People have always realized that certain objects are longer than other objects. For example, this segment looks like longer than this. But that is not enough. One wants to be able to measure them in order to compare. We want to be able to determine how much longer is the second segment of the first. And how do we proceed in this case? Well we define a single length. And if that’s our unit length, we say it’s one unit, then we can say how many of these lengths are applied to each of these sections. In this first section, it seems we can apply one of these units and then we can do it again, so it seems so far there are two units. While for the other at first glance it seems that we will be able to apply – let’s see, these are 1, 2, 3 units. We have three of the units. And here I just say units. We have agreed, for example, on the centimeter unit, which looks something like this. It will look different depending on the screen. Or we can have an inch that looks something like this. Or a foot I won’t be able to fit on this screen, given the length at which I drew an inch. So there are different units that we can use for measurement. But now let’s think about more dimensions. This is literally a one-dimensional case. This is 1D. Why is it one-dimensional? Well, I can only measure length. But now let’s go to the 2D case. Let’s go to a two-dimensional case where the objects can have length and width or width and height. Let’s imagine two figures that look like this. Let’s say this is one of them. This is one of them. And notice that it has width and height. Or it can be considered as width and length, depending on how we want to view it. So let’s say this is one figure here. And this is the other. I try to draw them relatively accurately. So, again, we have two dimensions. How much space does this take up in two-dimensional space? Or what area do these two occupy? Again, we can just make a comparison. Here the second figure, if we consider the figures as carpets or rectangles, the second rectangle takes up more space on my screen than the first, but I want to be able to measure it. And how can we do that? We will define a single square again. Instead of just a single length, we now have two dimensions. We need to define a single square. Let’s draw our unit square. And the single square, which we will define as a square, of which the width and height are equal to the unit length. This width is one unit of measurement and height is one unit of measurement. So we will often call it a square unit. We will often meet it as 1 unit. We put this 2 up here, which literally means 1 unit per square. And instead of recording a unit, this it could be an inch. So this will be 1 square centimeter. But now we can use it to find these faces (areas) here. And just like we asked how many times this single length can be applied in these sections, can we ask how many of these single squares can be plotted here? So here we can take one of the single squares and let’s say, well, it takes up so much space. We need more to fill it completely. Well we will add another single square there. We will add another single square here. And we will add another one here. Exactly 4 single squares fill the space. So we will say that the face of this figure is 4 square units or 4 square units. And this here? Here I can apply 1, 2, 3, 4, 5, 6, 7, 8 and 9. I apply square units. We continue like this. We live in a three-dimensional world. Why limit ourselves to just one dimension or two? Let’s now go to the 3D case. Again, when people say 3D, they speak of 3 dimensions. They talk about different directions, in which we can measure things. Here we have only length. Here we have length and width or width and height. And here we will have width, height and length. . We say again, if we have, for example, one body, and now we are in three dimensions, we are in the world, in which we live, which looks like this. There’s another body that looks like this, as if this second body takes up more space, more physical place than this first body And it seems to have a larger volume. But how do we actually find it? And let’s not forget that volume means how much space something takes up in three dimensions. The face represents the occupation of space in two dimensions. The length is related to taking up space in one dimension. But when we think of space, we we generally think of three dimensions. Then how much place will we occupy in the world, in which we live? And as we did earlier, we can define, instead of a single length or a single face, we can define a unit volume, or a unit cube. Let’s do it. Let’s define our single cube. And here we have a cube, so its length, width and height they will be equal to each other. My best attempt is to draw a cube. All squares in it will have sides of one unit. So the cube will be one unit high, one unit wide and one unit long. And to measure the volume, we can say, well, how many of these single cubes can be applied in these different bodies? Well here – we won’t be able to see them all; I can actually divide this whole thing into – let me see how well I can do it, so we can count them all. It’s a little harder to see them all, because some cubes are at the back, but if we look at this as two layers, then a layer will look like this. A layer will look like this. And let’s imagine that two such layers are stacked on top of each other like this. So here we will have 1, 2, 3, 4 cubes. And here we will have two such layers on top of each other. Ie here we have 8 single cubes. Or we have a volume of 8 cubic units. And here? If we try to fit everything – let saw how well I could make a drawing. We will have something like that. And obviously this is a rough drawing. If we try to separate this, we will essentially have 3 parts stacked on top of each other, each of which will look like this. My best attempt at drawing. Three parts that will look similar to what I will draw now. Each part will look like this. If we look at these three segments and gather them on top of each other, we will get something like that. And each segment contains 1, 2, 3, 4, 5, 6, 7, 8, 9 cubes in itself. 9, multiplied by 3, will contain 27 cubic units in this one here. And I hope that helps us think a little bit about how we measure things, especially how we measure them in the presence of different dimensions, especially in three dimensions, when considering volume.

Algebra