 Hi, I'm Zor. Welcome to a new Zor education. This is an introductory lecture dedicated to lengths and area measurements. I did touch what lengths actually is all about in some previous topics about segments, etc. I would like to approach this from a little bit more theoretical, more mathematical, if you wish, standpoint from the standpoint of the theory of measure. So all these concepts like lengths and area and volume and some others actually do have certain characteristics in common and as you know mathematicians usually would like to kind of extract what's the commonality among different objects and combine them into one abstract theory. So I'm talking about measure theory or theory of measurements. So let's talk about measurements. What actually measurements is? Well first of all if we talk about measurements we have to talk about what exactly we are measuring. We can measure the lengths of a segment we can measure the area of parallelogram, we can measure the height of a building, we can measure the height of a tower made of tele cubes. So first of all what's very important is to define the set of elements for which we introduce certain numerical characteristic which is a measure of each element. So every segment has a length every parallelogram has an area, every toy cube tower has a height. So my first point is that we have to be very strict about what exactly we are measuring. So we are talking about a set of certain elements and we would like to assign certain numerical characteristic to each of these elements. The question is how? And another question is what kind of properties this particular characteristic has. Well again combining a general experience about measurements let me exemplify it. For instance you have a tower which consists of a certain number of toy cubes, then you have another tower and then we put one tower on the top of another so we will get something like this, something like this, right? So how about this? Does it make sense? Oh absolutely. So not only every toy cube tower has certain characteristic like in this case number of cubes, in this case it's four, in this case it's two and the combination is six. But also our measurement actually satisfies exactly the same equation. Now what does it mean? Well let me abstract this particular thing. We have certain operation on the element of a set we are measuring. Now in this particular case the operation on two elements, two different towers built from the toy cubes is the operation is putting one tower on the top of another and that's what we did here. Well let's call it somehow this particular operation. I think it's appropriate to call it combining. So by combining two elements of our set and getting another element of the same set of all the towers built from the toy cubes we are actually defining an operation and our measure should somehow be consistent with this operation. And this is a very important thing because if you will go to let's say another example for instance if you go to segments, if you have one segment and you have another segment let's say this segment has the length five and this length let's say two. You can put segments together getting some longer segment of the lengths up just with seven. So that's what we kind of expect from the measurement if it's correctly made. Measurement of two different segments if we put them together then the new segment, longer one, will have the measure which is equal to some of the measures of these. So one extremely important quality of the correctly defined measure on any set of objects which are subject to our measurement. The property is that if we combine in some predefined way these two objects which we are measuring we will get another object and the measure should actually add up. So measure has a property of being additive relative to an operation of combination. Whatever the operation of combination is defined whether it's putting one tower on the top of another or stringing together two different segments or whatever else we can you know think about it. So we have a set of objects which we would like to measure. Then measure is some numerical characteristic of each object each element of this set. We also have an operation of combination, I call it combination, combining of two elements of this set to get the third one and the measure should be additive relative to this particular operation. Now what else is interesting. I think it's very natural to assign the measurement or measure of empty set to be equal to zero. Empty tower is basically the tower which contains zero cubes and obviously if this zero cube tower if you will put on the top of another you shouldn't really have the measurement different. So the addition should provide exactly the same results. The four plus something which is an empty measure should give the four because this empty tower doesn't really add anything. So this particular measurement of the empty element or empty set or whatever the empty actually is depends on the set. Measurement should be equal to zero because that's the property of zero to be added to some other number to get the same result. So that's very important. There are some other important qualities of the measure but these are probably the most fundamental. So if there is any element in our set which combined with another element doesn't change that other element and that's what I call basically an empty element. It should have a measure of zero. Now in the segments what segment has the length of zero? Well the segment when the left and the right boundaries coincide basically making it a point. So if you segment at a segment of which contains only one point you should have exactly the same result which means that the segment which has only one point left and right boundaries are coinciding together. It should have a length of zero. It's natural. Another very important property of the measure is positive because if one object has a negative length and another has a positive length we combining them together we kind of expect we will get it in some sense a bigger object. Bigger segment or taller tower. So that's why measurement is defined as a positive function, positive numerical function of the elements of our set. So what have we learned? There is a function which is defined on every element of our set of objects we would like to measure. This function has positive or zero value zero is also possible. Those objects which combined with other objects do not change them should have measure of zero and one of the probably most fundamental measure is additive. Two birds or relative to the operation of combination. So that's basically kind of a general thing about measurement and what we have to talk about right now is well what measure is we basically know but how to measure. How can we measure towers from the toy cube or segments on the line? Well I think very very natural way to measure the height of the tower is how many cubes does it have. I have already started. So basically if we consider the set of all towers then the number of cubes in each tower is a very good numerical characteristic which basically satisfies all our requirements for the measure. It's additive obviously. It's positive or zero for a tower which contains no cubes empty tower. What else? Basically that's what it is. So in this particular case all elements of our set will have an integer positive or zero number as the measure and it corresponds to the number of cubes this tower is built. Alright that's an easy part. Let's go into a little more complicated segments. How can we measure the segments? Well traditional way is have some kind of a measuring unit. One particular segment which we basically say okay this is the segment and its length by definition would be equal to number one. So in this particular case I can say that okay this is a segment which has the length of one. Now using the additive property of the measure I can say that this segment which is the combination of two units has the length of two three, four, whatever, etc. And that's how I can measure any segment which can be represented as a combination of the certain integer number of unit segments. Whatever number of unit segments fit into my segment that's its measure. Well that's great. It's more or less equivalent to my towers and toy cubes but does every segment have the property of the existence of this representation? Obviously not. But if I have this particular segment which is greater than one but less than two, what should I do with this? Okay, let's not forget that our measure is just a non-negative real number and not only integer numbers we have among these numbers but also fractions. Maybe the fractions will do. So if I can combine let's say one unit segment and half of unit segment which is this little piece, then I can say that the length of this should be one and a half. Now I can obviously expand this to any rational let's call it rational segments which are segments with rational lengths. It means the length is m over m means that I should have one nth part of the unit segment and add it together m times. That's basically a consequence of the fact that my measure is additive. So if one unit has the length of one then I can divide it into n equal pieces and say that each one has the length of one nth which is fine. If this has the length of one then this has the length of one quarter. Why? Because four of them make the unit one. That's why it's very natural to assign the measure of one quarter to every fourth part of the unit segment. And then again using the operation of addition I can edit the property of editiveness of the measure I can say that if I can take one nth of the unit segment combine them together m times then I will have the segment which has the length m over m. So that's how I cover a lot of segments. Not only those which can be combined from the integer number of unit segments but also from its fractions natural fractions like one nth. Are all the segments covered? No. We have segments which cannot be represented in this particular way. And here is the perfect example, very simple example. If this is a square which has a unit one on the sides then everybody knows that from the Pythagoras theorem the length of diagonal is square root of two. And it can be proven that this is irrational number. Actually I did prove it I think some time ago when I was talking about when I was introducing irrational number in some numerical systems lecture. I proved that one square root of two is not really an irrational number. So it cannot be represented as the rational fraction of unit segment. So what do we do in this particular case? How can we define a measure which is irrational? How can we measure irrational segments? How can we measure the rational segment? Well that's easy. If you remember we take the unit segment divided into n equal parts. So this is one nth and then we combine them together and we will get m times. If we get m over m rational length. So we can physically measure it using the circle, using the compass and the straight ruler. How can we measure this? It's not easy, right? Now let's remember the theory of limits. I do recommend you to go to Unisor to the algebra section and review those lectures which are dedicated to sequences, series and limits. Because measurement of irrational segments is actually, in a truly mathematical fashion, can only be considered as certain limit of rational segments. And it's based on the fact that any irrational number can be approached, can serve as the limit of certain process where rational numbers are involved. For instance, if you will take for instance a decimal representation of square root of 2, it will be 1.414 and I don't remember whatever else is. So you can always consider a sequence of rational segments. First rational segment is 1, next is 1.4, next is 1.441, next is 1.414, etc. So these rational segments, each of them, is actually rational because we have a finite decimal fraction so we can build it in some way or another. But their limit, and I'm talking about the limit theory, about sequences, about series and all this stuff, their limit actually is this particular thing. So we can only tell that the true measurement can be made not precisely but only as a process and as a limit of that particular process of approximating a particular irrational segment with its rational parts. First you represent like 1, then 1.4, then 1.41, then 1.414, etc. and only the limit of this would be this irrational segment. So measurement of all these irrational things is always related to some process where you're going to a limit. Now, if with segments it might not be absolutely obvious, then I will make it a little bit more obvious for you with another example. So let's talk about circle. Circle, and I will talk separately about how to do it but in theory what is a circle? Circle is not a straight line, right? It's curved. How can we measure a curve with a straight unit segment? Impossible. I cannot really put that onto this particular curve. It will not fit no matter how I try. So we have to define the lengths of these curved lines in some other way and again the theory of limits actually comes to mind because we can always approximate this particular curve with some kind of regular or not very regular polygon and the better approximation is the closer the length of this particular, the perimeter of this particular polygon would be to something which we can think about as the length of the circumference of the circle. So to define what is a circumference of the circle we really have to consider this approximation and go to certain limit under certain conditions more and more sides of this polygon would be closer and closer to the circumference of the circle. This is the process how we can define what is the length of a circle, what is the circumference rather of a circle and it goes not only to circles it goes to any line I can just talk about the line like this what is its lengths if you want to measure in these particular units and again the answer is you have to approximate it as much as you can and go to some kind of limit and that limit is by definition is the lengths of this particular curve or that particular curve. Now the only thing which I would like to talk about is how about area. Now if the length seems to be a little bit more complicated the levels of complications are first we are talking about integer lengths like height of the tower of the toy cubes then we go to internal lengths when you have to really divide the unit of the measurement into certain number of parts and then combine them to get to the irrational segments then to irrational segments the measure of which can be defined only as a certain limit of a certain process of approximation then even more complex you have to go to a nonlinear objects like circle for instance like circumference of how to define circumference of a circle then you have to really like approximate the whole curve with certain finite number of rational segments. Now we will go to areas with areas it is even more difficult because the shapes are much more complex. First of all let's talk about the unit of measurements. Traditionally we have a small square and if the side is equal to one in terms of segment lengths so this is the segment of the lengths one it's a unit of measurements of the lengths then we can say that this particular segment has area of one whatever the units are I mean I'm not talking about units like centimeters, meters, yards or inches or whatever else we are talking about abstract numbers so if a certain segment is assigned a number one as its measurement so it's a measurement unit then this particular square is assigned an area of one. Yes in case of the lengths if it's meter then in case of a square it's a square meter if it's inch it's square inch etc but let's just not talk about this only numbers. If this is the length of one then the area of this square is one by definition this is our unit of measurement. What can we measure using this particular unit? Well we can definitely measure something like this. A rectangle which has integer number of unit of lengths on each side because this is in this case it's measures two, this is measures three so it's six different squares like this which means the area is equal to six and obviously our editiveness is preserved in this particular case if we consider only these rectangles with integer number of unit segments on each side we get very clear picture of pure editiveness everything is defined quite well. Now the complications obviously the unit the lengths of these sides might not necessarily be an integer number of the lengths units right so it can be again rational it can be irrational etc etc so basically knowing the lengths of these two sides and noticing that in case of an integer the area is actually the result of multiplication I can always say that I can obviously prove that this is true for any rational segments because you can divide every rational segments into one ends of the unit of lengths and then the multiplication will be preserved. But even with irrational I can always say that okay since my lengths of irrational segment is a limit of corresponding rational ones and for every rational I do have the result of multiplication of lengths by lengths of the sides as an area I can actually say that this is a valid definition of multiplying the lengths whether it's rational or irrational to get the area of this particular rectangle. So it's very important to understand that in case of integer it's obvious in case of rational can be proved and in case of irrational we are using the fact that we're basically approximating irrational segments with rational and that's how we define area of any rectangle alright now let's talk about circle. Try to put this as a measure of this circle it's not easy it doesn't fit obviously I mean it fits somewhere in the middle but how about the edges so it's again the same kind of a difficult problem we have to gradually increase the complexity of different areas which we can, different objects which we can measure the area of so in the case of a circle for instance what we do first we do as much as we can integer squares like this right then when we have edges what we can do we can divide it like this let's say 10 vertical and 10 horizontal so each of them would be 100's of one 10's times one 10's and then we can use these small ones to inscribe into whatever the holes we have here but we still have holes closer to the edges well then each of them we will divide again in 10 by 10 into 100 different pieces and the process can be continued and that's how we will inscribe into our circle first integers then fractions then the fractions of fractions etc and by summarizing by using the edginess of the measure we can always measure now it's a different story that I can actually calculate the measure using some formula like pi r square where r is the circle's radius but this is all kind of a subsequent after I defined what the actual measure is because if you will tell me the area of the circle is pi r square well what is the measure I mean I can't really by definition say that the area of the circle is pi r square right so first we have to define measure in some basic terms like these ones and then prove that pi r square is actually the measure of the measure so these are most important elements of this is an introduction actually important properties of measure so you could feel that it's not such a simple thing as just applying for rectangles for instance the formula length times width or for a circle it's pi r square or something like this it's not just the formula it's the formula which is supposed to be well learned if you wish if you are talking about measure you really have to understand the theory behind it the property of the measure in an abstract sense that this isn't an additive positive function and it has certain properties etc one more very small measurement if you have two congruent geometrical objects and we measure their measure then it must be the same right if you have two congruent segments they must have the same length if you have two congruent triangles whatever the measure whatever the area of a triangle actually is the measure of two congruent triangles must be the same now this is again an axiomatic property of the measure and should be considered together with editiveness existence of the zero measure existence of the unit measure etc etc and that concludes my introduction into lengths and measure I will go into more details about what is the length of the circumference and how it can be measured using the theory of limits or area or whatever else that would be in the subsequent lectures this is just an introduction general theory of what actually measure is so you have some more appreciation to abstract approach to measure because we can measure all the different things not necessarily the length and the area as is typically but we can also measure for instance the towers of toy cubes or anything else thank you very much that's it for today