 My name is Zor. Welcome to Unizor Education. Today's lecture will be about rational numbers. In the previous lecture I was talking about integer numbers and how they evolved from natural numbers and operation of addition. We had to make this complete. We had to make this set a group, a mathematical group, which means we have to reverse our operation at will, and that makes it actually a beautiful, harmonious, symmetrical. Now let's move just one step forward. So let's assume we have our integer numbers and we have an operation of multiplication. Well, exactly as in the case of natural numbers and addition as not being fully sufficient to make the whole theory complete. What I'm telling right now is the operation of multiplication as far as integer numbers is not complete. Excuse me. Because you cannot inverse it always and you cannot, inverse operation cannot be applied to any integer numbers as we have. Well, you can, for instance, multiply 8 by 3 to get 24, and in this case you can inverse operation, which means you can divide basically 24 by 3, getting 8. But can you divide, let's say, 5 by 7? No, you cannot. So the reverse operation is not always possible, and that's what actually makes people mad. Mathematicians don't like to not be able to do something. So how can they solve the problem like this? Exactly as before. As far as addition was concerned, mathematicians have invented negative numbers and zero. As far as multiplication is concerned, mathematicians had exactly the same thing to do. They have invented new numbers called rational numbers. As far as notation is concerned, well, we know that among integer numbers we cannot divide 3 by 5. So they had, fine, okay, they invented a brand new number which is called 3 over 5. Purely abstract. Maybe it doesn't even occur in nature. Doesn't really matter. But what we can say is that among all rational numbers which can be represented as a pair of integer numbers with some divider in between. I mean, I can represent it this way or I can represent it in, I don't know. This way doesn't matter how I represent the object using some numerical notation. The matter is that this becomes a new element of a new set, set of rational numbers. So forget about practicality. If you can find some natural numbers like counting sheep or something like this, you always have a problem finding negative numbers. And obviously you will find some problem finding in nature, in the real life, some rational numbers. So you really have to go far to justify their existence formed from the practicality. You can, but it's not really easy. It's not what counts. But in mathematics it's very simple. You just invent a brand new abstract object and now we can research the properties of this thing. Again, the point of this lecture actually was to explain that certain theories are developed just based on its own theoretical necessities and maybe find much later some practical application. Correct. So in this case we have introduced brand new numbers which together with operation of multiplication make the whole set of numbers a group. Remember, we were using groups as an example of a mathematically symmetrical and harmonious object. Groups have unit element, if you remember, which being applied to any element leaves it alone basically as it was before. In multiplication we have number one, which plays this role of unit element. In group theory we always have a reverse operator which being applied to any number. If you apply the operation with the element of the group and it's inverse you will get this unit element. Well among rational numbers obviously if you have one rational number you can always have another rational number which being operated upon will give you the unit. Well there are some exceptions actually when designing it was equal to zero. Let's just not talk about it. This is a little difficulty in our theory. And by the way this little difficulty also by itself caused lots of different branches from the theory which we have kind of used to. So this thing makes rational numbers a nice group and well basically that was the subject of this particular lecture. We are introducing new numbers, we call them rational. We have notation how to write them and the necessity was to make the multiplication complete. Now just as a kind of interesting example of how the whole theory was developed a little bit further, let's consider the notation. As you know we represent our numbers in decimal system. So sometimes instead of using p over q as a representation of rational number where p and q are integers sometimes we are using decimal system as a decimal point. So let's say you have this. This is also a rational number but is it general enough representation to write down all possible rational numbers which exist in our theory? Well not quite, it's not really very easy. Let's start from a very simple example. Let's say you have one-third. This is a rational number, no doubts about that. But now let's talk about its representation in our decimal system. Well obviously this is 3, 3, 3, 3 etc. It's an infinite number of threes. Well it's not really so nice to have an infinite number of digits to represent the number. Yes I understand people can cut it off and say well approximately it's one-third but it's not one-third right? It's actually 3, 3, 3, 3, 3 over 10 to some degree. So there is a concept which mathematicians came up with which actually allows to make this notation more finite. Here is a very interesting story about rational numbers. Whenever you take two numbers to integer numbers which form a rational number and then convert it into decimal you will always have certain digits repeated in this particular sequence. Like in this case 3 is repeated. Sometimes you can have number like this 1, 2, 3, 1, 2, 3, 1, 2, 2, 3 etc. So 1, 2, 3 is a repeated group. So as far as notation is concerned, mathematicians came up with a little bit more well convenient if you wish notation using the decimal system. They can have something like this which basically means that 1, 2, 3 repeats infinite number of times. Well it's a notation right? I mean it's a written representation of some abstract concept. Whatever this rational number is, it means that 1, 2, 3 is actually repeated many, many times and together they form a rational number. But here's a very interesting story. You see, these are periodically repeated sequences of 1, 2, 3. And it turns out, and it's a very interesting theorem actually and not very easy one, that any rational number can be represented in decimal system as some kind of notation with some periodically repeated numbers, sequences of numbers. It's, as I was saying, it's not really a very easy theorem. It's still provable and that's what actually differentiates rational numbers from irrational numbers which will appear in one of the, will be introduced in one of the next lectures. So basically the periodic decimal fractions are representation of all the rational numbers and that's very important. Let me just give you a very interesting, funny in a way, example of properties of this type of representation, periodic representation of rational numbers. If you want to represent one seventh, it will be 1, 4, 2, 8, 5, 7 in period. Well, if you take two sevenths, it will be 2, 8, 5, 7, 1, 4 in period. Well, notice that it's exactly the same digits here and you have to start 2, 8, 5, 7, 1, 4. It's like a cyclical representation of the same thing. 3 sevenths will be, you start from 4 I think. Is that right? Or from 5. 4, 2, 8, 5, 7, 1. Yeah, that's what we'll do. So again, it's the same numbers. You can actually put it in a circle if you wish. 1, 4, 2, 8, 5, 7. And you always go into this direction. So 1 sevenths is 1, 4, 2, 8, 5, 7 in period. 2 sevenths is 2, 8, 5, 7, 1, 4 in period. 3 sevenths is 4, 2, 8, 5, 7, 1 in periods, etc. So all these 7s up to 6 sevenths are represented using the same numbers in the same cyclical sequence. Well, try to prove that this is really true. You can check it out as the first, but here's an interesting theory. If the number of digits in the periodicity, in the periodic representation, in this case 6, is 1 less than the denominator, then you will have this property. Well, I don't know. It's kind of a strange thing that that's the property of the periodic representation of rational numbers. It would be really very interesting if you think about how to prove this theorem. And maybe someday we will organize a kind of lecture about this particular problem. Until then, well, good luck with this particular thing and see you next lecture.