. So you would write to start 12.5 times 10 to the 31. Now you have a problem here and that is that 12.5 is not a proper scientific notation. That decimal has to go after the first non-zero digit to the left of it, okay? So you have to move the decimal over one. So you get 1.25 times 10 to the 32. You have to move it one to the left. That's the final answer, 1.25 times 10 to the 32. For division, it's a slightly different feel, but it's the same rules. So 2.5 times 10 to the 17 divided by 5.0 times 10 to the 14. You will divide the two decimals, 2.5 divided by 5.0, gives you 0.5. And 10 to the 17, 10 to the 14, you're gonna subtract 14 from 17 to get three. So you'd write to start 0.5 times 10 to the three. Then, since that zero is not after the first significant digit, because that zero's a placeholder, I have to move it to the right one spot, which would give me 5.0 times 10 to the two in proper scientific notation. And I want you to realize that if we look at a significant digit specifically, the actual thing we could calculate, the number we could calculate through dividing these decimals is actually 0.50. I could put another zero on there because I can have two significant digits, two significant digits here. So our final answer for this guy, if we use scientific notation properly, would not be 1.25 times 10 to the 32 because I only have two digits to start with. Significant digits in each of the original scientific notation numbers. So I have to round that 1.25 times 10 to the 32 to 1.3 times 10 to the 32. And one last thing I wanna show you about scientific notation is how can you be used to remove ambiguity in standard notation? Guys, look at these three numbers that I'm multiplying together. 34.6 times 12.1 times 1.2. Raw answer from the calculator would be 502.392. I can't have all those digits guys because one of my numbers has only two significant digits that's the smallest amount. So I have to round this number to two significant digits. It would round down to 500 without a decimal. One of the zeros being insignificant and one of the zeros being significant. That's a problem, okay? It's unclear to the reader or the person looking at your report or your calculation which zero is significant or if any of them are. If you leave the decimal off, both those zeros look insignificant. So we will put this number in scientific notation like so. 5.0 times 10 to the 2, retaining the zero that is significant. 5.0 times 10 to the 2, I'd have to move the decimal over two places to get it in that standard scientific notation form. And I have the 5.0 here, two significant figures. Okay, once you do the arithmetic, you might have to round, okay? It could be adding, it could be subtracting, it could be multiplying or dividing. You're gonna have to round so you have, sometimes all the significant digits are on the right of the decimal. You might have to round down so you got a bunch of zeros. You can rewrite the answer in scientific notation to make the number of significant figures more clear in case it's ambiguous like this. All right guys, thanks so much for watching this video. Stay tuned for the third lesson in chapter one. That's coming up next. For now, this is Falconator, signing out.