 Hi, I'm Zor. Welcome to Unisor Education. Today's topic will be something about lines in geometry. Well, let me start from just pointing to one of the previous lectures, which contains notes with the 2021 axioms of German mathematician Hilbert to basically replace Euclid's axioms. Hilbert's axioms put geometry in the very, very solid, rigorous, logical foundation. And about lines, I would like to remind three particular axioms, which are very important for just understanding how the whole theory is built. So here are just three different axioms about lines, which I would like to mention here. Number one is that if you have two points on the plane, there is a line, straight line, which contains them. Another thing is that if you have two points of one line coincide with two points of another line, I put them very, very close, but actually it's the same points. These points are the same and these points are the same. These two belong to top line, let's call it A, A1 and A2. And points B2, B1 and B2 belong to a line B. So if A1 coincides with B1 and A2 coincides with B2, then A and B coincide with each other. It's one in the same line. It's kind of obvious, but still it's the axiom. And yes, obviously most axioms are quite obvious, maybe except the first part of the parallel lines. But in any case, what it says is that if you have two points, there is only one line which can cross them. So that's number two. And number three is that two different lines can cross in no more than one point. Actually, they can have no crossing points as they are parallel, but at least no more than one. So these three statements about lines, which seem to be obvious, Hilbert had to actually put them as axioms if he wanted to build a logical and written rigorous generatrical theory. So just keep them in mind. Primarily, we will not actually be using explicitly these particular axioms, but it's a very good practice to logically understand that something like this should be in the foundation of geometry. Now continuing this topic, I would like to basically consider segments which are parts of the straight lines in between two different points. By the way, I'm using certain words like in between, which again need to be clarified in one way or another. And again, I should refer you to Hilbert's axioms where he quite extensively actually elaborates about what it means in between, et cetera. We understand it on a common sense level, which is sufficient for this particular thing. So speaking about segments, segments actually do have certain properties. And you can operate. You can do something with these segments. Now one of the obvious things which you can do is to state that two different segments are congruent, like this one and this one. Intuitively, we are talking about the lengths of the segments that I will address a little later. In theory, congruent means you can just have one segment and using certain non-deforming transformations, like shift this end to this point, to this end, and then turning. Now this will be something like this. And then turning, rotating the segment counterclockwise until this point will coincide with this. That's what basically congruent means. All right, so we understand what congruency among the segments actually is. It's the existence of the process of non-deforming transformation, which will make them coincide when left end of one segment coincides with left end of another segment and the right with the right. All right, is that the only thing we can do, just transforming one segment into another and then see whether they are congruent or not? No, there are certain operations which you can do with segments, actually arithmetic operations. Just as an example, let's talk about addition. Can I add two different segments? Well, obviously, again, intuitively you understand. You just take one segment, attach to it another segment, and you get something which can be called a sum of these two segments. All right, let's try to define it more rigorously. So if I have two segments, segments AD and another segment CD, what does it mean to add these segments? Now, by the way, I put them like they are located on the same straight line. That's absolutely irrelevant. I can put CD somewhere here. Doesn't really matter, because whatever I do has absolutely nothing to do with their location. It has something to do with their length, actually. All right, so what does it mean to add these two segments? Well, quite simply, it means that I have to find another segment called EF and a point in between called G in such a way that AB would be congruent to EG, this one to this one, and CG would be congruent to GF, this to this. So that's what it means to add these two segments. Well, I can define anything I want. The question is, does it make any sense? Can I really construct the sum of two segments? Well, apparently, yes, we can. You remember that all the constructions in geometry traditionally are made with two different instruments, straight ruler and the compass. So using these two instruments, I can construct this particular sum of these two segments using a very simple technique. Step number one, I draw a line using the straight line ruler. Number two, I pick the point, which is E. Number three, I take the compass and put one left in A and another in B. And then the one which used to be in A I put in E and what used to be the span of the compass, I will just mark the point called G. Then I repeat the same procedure with the segment CD. I change the compass from C to G. And using this span, I put my leg into G. And another leg of the compass will mark the F. Now, obviously, this procedure of taking the compass to a certain segment and mark it somewhere else on the line marks the new segment, which is congruent to the original one. Same thing here. So we have built the segment EF and the point in between G with EG congruent to AB and GF congruent CD. So we have constructed our sum. And basically, that's the proof of the fact that sum does make sense, that our operation, which is defined like we have to find a new segment and the point with this property of being congruent, that this definition does make sense. This segment does exist because we physically constructed this segment. OK. Now, we have constructed the sum of two segments. What it means, we can actually construct the sum of n segments, where n can be any integer number. Now, constructing the sum of n segments of the same, constructing a sum of n same segments, which means AB will be summed with AB, with AB, with AB n times. Basically, that's what people call multiplication. So what we can do is, not only we can do AB plus CD equals this particular segment, we can also construct AB times n. And how can we construct it? Well, very simply. If n, for instance, is equal to 2 for simplicity, we take compass from AB to AB, from A to B, marked here. Then from the game, the same span of the same compass, we mark the second point, which is equal to AB. And that would be EGF, where EG is equal to AB and GF is equal to AB. So we basically multiply our initial segment AB by 2. We add it to itself twice, and we can add any integer number of times. So we can add segments, and we can multiply them by integer number. Let's do some reverse operations, subtraction and division. How to do that? Well, it's quite elementary. Let's do subtraction first. How can we subtract from AB, CD? Well, we start with basically having a compass spanning the AB and marking it here. Then we have the compass set to CD. And then instead of going outside of the segment EF, we go inside. So we go this way. This will be G. So EF is congruent to AB, and GF is congruent to CD. Then this segment in G will be a difference between AB and CD. Why? Well, do you remember how to check that the difference between, let's say, 5 and 3 is equal to 2? What does it mean? Well, it means that if you will take the result of this subtraction 2 and add to the number which you subtract, you have to have 5, the original number from which you started. This is a definition of subtraction. Now, does it actually fold in case of our segment's arithmetic? Well, let's try. If we will add EG to GF, according to our rules of addition which we have already discussed, we obviously have a segment EF which is congruent to AB. So that's what actually is a proof that our subtraction does make sense. If I take the difference, EG, and whatever I subtracted, I have original AB from which I started. So this is basically the explanation of what the difference between two different segments is. By the way, I have to note that inability to subtract from a smaller number, the bigger one, from 3 to subtract 5, this inability led mathematicians to invent negative numbers, minus 2, for instance, in this case. Well, in this case, negative segments were not actually invented because they don't have any practical application. So we are saying that this subtraction works only if we are subtracting from a bigger segment to subtract the smaller one, which does make sense for some practical standpoint. We don't have any kind of negative segments. Segments with a negative length, so to speak. All right. So in all addition, we know subtraction in our multiplication. Well, division is obvious. We have a very, very briefly mentioned here. Again, let's consider what division actually is. What does it mean that 6 divided by 2 is equal to 3? What does it mean? It means that the multiplication of the result of the division by divisor gives the original number from which we started from. So division is validated by a multiplication. Same thing with segments. Since we know what multiplication actually is, if you want to divide by, well, let's say 2 in this particular case, what does it mean? Well, it means that the segment Cd multiplied by 2 would be equal to AB. So if I have a segment CE, which originally is congruent to AB, and I can find a point d, which actually gives me two equal segments, not equal, actually, congruent, Cd and de congruent to each other. Then, of course, as you understand, Cd multiplied by 2, which means added to itself twice, will give me the original segment AB. So that's what actually means to divide a segment by any natural number n. If you want to divide it by, let's say, n, it means that the result of the division multiplied by n should give us the original, which means that this point d should be somewhere here, where n is the number of these little segments which added up result in AB, in the original segment AB. So basically, we have to know how to divide a segment into n pieces, each one having plans of one nth of the original piece. And by the way, it's a very interesting problem, and we will address how to add is simply how to divide is not. So I'm not going right now into construction of the division. Right now, I'm just talking about the existence of this point d, which will produce the segment Cd multiplied by n, giving the original segment AB. So it's existence, but not the construction. Construction will follow. All right. So we have segments, and we have segments arisen. OK. Now, we know how to multiply a segment by an integer and how to divide it by integer, which means we can very easily introduce multiplication of a segment by any rational number m over n, where m are our natural numbers. Obviously, it means that by definition, I'm not really trying to prove anything. It's the definition. It means that you will multiply AB by m first, having summed AB m times. And then the resulting segment, you divide into n equal pieces, as we know how to do it actually. And that will produce this result of this multiplication. I'm not saying it's completely rigorous. It's probably rigorous enough for a certain level, which I'm trying to address. Because in theory, whenever I define something like multiplication of a segment by rational number, I have to address two very important issues. Well, existence of the result of this multiplication and its uniqueness. Because you don't want two different people using two different ways to come up with multiplication, to come up with different answers. So existence and uniqueness of the result of this operation are supposed to be addressed very, very rigorously and logically. I'm not doing this right now, because the level which I am trying to address actually suffices. But in theory, you have to understand it's really something which we might address maybe in the future. All right, so we know how to multiply by any rational number. Well, from this, it's really one little step to multiply by any real number, which includes rational and irrational number. And you can consider any irrational number as some kind of a limit of rational numbers, which are approximating this. For instance, irrational, the square root of 2, is approximated, for instance, to 2 decimal points with 1.41. This 3 is 1.41.4. I'm not sure I'm right, by the way. In any case, you can always approximate with a certain number of decimal digits. And each time, this is a rational number written in a decimal system, but it doesn't really matter what the system is writing in this case. So you can always multiply any segment by any approximate value. And then the result will actually go into a certain limit. And that limit is called the multiplication of AB by square root of 2. Again, it's definitely requiring certain knowledge of how irrational numbers are defined as limit of rational sequence of numbers. It's not an easy topic. And I don't want to address it right now. I just want you to understand that we can definitely use something like this approximation to make this calculation as precise as possible. And whatever the result of the limit of different results of the multiplication of AB by certain more and more precise approximation, that limit will be this particular result of this particular multiplication. All right, so we can do that. And now let's think about measuring. How do we measure the segments? Well, basically, we can do it if we know that segments can have this type of arithmetic. Now, how? Very simple. We take one particular segment as a unit. Doesn't matter. This is a unit. Let's call it x, y. Now, if you would like to measure any particular segment in these units, you have to basically find out what is this multiplier? Or if you wish, you have to be able to find the divisor, which will give you x, y. That's the same thing. So either you divide AB by r, and you get x, y, or you multiply x, y by r to get AB. If you find this particular r, then this r is a measure or lengths, basically, expressed in these units, in units of x, y. Now, again, there are very important logical steps, which must be went through if you want to do it really rigorously, really logically correct. We have to prove existence of this number and its uniqueness. And again, it actually follows from the definition of finding, for instance, among rational numbers, finding m over n, which will give this particular equality. Now, in case of irrational numbers, again, you have to consider the limit of certain approximations of the irrational number. And without going into all the details, I can just say that existence and uniqueness are important, and they can be proven, which means that the lengths of the segment expressed in the units of any units, basically, in this case, does make sense as a mathematical definition. However, as everything which is related to some fundamentals, it's always very difficult to prove because, you see, if you are trying to prove something which is much more advanced, you have all the baggage which preceded this, which has already been proven. If you are going into some very, very fundamental things, like lengths of the segment, you don't have much, except the axioms of Hilbert, basically, to manipulate this. And that's why it makes it a little bit more difficult. However, proper knowledge of Hilbert's axioms would suffice to logically step-by-step prove existence and uniqueness of this multiplier R, which makes the concept of lengths really making sense. And now, the very last, basically, topic I would like to address, what are these units of measurements? Well, everybody knows that there are many different units of measurements. Different people at different times invented different units of measurements from food to meter to astronomical units, whatever it is. So basically, it doesn't really matter what kind of unit you are choosing. I mean, it probably is important for physicists or somebody else because they have to deal with real world and real objects. We are dealing in abstract math, which means we can say that any segment can be chosen as a unit. And then based on this particular segment, we can measure any other segment in these units and compare the lengths. Basically, comparing the lengths means just comparing this number. So if one particular segment, so this is our unit, x, y. So if A, B is x, y times R, and then you have another segment, C, D, which has four lengths, S. So this one has a length, R, and C, D has a length, S, in terms of units, x, y. Well, then you can do anything you want with the segments. And the lengths will actually follow the segments. For instance, if you add two segments together, well, obviously you understand that using the construction of the addition which I have offered, their lengths will also be added together as two different numbers. And what it actually corresponds from here is that it's supposed to be R plus S. So lengths should be some of two lengths. But on the other side, if you just follow the purely syntactical expressions which I have here, I mean, you can always add these two together and you will have A, B plus C, D equals to x, y, R plus x, y, S. So these left parts of these two equations are the same, which means that this is the same, which means, as you see, we have a distributive law which multiplication by some number, multiplication with segment by certain number, really awaits. So x, y times R plus S is equal to x, y, R plus x, y, S. This is a pure distributive law of multiplication relative to addition. Well, it works with the segments as well. So that's quite interesting. And obviously you understand that not only addition works, but the subtraction works as well. And even the multiplication and division works. And without much of an effort, we can derive something like, if you have x, y times R, and then you get some segment and then you multiply it by S, it would be the same thing as x, y multiplied by R times S, which is associative, basically. Now, what's important here to understand that x, y is not a number. This is a segment. So this is an operation between segment and the number. This is an operation between a bigger segment and a number. In this case, it's an operation between segment and this number. But this particular sign is a multiplication between two different numbers. So you see how interrelated operations on numbers, which represent the length of the segments, is with operations or on segment and numbers. So I should actually differentiate this operation, which is, let me put it in a circle, which is a multiplication of a segment by a number. And this is as well. But this is actually a true multiplication in original sense, multiplication of two numbers. I'll just use that to differentiate them. So this associative law kind of works, except that in some cases, it's a multiplication of number and segment and another number and number. All right, so basically, that includes my little exercise about segments and their lengths and their measurements and arithmetic operations on segments. I would also like to point out to the website Unizor.com, where not only you can study this material, there are some notes over there. And what's important is that parents or supervisors can actually use this website to control the educational process of students which they supervise with corresponding exams, tests, scores, et cetera, et cetera. Thank you very much.