 Hi, I'm Zor. Welcome to Unisor Education. I would like to continue talking about measurements. In particular, today we will talk about lengths. The previous lecture was more about like abstract measure, what is measure, what's the qualities, etc. Like the property of being additive, for instance. In this particular lecture, I will talk about lengths. That's one specific type of measurement which we definitely think that we are familiar with. Alright, so let me start with something very, very simple. First of all, for any measurement we need a unit of measurement. Now, in case of lengths, the unit of measurement is some segment. Whatever that segment is, but we do assign this particular segment length of one. Doesn't matter what's the unit. It can be one meter, one kilometer, one yard, one inch, whatever. Now, after we have the unit of measurement, we can very easily measure any segment which has certain integer number of units in it. So, if a particular segment has one, one and one, three units of measurement in it combined together, then we assign the measure of three in this case. If you have M unit segments combined together which represent this particular segment or congruent to this particular segment if you wish, then the length is M for all these reasons. And another related to this, if for instance one quarter of a unit segment is congruent to our segment, then the length is one quarter. If one nth of the unit segment is congruent to our segment, the length is one nth. And finally, if M over M, which means M one nth part of the unit segment is congruent to our segment, then the length would be M over M. Now, these are all trivial and simple cases. And as usually, the theory is very simple for simple cases and significantly more complex when the cases are basically all other, not very simple ones. Well, obvious example is if you have a square with the side of one unit, then what's the length of diagonal? It's very easy to prove that there is no M over M which can be assigned to this. It's irrational, it's irrational number. So what did we do with measurement when the segment is not congruent to some M over M of unit segment? Well, we have a problem which we have to really spend some time and explain. It's all about definitions of real numbers, irrational numbers, how can we represent these numbers, etc. It's relatively advanced material and for obvious reasons, I don't want to overwhelm you with a really very rigorous, strict logic which leads to exact concept of what is the length of this particular diagonal in this particular case. However, I would like actually to explain basically how the whole process is going to. Well, every irrational number can be represented as a sequence of rational numbers. Now, what's interesting is that there are many different sequences which have the same limit. That's very important. To get to one particular irrational number we can have many different ways of approaching it and no matter how we approach it will be exactly the same irrational number. Same thing actually goes here. For instance, this particular length we can represent as a decimal fraction 1, 1.4, 1.421, 1.414, etc. I don't remember obviously many digits but eventually this particular sequence of rational numbers will lead us to the square root of 2. I'm not afraid to put it here. That's an irrational number. So any process which would actually lead us to a particular irrational segment will do and the limit will be exactly the same and basically the limit which can be obtained or defined if you wish in this particular way is called the length of this segment. The limit of its approximations, if you wish in a similar format as irrational number can always be defined as a limit of rational numbers. So lengths of segments we basically covered. In simple cases it's a simple thing. In more complicated cases we have to use the theory of limits of certain process of approximation to get to the concept of the lengths but what's important is and that's really very, very important any segment, any segment rational, irrational, whatever any segment has this length and it's a unique number. The theory of existence of the lengths basically is related to existence of the limit of approximations and uniqueness is also related to this particular theory. So the existence and uniqueness these are two very, very important qualities of the lengths for any segment. So no matter how we measure it no matter how we approximate its lengths we will always get exactly the same number which is defined as the lengths of this particular segment. The existence of the lengths which means we can measure any segment and uniqueness which means no matter how we measure we still get the same result. That's very important qualities. Okay, next. Next is a simple topic. If we are more or less comfortable with the concept of lengths I mean I'm sure we are very comfortable with the rational lengths but we are a little bit less comfortable with irrational lengths but still we kind of understand that it goes through the process. Next is a simple topic. If you have a polygon what's the perimeter? Well, the perimeter is just the sum of lengths of each side. Since every length, since every side has a length then we can always define perimeter of the polygon as the sum of its sum of the lengths of its sides. Simple topic, there's nothing to talk about this. Every polygon has a finite number of sides. Every side has exact lengths. As we were explaining before the lengths exist and it's unique so it's one and only one number and then we can just sum them together to get the perimeter. Simple. Much more complex case is a circle. Now, the first and obvious complexity of the circle of the circumference to measure the circumference of a circle is the following. Our unit of measurement is a segment which is a piece of a straight line. No piece of a circumference of a circle is a straight line. I mean, it's curved so no part of this segment can be actually can fit to any part of a circle. So, what do we do? How can we define the circumference of a circle? Or if you wish, the lengths of a circle which is basically the same. Well, it's not easy as I was saying and let's just think about well, similar process to measuring irrational segments. We are approaching through approximations through the theory of limits and here is how. First let's have two diametrically two perpendicular diameters and then we connect these points. It's very easy to prove that we have a square. Now, we do know what the perimeter of this square is because we were just talking before that any polygon has a defined perimeter. Now, is the perimeter of this square a good approximation for a circle? Well, quite frankly, not such a good one but we can improve it. How can we improve it? Very simply. Let me just wipe out these original diagonals and instead we will do the following. Now, this is the process. We will draw a perpendicular to each side and instead of having the straight line we will have two new ones. So I wipe out the old square and replace it with eight sided polygon. Now, is this a good approximation? Well, I think we all feel that this is a better approximation for a circle. The octagon looks like it's much closer to a circle. Now, what's the perimeter of this octagon relative to the previous perimeter of a square? Well, let me restore one particular line. Now, as you see this line which was the side of a square it's shorter than some of these two. Well, that's the inequality of triangle, right? In any triangle, some of two sides is longer than the third side. So, when I replaced my square with an octagon my perimeter has increased since every straight line between these two points I replaced with two lines which have a greater length. Similarly, next one and next one and next one. So we are increasing our perimeter. That's good because we can continue the same process again. So, we will draw from a center perpendicular to every side and take the midpoint on the arc and replace this side with two smaller ones but some of which is greater than this one. As you see, we are more tightly approximating our circle doing this particular operation. So, again, this can be replaced with this and replaced with this, etc. So, again, we are increasing the perimeter from four-sided polygon square we went to eight-sided octagon then to sixteen-sided to thirty-two-sided, etc. So, we can continue this process up to infinity and every time, on every step we are tighter and tighter approximating our circle and what's important is we are increasing perimeter. Now, the perimeter is actually bounded from the upper, from the top. Well, obviously, I mean, we can probably draw some kind of square, a very big one, to have perimeter greater than perimeter of any of these polygons which we have. It's not obvious, but I just take it for, you know, just believing it. I think you feel that this is right. We can draw a really very big square which would definitely have a perimeter larger than any polygon which is inscribed into the circle. So, what it appears to be is that the perimeters of these four-sided, eight-sided, sixteen-sided, thirty-two-sided, etc. polygons this is a sequence of increasing numbers monotonically increasing numbers which is, which has upper bound. It's bounded from the top. Now, if you remember from the theory of limits that if you have a sequence which is number one monotonically increasing and number two has an upper limit, it has a limit. And this limit in this case, as again you feel, but I'm just telling you as a definition basically would be, by definition, a perimeter, a length of the circle, circumference of the circle. So, we very much rely on the fact that our sequence of perimeters is monotonically increasing sequence and it has an upper bound. And that's enough to say that, okay, there is a limit to this particular sequence. Alright, so is it really a correct definition? Not exactly, because what if I will draw my circles somehow, my initial square slightly differently? Would it go into the same limit when you will try to implement this particular process of dividing every arc in two? What if I will start instead of a square, what if I will start with triangle and then divide each line in two? So triangle will be replaced with hexagon, hexagon with a 12-sided then 24-sided polygon, etc. It's a different sequence of polygons. Would it go into the same limit? So, what's important is we can really approach differently the same problem of approximating the circumference of a circle and the obvious question is will we obtain exactly the same result? And that's the beauty of it because in theory we must, right? No matter how we do this we must go into the same result if we believe that there is such a thing in the circumference of a circle. And yes, there is again a relatively more complicated story about this. The theory is relatively deep and what it says is the following. We can prove the theorem that if you have a sequence of inscribed polygons it doesn't really matter that they are regular like in my case, like equilateral triangle or square as a starting point and then you double the size. No matter what the sequence is as long as our inscribed polygons have a very important property that the size of the maximum side is going down to zero. So, if the sequence of the inscribed polygons is such that the maximum side the maximum side whatever the maximum is doesn't really have to be a regular polygon it can be the polygon with different size as long as the length of the maximum side goes to zero it's enough actually for the process of the sequence of its perimeters to go to the same limit always the same for a particular circle and that limit can therefore be legitimately called the length of a circle its circumference. This is not a simple thing lots of things, whichever I was talking about I'm just asking you to take for granted because it's difficult to prove and it's basically outside of the scope of high school material However, I would like you to feel that these are correct principles Now, in this particular case I think you kind of intuitively feel that this is the right thing that no matter how I inscribe the polygons if they are smaller if they have smaller and smaller maximum side it means they are tighter and tighter approximate the length the circumference of a circle and if you feel it this way then you feel that this is supposed to be the right statement that no matter how we inscribe as long as the maximum side goes to zero converges to zero then the perimeter will converge to something and that something should be exactly the same thing which is circumference of this circle So, this lecture was not really very I would say rigorous as I would prefer it to be but it's just because the theory is relatively complex like measuring of a curve with a straight piece of straight line, a segment However, I think I was trying to convey certain intuitiveness of this process So, whatever we cannot measure directly like irrational segment or curve then we are using the process of approximation and what's important is that there is always a limit of this approximation it exists this limit and it's unique no matter how we approach this particular curve or irrational segment as long as it's reasonable Now, in case of a curve, the reasonable means your largest side of the inscribed polygon converges to zero Okay, that's it for this particular lecture Thank you very much Don't forget if you register to Unizord.com you will be able to take tests and also another person can really be involved in supervision of the whole educational process Thanks again and I will continue talking about measurements in different cases including errors, for instance in the future lectures Thank you