 We're still talking about making measurements. What I want to talk to you now about is something called a measured number. So measured numbers are numbers that you get when you use a device to make a measurement. So that may or may not be obvious to you, I'll try to make it obvious here. One example is if you use a ruler to measure height, the ruler is a device and the number that the ruler spits out is called a measured number. A while ago I told you that I was about 71.5 inches tall. This 71.5 is a number and it is a measured number because I got this number by using a device called a ruler. So that's an example of a measured number. Another example of a measured number is when you use the odometer, the little mileage meter, on your car to measure the distance to the grocery store. So if your odometer says that it's 2.8 miles to the store, this 2.8 is a measured number. The reason it's a measured number is because you got this number from a device called an odometer that was built into your car. Or Google told you on its maps and Google's cars figured out how far it was to the grocery store. But somewhere along the line there was a device that made the measurement. Another example is if you use a balance to measure how much you weigh. So let's pretend in some fantasy world that I weigh 205 pounds, I don't really weigh 205 pounds, I'm way past that but let's pretend, humor me. This 205 is a number and it's a measured number because I had to use a device called a balance to actually get it. They're all measured numbers so whenever you use a device to get a number from a measurement, that number that you get is called a measured number. The punchline however is that measured numbers are never perfectly precise or maybe a better way of saying that is the only time measured numbers are perfectly precise is when you get lucky. So the example that I'm going to give is with weight. A few minutes ago or a minute ago I said that I weighed 205 pounds, sometimes I will ask my students if I weigh exactly 205 pounds, sometimes they will say yes and what I will tell you is that no, I probably don't weigh exactly 205 pounds, it's just that my balance, my scale is only good enough to tell me down to the single pound. That's certainly possible that I weigh 205.2896 pounds, something like that. But my scale just isn't good enough to tell me these four digits past the decimal place so it says ah you weigh about 205 pounds. So you can see in this example 205 pounds is not a perfectly precise measurement. Now you can sit there and say well okay but if you get a fancier scale, a fancy scale that can measure down to four decimal places then that scale is perfectly precise because it will tell you that you weigh 205.2896. But what I'm going to tell you is that even then that scale is probably not perfectly precise because maybe I actually weigh 205.289611734 pounds. If you see where I'm going with this, no scale that you can come up with no matter how sophisticated is going to be able to tell you as precisely as possible how much you weigh. There may always be numbers hidden, which I'm showing with question marks here, hidden past the last decimal point that your scale or your device is good enough to measure at. So that's the punchline. Whenever you make a measurement, your measurement is not going to be perfectly precise, or another way of saying it is the only time it's going to be perfectly precise is if you get lucky. If I happen to weigh 205.000 going on forever pounds, if I was just lucky enough to weigh that, then the scale might just get lucky and say that I and correctly say that I weigh 205 but under that's a very rare situation. It's much more common that whatever measurement you make is going to be not perfectly precise. So keep that in mind. So that's what a measured number is. There's something that there's a different kind of number called an exact number. You can think of this as kind of like being the opposite of a measured number. There are two types of exact numbers. There are exact numbers that you get when you count things. So if I say that there are five apples on my kitchen table, this number five, not a measured number, it's called an exact number. The reason it's exact is because I didn't use a device other than my eye to actually get this number. I just counted. So whenever you're just counting things, assuming that you can count correctly, and nothing's being hidden from you, the number that you get is an exact number. That's exact number type one. Exact number type two is a number that you get when someone or some group of people make up a rule that compares to units of measurement. Now, you may or may not know this, but 2.54 centimeters of length is equal to one inch of length. The reason that this is true is because some group of people, some committee got together a long time ago and said, look, we need to be able to relate these two units of measurement. We need to tell people how many centimeters there are in a single inch. So let's make up a number. And they basically said it's going to be 2.54 centimeters. I'm sure there were good technical reasons for coming up with that number. But it's basically a made up number. Because it's a made up number, because some people just came up with a rule. This is also an exact number, even though it looks a little bit messy. That's just a number that was that was made up as part of a rule. And because of that, it's considered an exact number. So two types of numbers measured and exact. On this slide, I want you to pause the video in a minute and basically look at the numbers that I'm circling and make a guess as to whether they are measured numbers or exact numbers. In other words, what I want you to do is I want you to look at this 12 and 12 dozen eggs. Tell me whether it's measured or whether it's exact. Look at this 128 and 128 students. Tell me whether it's measured or exact and do that for all of the numbers. And you can pause the video. And when you unpause it, I will work through this. Okay, on pausing, let's look at oops, let's go back. If I say that I have 12 dozen eggs, that is an exact number. Because the only device I used was my eyes. And I was just counting. Same thing with 128 students. Assuming that I can count correctly, this is an exact number because that is all I'm doing is I'm counting students or counting names, maybe. If I go to the gas station and the pump tells me that I put 24.3 liters of gasoline in my tank, this is a measured number. The reason it's measured is because I used a device, I used a pump. And the pump has a little meter and it says how much gasoline you put into the tank. And but it's not perfect. Maybe you put in 24.292 liters. But the pump isn't good enough to tell you down to three decimal points. It's only good enough to tell you down to one decimal point. So the pump is not perfectly precise. And it just says, ah, you pumped about 24.3 liters. Same thing here. If I tell you that it's 100 meters to the grocery store, it's a measured number. Because I had to use a ruler or a car or something to make the measurement. I used a device. I picked the number 100 because it sounds kind of round and it sounds like it kind of should be exact. But it is not an exact number if I say that it's 100 meters to the store. If I say, this relates to the previous slide, if I say 2.54 centimeters in an inch, this is kind of a messy looking number, but it's considered to be exact because it's part of a rule that people made up. And because it's just a number that people sort of picked, it's considered to be exact. If I go to the grocery store in Europe and I buy 80.00 grams of salami, somebody had to actually put the salami on a scale and measure it. And so this is a measured number because of a scale that I used. And again, I put the extra zeros in there to make it maybe look a little more exact. But you have to realize that you could get a really fancy scale that could measure down to two decimal places, or even more. But that scale is not going to spit out a number that is perfectly precise because there may always be numbers that are hidden past the last decimal place that you're not aware of. And then finally, 14.3 seconds to run the 100 meter dash. This is a measured number because I used a stopwatch or some other type of clock. And that clock or that stopwatch is a device that is measuring time. And so this number is not perfectly precise. It's measured. So there you have it measured and exact numbers.