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## Topic C: Foundations

## Video transcript

Human beings have
always realized that certain things are
longer than other things. For example, this
line segment looks longer than this line segment. But that's not so satisfying
just to make that comparison. You want to be
able to measure it. You want to be able to
quantify how much longer the second one is
than the first one. And how do we go
about doing that? Well, we define a unit length. So if we make this
our unit length, we say this is one unit, then
we could say how many of those the lengths are
each of these lines? So this first line
looks like it is-- we could do one of those units
and then we could do it again, so it looks like
this is two units. While this third one looks
like we can get-- let's see that's 1, 2, 3 of the units. So this is three of the units. And right here, I'm
just saying units. Sometimes we've made conventions
to define a centimeter, where the unit might look
something like this. And it's going to look different
depending on your screen. Or we might have an inch that
looks something like this. Or we might have a
foot that I won't be able to fit on this screen
based on how big I've just drawn the inch or a meter. So there's different
units that you could use to
measure in terms of. But now let's think
about more dimensions. This is literally a
one-dimensional case. This is 1D. Why is it one dimension? Well, I can only measure length. But now let's go to a 2D case. Let's go to two
dimensions where objects could have a length and a
width or a width and a height. So let's imagine two figures
here that look like this. So let's say this
is one of them. This is one of them. And notice, it has a
width and it has a height. Or you could view it as
a width and the length, depending on how
you want to view it. So let's say this is one
figure right over here. And let's say this
is the other one. So this is the other
one right over here. Try to draw them
reasonably well. Now, once again, now
we're in two dimensions. And we want to say, well,
how much in two dimensions space is this taking up? Or how much area are each
of these two taking up? Well, once again, we could
just make a comparison. This second, if you viewed
them as carpets or rectangles, the second rectangle is
taking up more of my screen than this first one, but I
want to be able to measure it. So how would we measure it? Well, once again, we would
define a unit square. Instead of just a unit length,
we now have two dimensions. We have to define a unit square. And so we might make
our unit square. And the unit square we will
define as being a square, where its width
and its height are both equal to the unit length. So this is its width is one
unit and its height is one unit. And so we will often
call this 1 square unit. Oftentimes, you'll
say this is 1 unit. And you put this 2 up here, this
literally means 1 unit squared. And instead of
writing unit, this could've been a centimeter. So this would be 1
square centimeter. But now we can use this
to measure these areas. And just as we said
how many of this unit length could fit
on these lines, we could say, how many of these
unit squares can fit in here? And so here, we might take
one of our unit squares and say, OK, it fills
up that much space. Well, we need more
to cover all of it. Well, there, we'll put
another unit square there. We'll put another unit
square right over there. We'll put another unit
square right over there. Wow, 4 units squares
exactly cover this. So we would say that
this has an area of 4 square units
or 4 units squared. Now what about this
one right over here? Well, here, let's seem I could
fit 1, 2, 3, 4, 5, 6, 7, 8, and 9. So here I could fit 9
units, 9 units squared. Let's keep going. We live in a
three-dimensional world. Why restrict ourselves
to only one or two? So let's go to the 3D case. And once again,
when people say 3D, they're talking
about 3 dimensions. They're talking about
the different directions that you can measure things in. Here there's only length. Here there is length and
width or width and height. And here, there'll be
width and height and depth. So once again, if you have,
let's say, an object, and now we're in three dimensions,
we're in the world we live in that looks
like this, and then you have another object
that looks like this, it looks like this second
object takes up more space, more physical space than
this first object does. It looks like it
has a larger volume. But how do we
actually measure that? And remember, volume is just how
much space something takes up in three dimensions. Area is how much space something
takes up in two dimensions. Length is how much
space something takes up in one dimension. But when we think
about space, we're normally thinking
about three dimensions. So how much space would
you take up in the world that we live in? So just like we did before,
we can define, instead of a unit length
or unit area, we can define a unit
volume or unit cube. So let's do that. Let's define our unit cube. And here, it's a cube so its
length, width, and height are going to be the same value. So my best attempt
at drawing a cube. And they're all
going to be one unit. So it's going to be one
unit high, one unit deep, and one unit wide. And so to measure volume,
we could say, well, how many of these
unit cubes can fit into these different shapes? Well, this one right
over here, and you won't be able to
actually see all of them. I could essentially
break it down into-- so let me see
how well I can do this so that we can count them all. It's a little bit
harder to see them all because there's some
cubes that are behind us. But if you think of
it as two layers, so one layer would
look like this. One layer is going
to look like this. So imagine two things like this
stacked on top of each other. So this one's going to
have 1, 2, 3, 4 cubes. Now, this is going
to have two of these stacked on top of each other. So here you have 8 unit cubes. Or you could have 8
units cubed volume. What about here? If we try to fit
it all in-- let me see how well I could draw this. It's going to look
something like this. And obviously, this is
kind of a rough drawing. And so if we were to
try to take this apart, you would essentially have a
stack of three sections that would each look
something like this. My best attempt at drawing it. Three sections that
would look something like what I'm about to draw. So it would look like this. So if you took three of
these and stacked them on top of each other, you'd
get this right over here. And each of these have 1, 2, 3,
4, 5, 6, 7, 8, 9 cubes in it. 9 times 3, you're going to
have 27 cubic units in this one right over here. So hopefully that helps
us think a little bit about how we measure
things especially how we measure things in
different number of dimensions, especially in three dimensions
when we call it volume.