 I am Zor. Welcome to Unizor education. I would like to continue to talk about rational numbers. Rational strings will be subject. When I was introducing integer numbers, besides something which is, I would say, a homemade definition of minus 3 as the number which if added to 3 results in 0. This is not a mathematical rigorous definition because maybe there is no such thing as negative 3 which will result in 0 if added to 3. So instead I have offered a string-based notation. I have basically constructed new objects which looked like 4 plus 3. And defined operations of addition among them and it actually followed that in that particular universe of integer numbers which I have built using these abstract strings I can basically do all the arithmetic operations. That was fine for introducing an operation of subtraction and now integer numbers are complete because operation of subtraction is defined on any two numbers in as much as operation of addition was defined on the amount of the natural numbers before them. Now, we have a similar situation when we are introducing rational numbers. These were introduced as a result, as a necessity to be able to divide any two numbers into each other. We know that we can multiply two integer numbers but we cannot always divide one by another. So if I'm saying that I can multiply 3 by 5 that's no problem but if I want to divide 3 by 5 among the integer numbers that is not possible. So to overcome this, again, the homemade definition was we introduced number 3 fifths which if multiplied by 5 gives 3. Well, it's not good enough for definition because maybe there is no such number as 3 over 5 as 3 fifths. We say a property of an object is defining an object. That's necessarily true. Sometimes you can define the property of an abstract object when it doesn't exist at all. To constructively approach this particular problem is to basically create new objects from whatever elements you have and in as much as I created integer numbers using these strings we were talking about with curly brackets, curly braces, whatever. I will try to do very much similar thing to introduce rational numbers. So if these are integer instead of expanding the set of integer numbers into a set of rational numbers I'm saying that rational numbers will be a completely different set. However, there will be a subset which we can call SUDA integer. This is the subset. And every element of integers, every element of integer set, every integer number should have a corresponding element among SUDA integer numbers which are part of subset of rational numbers. And operations among integer numbers like multiplication in our case and division whenever it's possible should really correspond to this similar operation among the SUDA numbers. So that's the goal. And let me start walking along this way and I will constructively introduce the rational numbers by basically saying these are new objects which I'm going to construct. They are called rational. Okay, so here is the definition. Rational numbers are strings of curly braces and in between you have two integer numbers separated by vertical bar. Well, obviously you understand that in the future we will probably write something like this which is much more traditional but I would like to break this tradition because this is a formal definition. The rational number is not a number, so to speak, it's a string which consists of two braces on both sides, vertical bar in the middle and two integer numbers on the left and on the right from the vertical bar. All right, so I have introduced these numbers and now I have to operate upon them. So I have to define an operation of multiplication because that's basically the ultimate goal to operate future numbers freely as far as multiplication and division are concerned. So let's define an operation of multiplication. Well, there is one more thing which I actually did not say yet and should not be equal to zero. This is part of the definition. So my new set contains all strings of this type where n is not equal to zero. And obviously you understand that you cannot divide by zero for whatever reasons and that's why I have restricted my constructed number of constructed objects to those where denominator, I will call the denominator and numerator obviously in the future where denominator is not equal to zero. Okay, so these are strings, nothing more than that. So first what I'm saying is that there is one particular string which is one over one, which is a unity. Unity means that being multiplied by any other rational number in this particular incarnation as a string, it will result in the same string. So if I will do, I'm introducing an operation of multiplication. So this is the multiplication. If I am doing something like this, I will always have this. And vice versa. Vice versa means I'm basically defining the commutative law in this particular case. So that's the definition of the unit actually is. If multiplied by anything else, it will result in the same, that same anything else. Left or right or right or left. Okay, so this is one thing which we have defined by definition. So this new set of elements where m and n are any numbers, any integer numbers, one of those is this particular element. So I'm saying that this is it. This is the unit which the unity which will result in this rule of multiplication. Another thing which I would like to say again by definition is by definition that if you have two rational numbers, rational string, whatever you want to call it, of this type where m is any integer number, this will be the unity. What this basically means is, well, obviously as you understand, in the future an integer 123 will correspond to a pseudo integer 123 over 1. So this is 1 over 123 which is an opposite number result of the multiplication of 123 and 1 over 123 should be equal to 1. So by definition I actually say that this is the rule of multiplication of my numbers, my new numbers, my new rational numbers which are actual strings. So for any number of this type there is always an opposite of this type. Okay, saying that I have to really introduce the rule of multiplication between two rational numbers as introduced here. And the rule is very simple. If I have a over d and I'm multiplying c over d where a, b, c, d are a integer numbers. By definition this is the result of multiplication of a by c and we know how to multiply two integer numbers. That would be my numerator and my denominator will be d times d. So this is the real multiplication and this is the real multiplication between two natural numbers. So that's how I define my multiplication between two rational numbers and multiply numerators and multiply denominators. This is a rule. This is the definition of multiplication between two strings which represent my natural numbers. Well, as a consequence of this let me just prove a very simple property of these numbers. You know that if you have a fraction let's say 4 over 8 you can always reduce it to 1 half. So if there is a common multiplier in numerator and denominator you can basically reduce the fraction to get smaller numbers in both numerator and denominator. So let me do something similar with this. My property which I would like to prove is the following. Let's assume that there is a common denominator between multiplier k which is included in numerator and denominator. I would actually like to prove that this is the same as m over m. So that this string is equal to this string using these rules. Okay, how can I prove it? Well, very easily. I will use just these two rules basically. And here is how. I can say that this particular rational number expressed as a string of this type of quotient is equal to result of multiplication of m over 1 times k over 1 times this is multiplication of rational numbers. 1 over n times 1 over k. Well, why is it? Well, obviously if you multiply m over 1 times k over 1 according to this rule m over 1 k over 1 it will be mk over 1 1. So this will be mk over 1. And this similarly will be 1 1 times 1 over n times k. And the result of multiplication of these two will be mk times 1 on the top and 1 times mk on the bottom. So that's why I can do this. But now considering I have defined my operations in such a way that obviously I would like to say that all the commutative and associative laws of multiplication should be preserved. Basically I can use the commutative law and change the order of these two things so instead of k over 1 and then 1 over n I can put 1 over n first and k over 1 second. That's just the result of the commutative law which should be part of the definition. Now here using the associative law I can say k over 1 and 1 over k. That results in according to rule number 1 and 1 over k. That's 1 over 1, right? So I can say that the whole thing is equal to m over 1 times 1 over n times and these two guys will give me 1 over 1. Now multiplication by unity actually results in the same number I started with so I can just completely wipe out this multiplication by 1 over 1. And this thing again using this rule is equal to m times 1 on the top which is m and 1 times m on the bottom. Well, a couple of lines of proof basically shows that we can reduce our rational number by the common multiplier. So that was basically the purpose. Now as you probably have guessed correctly all I have to do is to put into correspondence my integer numbers and my new rational numbers. So again, these are integers. These are rational numbers. This is a subset of pseudo integer numbers and I have to build the correspondence but in such a way that if I have an image of one integer number as a pseudo integer and I have an image of another integer number as a pseudo integer then the result of the multiplication of these two integers should map into the result of the multiplication of their images, pseudo integers. Otherwise this correspondence is not really well defined. Well obviously any integer number n will correspond to, as you understand a rational number, a string which contains one as the component on the right from the vertical bottom. Similarly to the fact that 5 over 1 is actually 5. So that's the correspondence and obviously there is a one-to-one correspondence between all integer numbers and all rational numbers of this type. Now how can I prove this type of correspondence? Let's take number n which maps into rational number n over 1. Now result of their multiplication here is m times n. Result of the multiplication of these two things it's premature result of multiplication of these two things according to the rules is you remember numerators should be multiplied by themselves as two integer numbers and denominators should be multiplied as integer numbers. But this is one of this, right? One times one. And this is exactly what should be expected from the image of the mn. So mn maps to this. So either we do it this way so we multiply and then map or firstly map and use the images and multiply according to the rules of the multiplication of rational numbers we get exactly the same thing. So that proves that our representation of rational numbers in this way allows a very well-defined mapping between integers and a subset of the rational numbers which we can call to the integer numbers and this correspondence is really well balanced, etc. But now what's the advantage of reducing these numbers? Well, obviously now we can divide anything into anything and this is basically how. We can introduce operation of division I'm introducing operation of division if I have rational number AB and I have another rational number C over D and if you remember neither B nor D are zero then I can define operation of division between them as multiplication by reverse of this particular number and what is the reverse of C over D? Well, it's D over C which by the way indicates that C also should not be equal to zero otherwise this division is not defined. Now why CG reverse is GC as I was saying well, for a simple reason if you will multiply CG times DC this is CG over DC and you know we can reduce the rational number by the common multiplier common multiplier is CG obviously so this is the same thing as one over one which is a unity so if the result of multiplication is unity that's what we call that one is the reverse of another Alright, so this is a definition of the division and as you see any two rational numbers as long as BC and D are not zero can be divided by another Now, going back to the integer numbers and see the integer representation of our integers among rational numbers what we can say is that one twenty-three over one which is actually one hundred and twenty-three in integer number if you will multiply it by one one twenty-three you will get one twenty-three times one one twenty-three and one times one twenty-three again one twenty-three which is reducible to one over one so we have two different numbers which if multiplied result by one what does it mean? it means that this number one over one twenty-three can be called an inverse of original number one twenty-three over one so if among the integer numbers you cannot find anything which is an inverse of one hundred and twenty-three among these rational numbers it's very easy to do you just change the order of left and right part in this string representation and you got a new number which serves as an opposite as a reverse of the original number well that basically concludes the introduction of rational numbers using these strings now what's the advantage let me just emphasize it again I have constructed rational numbers I'm not saying that let's introduce a number which if multiplied by one twenty-three gives one this is not the proper definition because maybe there is no such number and if there is one we have not really defined what's the operation of multiplication among them so who knows in this way I have basically constructed a new set of objects which is a set of strings representing rational numbers if you wish this type and among these strings I can very easily define this operation of division or a reverse element to anything in very defined and constructed way so that's very important this is a constructive approach which allows to basically say I'm not defining some abstract object which has certain properties I'm saying this is an object and these are the properties so from the strict position it has much more weight and that was the purpose basically of this lecture to satisfy those people who feel that definition of one hundred and twenty-three reverse as just the result of multiplication of some number which gives one that this type of definition is really lightweighted this is much more rigorous and it's definitely not the only one which people can use to more or less rigorously introduce the rational numbers but that's just one of those approaches and the one which gives you at least some positive feeling that certain level of rigorousness is really needed this is the way to basically achieve it thanks very much