 I'm Zor. Welcome to Unizor Education. We have already spoken about rational numbers and their more formal introduction by using rational streams of form where m and n are two integers. Basically, this is just a formal introduction. In the real life, we're using slightly different notation, but I want to stick to this for metaformalism and further define arithmetic operations with the rational numbers. Basically, we have defined rational numbers in this formalism based on pure expansion to these streams, the regular laws of multiplication of integer numbers. If you remember, these are representing actually numerator and denominator of the rational fraction. But in any case, right now, we are about to define some other operations, which are possible with rational numbers. First of all, very briefly, what we have defined as operations available with these numbers are the following. If you remember, there is a number one, which plays the very important role that multiplied by any rational number, it basically results in exactly the same one. Another operation which we have defined was an operation of multiplication. When we have two different rational number and multiply them, we will get ACVD. This is just the definition of the operation of multiplication on these formal strings. Forget about the real multiplication, as you know it in the realm of operational numbers. This is just a completely different approach with strings A and B and C and G are integer numbers which we can multiply. Based on these, the multiplication laws of integer numbers we have defined the operation for rational. Obviously, we introduced it in such a way that the laws of commutation and association will be observed. So you can change the order of multiplication, it will be the same or if you have three different strings which you are multiplying by each other, the sequence is not really important. This is just a brief repetition of whatever it was before. Now let's continue moving forward towards other arithmetic which we can do with these rational strings, let's call it rational strings, it's a nice name. Okay. So first of all, I would like to talk about reduction of these strings. Reduction, I mean is the following. Let's consider the integer M is the result of the multiplication of two other integers, and integer N is the result of multiplication of these integers. What it means is that the K is common multiplier, common factor if you wish for M and N. Simple examples are six and four. Six is two times three and four is two times two. Which means that the two is common. So I'm trying to approach the problem of reducing the rational number, something like in this particular case, if you have six fourths, we know that this is three seconds, right? So I would like to formally introduce this type of operation. So to do that, I actually have to prove that the rational numbers upper case M over N and lower case M over N are exactly the same. So if K is common factor, common multiplier in both denominator and denominator, then I can reduce by K and what's left is whatever the results of the division, capital M by K which is M and capital N by K which is lower case M. So how can I prove that? Well, very easily. Since I know that capital M is lower case M by K, I can put it here, same for N. Try a little bit more room for this. That's what I have. So how can I prove that? Well, very easily. As we know, the operation of multiplication among these rational streams was defined in such a way that M over K and K equals 2 M K 1 times 1 over M K. So instead of M times K as denominator and M times K as denominator and denominator, I'm writing the multiplication of these two rational streams. In this case, denominator is the same as this and denominator is one. In this case, denominator is one and denominator is this one. Well, indeed, the multiplication law of these two rational streams states that I have to multiply denominator by the left-hand by left-hand, right-hand by right-hand of the vertical bar. So M times K times 1 will be on the left of the bar and one times M times K will be on the right, which is M times K, will be on the right of the bar. So this representation is perfectly lawful without any problems. Similarly, M times K over 1, I can represent as M over 1 times K over 1. Right? Because again, the multiplication will be M times K over 1 times 1, M times K over 1. Similarly, the next one will be 1 over M times 1 over K. So far, I did not break any laws. I just spread all these multiplications into individual pieces. So again, multiplying M times K times 1 times 1 will be M times K here. 1 times 1 times M times K will be here. Great. Now what do we do? Well, very simple. Using the commutative law, I can combine this and this together and oh, I'm sorry, this is supposed to be a vertical bar. And again, if you remember, one of the definitions of these rational strings, when I introduced that this formalism was that any number over 1 times 1 over any number will give 1 over 1. So K over 1 and 1 over K, as it was stated in the definition of these rational string, my definition is 1 over 1 and 1 over 1 multiplied by anything will be 1 over 1. So basically, I'm just gathering all the pieces together but in a different order. Same thing, M and 1 and 1 times M. So this will be M over M, which is exactly what we wanted to prove, that this is equal to this. So we can reduce. So using the laws which we have introduced when this formalism was first presented, the laws of what is the unit 1 over 1, what is an inverse number, K over 1 versus 1 over K, all these were in this previous lecture when I introduced these rational strings. So I'm using whatever was defined over there as the laws of transformation. And basically that proves that we can reduce the rational string the same way as we used to think about reducing any rational fraction. So there is nothing new here. However, what is important is that we have established this based on a relatively rigid foundation. Okay, let's move on. Next is addition. So far, I have not talked about operation of addition with rational strings. Well, again, we have to define something, right? And operation of addition is not really defined at all among these rational strings. But I have to define it in such a way that while it would be a nice definition which doesn't break any laws and it corresponds to whatever we intuitively think about addition of rational numbers. You know, it's common denominator thing and et cetera. So that's what I'm trying to approach right now. Okay, let's start with the point that all integer numbers are well mapped into rational numbers using this method. Any integer number corresponds to a rational number of this type when they're right hand side, right integer number in this recreational string. The string is one. Now that means that if we want to build a nice theory which looks reasonable then m plus n should map into m plus n over one, right? So m maps to m over one, m maps to m over one and the reasonable definition would be that the sum will map into this particular rational number. So what we actually define right now is the law of addition of certain rational numbers, primarily these two numbers. So instead of corresponding, I will put an equal sign and this is the definition. So to add two numbers which have the same right denominator equal to one, I just have to add their nominators, left hand numbers and retain one on the right hand side. This is a definition. But now this is just the definition of numbers which have one as the denominator. Okay, how can we expand it to something else? Well, do you remember the very nice distributive law among the integer numbers? This is distributive law. Sum times sum number is the first of these two times this multiplier plus the second one. So this is called the distributive law among the integer numbers. Obviously, I would like to introduce my rational arithmetic in such a way that nice laws which we had in integer numbers will be preserved, like the sum for instance. So I will try to do the same here. If I have this type of equation, I will multiply both sides by some rational number and I should actually have this distributive law observed. What it means is the following. If I will multiply both sides by one over k, so it will be like this. m over one plus m over one times one over k should be equal if all the laws are correctly defined. Correctly defined. So I just took this equation and multiply by, sorry, I mean it should be over. By one over k. So if this is true by definition, this should be true if we want our distributive law to be correct, right? And I have to put some extra parenthesis around this, right? Okay, now distributive law says that I can multiply the first m over n by this and that will be what? It will be m over k. So m over one according to distributive law and multiplying the first out of these two numbers by the factor, by the multiplier plus the second one. n over one times one over k by the definition of multiplication is this. So the left part is this. What's the right part? Right part is this, right? m plus n times one, one times k, so it's m plus n. So this is a definition of the addition of two numbers which have the same denominator. So this was a definition for two numbers, an addition of two rational strings with one as the right hand side, the denominator. But using the distributive law, we basically expanded this definition into the definition of addition of any two numbers, any two rational strings with the same right hand denominator. You just add the left hand sides together and you retain k, k, k. You retain the denominator, the right hand side. So this is a definition, but definition, not just any definition, I can put m over k plus n over k equals, for instance, two m plus 10 n over k because that would not be reasonable than the distributive law would not hold. We would like everything to be nicely defined, so our laws of arithmetic which we used to use would be held. So that's the right definition. And now there is only one step to define addition of any two numbers. That's how we will do it. If you remember, we started with the law of reduction. So if we have common multiplier between left and right parts of this number, we can reduce it. What it means is if I have a over b, if I have to define a over b plus c over d, I can say that this is the same as common denominator, b times d. So by definition of, I mean by the fact that we have proved the reduction of this thing, I can say that instead of a b, I can put a times d, b times d because it can be reduced by the multiplier, common multiplier d. So that's the same as a over b. Same thing here. But here I will introduce the common multiplier b. So it will be b times c over b times d. So instead of this, I can write this because it can be reduced by b. But now, going behold, the right-hand sides are the same. So I can use this particular law where m is a times g, n is b times c, and k is b times g. So it will be a times g plus b times c over bd, which defines operation of addition among rational strings over b and silver d, which exactly corresponds to your intuitive knowledge about rational numbers and the law of addition by looking for some common denominator. In this case, I just chose b times g if b is one denominator and g another and b times g is the common denominator, obviously. And then multiplying correspondingly the left parts, a times g and b times c. Okay, so this is exactly corresponding to our not-so-rigid, if you wish, definition of the rational numbers. Well, but nevertheless, it's basically a more formal introduction of rational strings. And if people feel the necessity to put this more formal and firm foundation behind the number just a little bit beyond their intuitive meaning, that's the one. There are others as well. This is just one of the things. All right, so we have covered addition. Multiplication was done before when we introduced these rational strings. How about division and subtraction and zero, basically? That's the only thing that remains. Let's start with zero. Rational zero is any number of this type. Z is any integer number and this is zero. So any rational number with denominator not equal to zero, obviously, but numerator equal to zero plays the role of zero in rational numbers. How can I prove it? Now, what is zero? We add to any other rational number result in that other rational number. So let's just add a over b plus zero over z. It's equal to a common denominator bz. So it will be az a times z plus zero times b over bz. Equals zero times b is zero. So it's a times z over b times z. Z can be reduced by which we retain results in a over b. As is supposed to be if we are adding zero. So as I said, zero over z plays in these formal rational strings the role of zero. And again, obviously, we all know that in our intuitive rational understanding zero over any number in numerator and denominator not equal to zero is zero. So that basically kind of introduces this number zero and proves that the concept can be formalized relatively well. Okay, what's remaining? Subtraction and division. Okay, that's simple. Obviously, we define these two operations as reverse to their counterparts. So to subtract a over b minus c over b in theory what we have to receive is some kind of a number x over y in such a way that x over y plus c over d will result in a over b. So we have to define it in such a way this x over z that this equation holds. Now, from this equation obviously it's what? x times d over plus c times y over y times d common denominator y times d x I multiply by d and c I multiply by y, right? So we have to have this to be equal to a over d. Now, as you understand we would get exactly the same thing if I would do a over d plus minus c over d equals to x over y. Because, again, this is in addition so now we can multiply a times d minus c over minus c times b over bd. So we will basically have transformed this particular equation into the same one which we had before the same equation between x, d, c, y, and a and b which basically proves that to subtract two numbers we can instead have addition of another number where the numerator just changes the sign. Now, I probably would not do these calculations. They're kind of tedious and obvious but if you will just use the common denominator in this particular case you will get exactly the same type of equality for x and y and a, b, c, and d. So that's for subtraction. How about division? Again, very similarly we will substitute division with multiplication. So in this case if this is division what it means? It means basically that x over y times c over d should be equal to a over b. That's what divide actually means. So we have to define it in such a way that this is a correct equation which means now according to the multiplication it's x times c over y times d should be equal to a over b. Now, again if we will change our original equation instead and put a over b multiplied by d over c. I'm reversing. In addition I have to reverse the sign in multiplication I have to reverse numerator and denominator. Now, that actually should be equal to x, y. And again if you will do all these multiplications this and this would mean exactly the same thing. So that's the definition of division. Well, we have defined an object which is a rational string operations of multiplication and division with one over one being a unit for multiplication which contains the number as was. We have defined addition and subtraction and the number zero. We have defined an operation of reduction of rational strings. Well, that seems to basically complete the whole thing except one little detail which I would like to it's kind of a subtle detail but still important we know that numbers let's say three fifths and six tenths are the same because they can be reduced one to each other. Now, among strings we also talked about reduction and now what I believe makes sense is to slightly change our definition of the rational number. Rational number is not just any string of this type where a and b are any integer number. I would actually like to have something like this I multiply both a and b by the same factor with the same modifier. I would like this which is obviously equal to this one not to be another rational number which is equal to the first one but rather to be the same object. So I do not want in the formalism which we are trying to build I do not want to consider these to be too different they are different strings by the way but I don't want them to be different rational numbers I would say that these are the same so rational number is not just any string of this type but a set of strings with this property so a over b string over b and a times k over b times k these are actually representations of the same rational number rather than two different objects which are equal to each other again it is a little subtle but if I will ask you what is the result of a division one number over another by another ok this is that number but then I will tell you how about if I will multiply numerator and denominator by 2 well you can say this is the same number but it is represented differently right so that is why I don't want to say that the result of division a over b over by c over g is whole infinite set of different rational strings which are equal to each other I would like to say this is some rational number so rational number is a whole set of strings infinite set of strings which are reducible to each other using the laws of reduction yes I understand this is a little again subtle and it is more of a formal rather than something which will introduce any difference in this theory but still I would prefer this to be defined this way just to have a uniqueness of the rational number so these are actually the same rational number well that concludes our lecture about the arithmetic of the rational numbers thank you