Topic C: Foundations
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.