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## Algebra 1

### Course: Algebra 1 > Unit 3

Lesson 2: Appropriate units# Reporting measurements

CCSS.Math:

We should think about the appropriate level of precision of different measurements in modeling problems. Created by Sal Khan.

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## Video transcript

- 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 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 like, okay, well let
me get out of 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, to the 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 and 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 get? 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
wanna make multiple copies of maybe jewelry of this little car gear, so we're gonna wanna 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 spacial anymore, but some finally 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 could even theory
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, and actually, 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're round 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 meter. And then there's reasons why you might wanna get even more precise. Maybe you wanna 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 on 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 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 gonna 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.