 I'm Zor. Welcome to Unizor Education. Today's topic will be a continuation of rational numbers, but primarily their representation as decimal fractions. Well, first of all, what do we actually mean when we are saying that something is a decimal fractions? Well, everybody kind of has an intuitive understanding that 2.51 is some kind of a number which is expressed as a decimal number with fractions. Let's be a little bit more concrete with our definition. Well, first of all, we see that any decimal fraction, decimal number with fractions, contains certain number of decimal digits from zero to nine. A delimiter, if you wish, which separates integer part from the fractional part. Usually, it's a dot in the United States and in some other countries. In Europe, in many countries, it's a comma. Well, whatever it is, let's just call it a separator that will be a username. Then there is an optional sign, which you can put in front of it. But again, considering that this is, if there is no sign, then the plus sign is assumed. We can always say that in the beginning, we always have some kind of sign, plus or minus. Now, what's on the left is pretty much defined. What's on the right of the separator, which separates integer from the fractional part, is much more interesting. And here is why. We are talking about rational numbers. And if you want to represent a rational number 1, 7, or in my more formal description, representation rather, it's a combination of two integers with a bar in between. If you'll try to express this particular rational number as a decimal fraction, you will actually see something like this. 1, 4, 2, 8, 5, 7, and then we'll repeat 1, 4, 2, 8, 5, 7, et cetera, et cetera. So a certain number of digits is repeated infinite number of times. So first of all, what we can say about decimal numbers is that to the right, we can have an infinite number of digits. To the right of the separator, which separates integer from a fractional part. So infinite number of decimal digits, that's very, very important. And here is why. We will use these decimal fractions not only to represent rational numbers, but also non-rational, irrational actually, numbers, which I actually have another lecture which is dedicated to irrational numbers. Irrational and irrational altogether are called real numbers. So basically, representation using decimal fractions is, it allows you to represent more than just rational numbers. It also allows to represent non-rational ones. So all rational are basically expressed in this particular form or in this. Irrational are not expressed in this particular form, but what's interesting is that both of them are represented as decimal fractions. But here is an important difference, and that's what I'm going to actually dedicate this lecture and the next one too. If we are dealing with rational numbers, then even if this sequence to the right of the separator is infinite, you can always find a period which will repeat many times, infinite number of times actually, which is certain number of decimal digits, which is repeated one time right after another. Now, I will spend some time in this lecture precisely for these type of cases. Okay, so first of all, we have defined what is a decimal representation using decimal fractions. So secondly, what we want to make sure that we understand that if this number of digits to the right of the separator is finite, then it's definitely a rational number. And here is why. When we are dealing with this representation of the number, what's important is that every digit based on its place has certain weight. So digits to the left of separator have weight 1, 10, 100, etc. Every position multiplies by 10. Digits to the right of the separator have weight 110, 100s, 1000s, etc. Now, this does not go to infinity. This is a finite number of digits because we are not dealing with infinite numbers which do not have a specific value. But to the right, we can have an infinite sequence. And let me give you an example. Let's say we are dealing with 0.1111, etc., etc. What does it actually mean using this decimal representation and using these weights? It's basically, well, integer part is 0, so let's just talk about the fractional part. It's 1 with the weight 110s plus another one with the weight 100s plus another one with the weight 1000s, etc., up to infinity. That's what it means. Well, obviously, we can draw these once. So it's a sum of certain numbers. Each one of them is exactly 10 times smaller than the previous one. Now, here is some kind of a generalization of this. If we want to deal with these sequences which contain numbers which are decreasing by certain ratio, in this case by ratio of 10, we can actually summarize to get the value of this sum even if the number of these elements is infinite. And here is how. Let's consider a more general problem. The problem is if you have to summarize these numbers, q, q square, q, q, etc., etc., and let's assume that the q is less than 1. Like in this particular case, q is 1 tenth, because every time next number is exactly multiple of the previous times q. So we start with 1 tenth times 1 tenth, 100, 100s times 1 tenths, 1000s, etc. Now, let's assume this is a finite sequence. So the end is q to the nth degree. Now, I can give you exactly the formula for what this particular sum is, and I can ask you to prove it as exercise for mathematical induction. And for those who are interested, there is another lecture on mathematical induction which Unisor has. Now, I will do it slightly differently. Just a little trick if you wish. Let's do it this way. We will multiply this sum by q. We will have s times q equals... Now, since q multiplied by a sum, you can break it down using the distributive law of multiplication over the summation. You can multiply each element by this q. So you will have q square, q cube, q to the fourth, etc., q to the n. And the last number will be n plus 1. q n times q will be n plus 1. Right? So we multiply the left and right part of this equation by q and that's what we get. Now, if I was subtract from one element, what will be? s minus sq will be s times 1 minus q. Right? Now, but here what's interesting is this and this, this and this, this and this. They will nullify each other. So with left, we have q minus q to the n plus 1. Now, again, what's important is that q is less than 1. As n tends to infinity, this particular element, since q is less than 1, and we are multiplying by itself, so the number is multiplied by something which is less than 1. By the way, when I'm talking about less than 1, I still mean positive. So it's from 0 to 1. Let's say it's one half. Something is multiplied by one half, which means it's getting smaller and smaller and obviously tends to 0. So this particular number tends to 0. So what's left? For infinite sum, we have only this part. So let me put it here. s times 1 minus q equals q and s is equal to q divided by 1 minus q. So for an infinite number of elements in this sequence, like in this particular case, this is the result of summation. Now, using this, let's apply it to this particular case, one tenth. Well, if q is one tenth, then we will have what? We will have s is equal to one tenth over 1 minus one tenth, which is nine tenths, which is one ninth. Well, obviously, for everybody, 0.111111 is exactly one over nine, because if you multiply it by nine, you will have 0.999, infinite number of nines, which means that the whole number is infinitely close to one. So if you have really going to the limit, you will get basically one. So that's why the representation of the number one as 1.0 or 0.99, etc., infinite number of times, these are actually equivalent, but this is a decimal fraction, which has a period, and nine is actually a period. This is another representation. So we have certain duality, I would say, of representation of certain numbers. They can be represented one way or another in decimal. Yes, obviously, this one is preferable, and that's how people do. However, this one still has its own right to existence. Now, there are certain numbers which you cannot represent as finite decimal number. For instance, I was telling you about one seventh. Now, we were talking about infinite fractions and fractions which have period, but I would like to prove right now that any periodic decimal fraction or decimal number with fractional part, which has a period, is always rational, which means it's always represented as basically some kind of a one over something else to integers. Okay, how can we prove that easily? Let's consider a general case of a decimal number with a fractional part, which has a period. Probably an example would be a better case, and here is why. For instance, you have 0.2 and then the period. I will use the same sequence of 1, 4, 2, 8, 5, 7, because I remember as well. Maybe I will tell you why sometimes. But in any case, this is a typical representation of a decimal number, which after a certain point has a period. And by the way, I have actually calculated it beforehand. I put a little plan for myself here. This is 15 over 70 in my formal representation. Or if you wish, it's 1570s, which is the same, by the way. You can reduce it by 5. It will be 3 over 14. So for those who are interested, you can start and divide 3 over 14, and you will get exactly this. First, you will have 0 of this with then 2, and then 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, etc. They will repeat up to infinity. So anyway, this is an example of a periodic decimal number. And before proving that this is a rational number, I would like to simplify my job a little bit. First of all, is it important that there is this 0.2 in the beginning? Well, actually not. Because obviously it is 0.2 plus 0.0 and then the period. Now, any finite decimal number is definitely rational. Because as I was saying, it's represented a certain number of digits with a certain number of weights. Every digit is like 0, for instance, has a weight 1 plus 2 has a weight 1 tenth, right? Because this is on the left of the decimal separator, and this is on the right from the separator. Then whatever else doesn't really exist, but weights are 100s, 1000s, and in this case, weights are 10 and 100, etc. So, since each member is rational, and as you know, rational numbers are completely closed as far as their summation and multiplication are, obviously the whole thing is rational. So every finite decimal number is rational, so this is rational. So to prove that this one is rational on the top, I really have to prove that this one is rational, right? Because the sum will be rational. Now, again, to prove that this is rational, is this 0 important here? Well, actually no, because as we know, every position has a weight. This is one tenth, this is one hundredth, etc. What if I will consider 0.142857? Now, in this case, one used to have weight one hundredths, now it has weight one tenths, 10 times more. And every digit used to be 1000, now it's 100. So the weights, when I'm shifting the whole number to the left by one position, all weights are actually multiplied by 10, which means that if this is rational, then this will be rational as well, since, again, rational multiplied or divided by 10 is rational. So, again, to prove that this one is rational, it's quite sufficient to prove that this is rational. Why am I talking about this? I would like to reduce my problem of a general periodic decimal number to be reduced to the one which has zero point and then the period, basically, which is a certain number of digits which repeated infinite number of times. So, generally speaking, I can actually tell that let's consider the number which looks like this, p1, p2, etc., pn. And parenthesis means that this is repeated infinite number of times. p1, p2, pn are some digits and each digit actually has a weight. This one has weight one over 10, this has one over 100, and this one over 10 to the nth degree, right? Because we can n digits and each position shifted to the right is, multiplies the weight by one tenth. So, this is a representation of an infinite number, decimal fraction, which has a period of length n, n digits, and each digit in the period is p1, p2, etc., pn. This is a general concept of as general as it can be a periodic number. And if we prove that this one is rational, then obviously all other numbers, something like, I don't know, d1, d2, decimal separator, d3, d4, and then the period, p1, etc., pn. This is more general and any sign plus or minus. Obviously, this also will be rational because we have already proved that this piece is general. So, all these numbers are different from this by basically aging something or multiplying by something. And that does not change the ratio analogy. So, the most difficult part of the whole proof that any periodic decimal fraction is rational is to prove that this is rational, this type of decimal fractions is rational. Well, actually, this one is also not very difficult. Let me go back to this progression, some of the progression which we already considered before. Now, actually, I will use a different letter here. That will be better because n is already taken as the number of digits in this thing. Well, here is what's important. Number which is written in this particular form where the sequence p1, pn is repeated infinite number of times can be represented not only this way using the following. p1 times 10, 110 plus p2 times 100 plus etc., plus pn times 1 over 10 to the nth degree. Plus, this is a periodic fraction, right? So, I am continuing. Plus, p1 again, because that will be again, p1 10 to the n plus 1, right? After 10 to the nth, next weight is 10 to the n plus 1, plus p2 times 1 over 10 to the n plus 2, etc., plus pn times 1 over 10 to the 2n, n plus n. But this is not the end of it, right? It's an infinite. So, again, there will be another one and another one and another one. Every time weights will be, you see, this is 10 to the first degree, this is 10 to the n plus first degree, then it will be 10 to the 2n plus nth degree, p1 over 10 to the 2n plus 1, plus p2 over 10 to the 2n plus 2, etc., plus pn times 10 to the 3n, etc., up to infinity. This is actually what this means. Now, how can we summarize this? Well, it's actually the same thing as this, because if you will consider this piece, which is obviously a rational number, which, by the way, is less than 1, because p1 is definitely, it's a digit from 0 to 9. So, maximum what it can be, p1 over 10 will be maximum 9 tenths, right? And this will be maximum 9 hundredths and this will be maximum 9 over 10 to the nth degree. So, the whole thing, as a maximum when all digits, finite number of digits and digits are equal to 9, the maximum number can be 0.99, etc., 9 times, which is less than 1. So, basically, what I'm saying is that this number is our Q. Well, let's not call it a Q. Let's call it, let's use a different number, let's use a different type. Let's call it A. Okay. If this number is A, now, what is this number? Well, look at this. This is p1 times 1 over 10 to the ferris degree and this is p1 over 10 to the n plus first degree. This is 100, which is 10 square, this is 10 to the n plus 2. So, every member of this sequence is different by every member of this sequence by 10 to the nth degree. So, we basically have to multiply this A times 10 to the nth degree. That's what my point is. So, if this is A, then this is A times 1 over 10 to the nth degree because each member, as you see, is multiplied by 1 over 10 to the nth. How about this one? Same thing. We multiply again by 10 to the nth degree. So, it's A times 10 to the 2m. That's what this number is, et cetera, up to infinity. So, if I will want to represent it in this particular way, I will represent it as the following. s is equal to A plus A times q times A times q square because that's what 10 to the 2n degree is plus A to the q q up to infinity, where A is this expression, finite, mind you, and q is, this is q, 1 over 10 to the nth degree. Now, having said this, I can definitely say that this is A times 1 plus q plus, et cetera, up to infinity. And this is just another particular, let me just put a couple of more members here, q square, et cetera. So, where A is 0.P1, P2, et cetera, Pn, finite, I'm not putting this as a periodic fraction, and q is 1 over 10 to the nth degree. So, that's what my number, this actually means. The first period is this, A, which is A. The second period, P1 to Pn, is represented by this, which is A times q, because it's shifted to the right by n positions, which means it's multiplied by 1 over 10 to the nth degree. The third one will be q square, et cetera, et cetera. So, how can we summarize this? Well, using exactly the same principle as before. The only difference is this multiplier A, which is a finite, rational number. So, it's very simple. So, you have S over A is equal to 1 plus q plus, et cetera. Then you multiply it by q. So, S times q divided by A is q plus q square plus, et cetera. Now, for finite number, it will be q to the k, and this will be q to the k plus 1, and then we will say that the k tends to infinity. Anyway, right now, let's subtract one from another, and what we will have, we will have S over A times 1 minus q equals 1 minus q to the k plus 1, S over A, S over A times q. So, that's the left part, and on the right, this one will be reduced. So, only the first and the last will actually remain. This is the plus sign, this is the minus sign. Now, S k tends to infinity for a q like we have here. Obviously, if you multiply q by itself, infinite number of times, it tends to 0. So, the ultimate formula is S is equal to A over 1 minus q. That's what it is. Where A is this, and q is this. Now, and this is obviously a rational number, by the way, because it's the result of some elementary operation over rational numbers which we had before. So, what we wanted to prove is that any periodic fraction actually represents a rational number. Now, two things I would like to make. Number one, let's make a quick check that the number which I liked so much, 0.142857, period, is indeed 1.7. Well, let's use this formula. A is 0.142857. No parentheses. It's a finite number. q is 1 over 10, how many digits? Six digits. So, it's 1 over 1 minus 1 over 10 to the sixth degree. Right? Okay, let's calculate it very quickly. But 10 to the sixth is a million, right? So, it will have 0.142857 divided by 10 times 1 million minus 1. It's 9, 9, 9, 9, 9, 9, 9. 10 to the sixth goes to the top. This is exactly number of positions which we have here. So, this is 10 to the sixth makes this fraction an integer number, right? We shift the decimal separator, sixth position to the right. So, it's 1, 4, 2, 8, 5, 7 divided by 9, 9, 9, 9, 9, 9, 9. Well, I'm not going to do it, but I believe anybody can divide 9, 9, 9, 9 by 1, 4, 2, 8, 5, 7 and get 7. So, this is equal to 1.7. So, this periodic fraction 1, 4, 2, 8, 5, 7, in period is exactly 1.7. And this is just an example of a periodic decimal fraction which represents a rational number. So, one thing is that we have proved that any finite or infinite periodic decimal number with fractional pot, obviously, represents a rational number. Next lecture, I will prove the opposite that any rational number of this type of any other rational number can be represented as a finite or infinite periodic. The word periodic is extremely important, decimal number. Now, and here we have basically a different approach to rational numbers. We can define them as something like this or more formal representation as I showed before with the two integers separated by bar. Or we can say that rational numbers are represented in the way of the decimal fractions with certain weights assigned to each position. But all rational numbers are either finitely represented by the decimals or if it's an infinite representation with a fractional pot going to infinity, it must be periodic. That's what's very important. So, periodic decimal is exactly the same as rational and vice-versa. Rational is the same as periodic. So, next lecture I will spend for proving that opposite pot that any rational number is actually represented by periodic decimal. Actually, I would like to make one more point here and this is more philosophical. I can definitely just tell you the fact that rational numbers can be represented by decimals with certain period. And all irrational numbers are periodic, non-periodic decimal fractions. This is a fact. Now, why did I bother proving that thing? Obviously, somebody has already proven it. Well, to tell you the truth, I think that the fact by itself doesn't mean much. In your life, you will most likely never use it at all or very rarely. What is important is to follow the proof. And just the fact that you're logically trying to prove something, you're trying to not only say that, okay, I live in a place A and I know that there is a place B where there are certain things which I can find if I will go there. What's important is really to go that way, really to reach that point B, to find the road which goes from A to B, because these are actually educational purposes of all these lectures. Not just presenting certain facts, but explaining certain things which will lead you to thinking about how to achieve something, how to prove something. And that's why I will spend a lot of lectures or explanations for just different problems, because solving problems is very important. Now, proof is also a problem solution, if you wish. The problem is the representation of the rational numbers. And we have proven that this can be represented in such and such a way. So, for those people who are looking for facts, you will definitely get it from here, but you most likely will be bored with the proof. Actually, I think much more beneficial is really to follow the proof, regardless of what exactly we are trying to prove. It's the logic behind it. It's the steps which we are making. And that's what actually achieves this educational purpose. And that's the goal of all these lectures. I made this little philosophical essay just because it's one of the first actual proofs, if you can call it a proof, which I was trying to present in my lectures. And again, I understand that for certain people it might be a little bit boring. Just don't forget, the purpose is not the fact which I'm proving. The purpose is the logic of the proof itself. That's where educational goal is. Okay, thank you. And again, next lecture, I will spend proving, by the way, proof again, that all rational numbers are represented as a periodic decimal. This is the converse theorem. This one is the any decimal fraction which is periodic is rational. Next one will be the other way around. Thank you very much. That's it for today.