Reporting measurements | Working with units | Algebra 1 | Khan Academy

• In this video, we're going to talk a little bit about measurement and the idea that you really can't measure exactly the dimensions of something. And I know what you're thinking: you're like, well, no, of course, we can measure the dimensions of something.

Let's say I have some type of a gear over here. So let me draw my gear. And if I were to ask you, that's not the best drawing of a gear, but if I were to ask you, what's the inner diameter of the hole of the gear right over here? Maybe you take a ruler out, right over here. So this is my ruler. And that you are able to see when you measure it, that it is one centimeter in diameter.

But then I say, is it exactly one centimeter? And then you realize, well, yeah, let me get a little bit more precise. Maybe you get a magnifying glass out here. So this is the lens of my magnifying glass. And you zoom in a little bit. Maybe you get a better ruler that marks off the millimeters, and you actually say, "Oh, well, when I look a little bit closer, it actually turns out it's not exactly one centimeter. It's actually closer to 1.1 centimeters."

And then I ask you, is that exactly the inner diameter of this gear here? And you're like, okay, well, let me get out a microscope. And then you realize, oh, you're right, it's actually 1.089 centimeters. And then I ask you, is that exactly right? And then you're like, yeah, I guess you're right. I haven't measured to the nearest height or the width of an atom. To do that, I would need a lot more precision right over here.

And so maybe I need some type of an electron microscope. But even if you're able to do that, and that would be many decimal places to the right of the decimal point here, if you're measuring in centimeters, you can still ask, was is that exactly right? Maybe you can measure the parts of an atom or to a measurement even smaller than an atom.

And if later on, you might study quantum physics, there are some levels of granularity where you can't get a true measurement below that. But as you can see, it is somewhat arbitrary for our everyday life.

And so the question is, which one do you pick? Or how much trouble do you take to get to these different levels of precision? And the answer is, it just depends. If the goal was, hey, we just want to make multiple copies of maybe jewelry of this little car gear, so we're going to want to put some type of, I don't know, gold chain through it. And we say, hey, we need at least three quarters of a centimeter in order to get the rope or the chain through it.

Well, then this first measurement, that's enough precision. But if I told you this gear is going to be an essential part of the space shuttle, or some type of really important machinery that has really fine tolerances— I guess people aren't using the spatial anymore, but some finely engineered automobile or something that's going to have a lot of needs— really close tolerances it needs to be really, really precise.

Well then, even this 1.089 centimeters might not be enough. You might have to get to something like it's 1.089203 centimeters, to be able to be really, really finely crafted. We're nowhere close with our everyday tools to get anywhere close to say the width or the height of an atom, and you can even theoretically measure within the atom.

And so you just have to think about what the measurement is for. I'll give another example: this right over here is a picture of Mount Everest. You might know it as the tallest mountain in the world. And if you were to ask someone, how tall is Mount Everest? If you were to do a web search for it right now, you would find that it is 8,848 meters tall.

Now, this is clearly rounded to the nearest meter because if you were to go to the top of Mount Everest, you'll see little pebbles. In fact, those pebbles might move around. And so the actual precise height of Mount Everest might change actually second by second, depending if rain is falling, snow is falling, how the wind is moving different pebbles around. But for most of our daily purposes, this is sufficient.

In fact, for a lot of us, we might not even need this level of precision. We might say, hey, it's roughly or it's approximately, we'd estimate that it's about 9,000 meters. But there are applications where you would need at least this level of precision, or maybe something even more precise.

For example, if you wanted to compare it to another mountain, say K2, which is the second tallest mountain in the world. And let's say they are close in height. If you were to do a Google search, you would see that K2 has a height of 8,611 meters, rounded to the nearest meter. You'd see that, that 9,000 meter approximation: it wouldn't be enough.

If you rounded to the nearest kilometer, I guess that wouldn't be enough to be able to compare Mount Everest to K2 because rounded to the nearest kilometer, they're both approximately nine kilometers. So this is approximately 9,000 meters as well. So you would need more precision.

If you wanted to answer which one is taller, you'd have to get at least to the closest hundred meters. And then there are reasons why you might want to get even more precise. Maybe you want to create a slide from the top of K2 to the bottom of K2.

And so you can imagine if your slide is too long by, let's say, three meters, what's going to be hard to get on that slide at the top, or it's going to dig into the snow at the bottom. And if your slide is too short by three meters, that's a pretty unpleasant thing to have happen. You go on this seemingly super fun slide, you have to drop nine feet at the end, or really, if you're off, what if you're off by 10 meters and you're going to drop 30 feet off the end, which could really break some bones and be unpleasant.

So the big takeaway is, it's very hard to measure anything perfectly precisely. And you have to think about what's the application? What are you trying to answer? What are you trying to judge about those things? To determine how much precision you need in your measurement.