 I'm Zor. Welcome to Unizor Education. I will talk about the problem number one in the category of rational numbers, especially the decimal representation of the rational numbers. The problem is about the fact that rational numbers are represented as decimal fractions as either finite or internet periodic fractions. Now, the problem is I would like to prove that some of two periodic decimal fractions is still the periodic one. I used that particular result in the previous lecture where I was trying to prove that any rational number can be represented as a periodic decimal fraction. So it's a very important, albeit simple, fact that some of two periodic decimal fractions is still a periodic decimal fraction. I was thinking about proving this basically a piece of paper for myself and then present a nice and smooth kind of a proof for general audience. But then I decided I will try to explain the solution to this problem, or proof if you wish, in the same way I was thinking myself. Because again, the purpose of these lectures is strictly educational, so I think it's very important for anybody who used these lectures to see not just the final result, but the way which we are using to approach that final result. So let's consider I never actually thought about this particular problem. I present it right now. I have to prove that some of two periodic decimal fractions is also periodic. And how can I approach this? Well, first of all, as any person probably would do, I would just check what happens if really I will add two periodic decimal numbers together and see what happens. All right, let's just, for an example, take one periodic number 0.030303, etc., which is 0.03 in period. And another one, which is, let's say, 57, 57, 57, etc., which is 0.57 in period. Well, if I will add them up together, well, that's easy, because as you understand, every period is basically self-contained and the result will be 60, 60, 60, etc., which is 0.60 in period. That's a simple case. Now, what's simple about this case? First of all, the lengths of the period is the same, and secondly, I don't overflow the period. Well, these are basically two main difficulties which we are facing when dealing with this particular problem. How to deal with periodic decimal fractions which have different lengths of the period and how to deal with overflow? Well, let's just consider them one by one. First of all, what if I have two different lengths of the periods? Let's say I have a fraction which has 0.142, 142, 142, which is 0, etc., which is 0.142 in period, and another function which is 75, 75, 75, etc., which is 0.75 in period. Well, this is actually a relatively simple thing because you see if this period is three decimal digits and this period is two, it's quite obvious that any multiple of these lengths will also be a period. Like in this particular case, this 2142 in a row is also a period because 1442 is repeated many times. Now, in this particular case, 375s in a row is also a period. So basically what I'm doing here in the general case is the following. If one periodic decimal number has n digits in the period and another n digits in the period, then obviously the first number can be interpreted as the one which has n times n digits in the period because if n is a period, then any multiple n is also a period. Now, this having n, I can also say that n times n number of digits represents the period for this number as well because again, if n is a period, length of the period, then any number multiple of m would also represent the length of the period. Like in this particular case, therefore, I can represent these numbers as these. 142 period is the same as 142-142 in a period. 75 in a period is the same as 75-75-75 in a period. And these two periods have the same lengths. That was the purpose. So I have reduced my more general problem of having different lengths of periods of two different numbers to the simpler problem where I have these two numbers having exactly the same lengths of the period. Okay, basically that's the general way of approaching any kind of a problem. We first simplify it as much as we can. The second problem which we dealt with was what if it overflows the period. Like in this particular case, obviously I am staying within the period if I will just summarize each period independently. Like in this particular case, it would be what? 7, 1, 7, 9, 9, 8. So the sum of these numbers is actually, again, a periodic decimal number and it will have this as a period. And the number of digits in the period is exactly the same. And obviously this piece is repeated infinite number of times and then this piece will be repeated infinite number of times because summarizing can be done completely independently within the lengths of each period. Alright, so now we have to consider the case when we do have an overflow of the periods. Alright, let's go again to a simple numerical example and then we'll see what happens. I have to wipe out these long six decimal digits periods and go to a kind of shorter one. So let's say I have 1.87 in the period and 0.75 in the period. So that overflows the period lengths because the sum will not be less than 99,000. Well, let's do it this way. Let's just write down exactly how each is as an infinite decimal fraction. So this will be 8.7, 8.7, 8.7, 8.7, 7.5, 7.5, 7.5, etc. Now, let's start somewhere and let's not think about this scale. What happens if I just summarize? 5 and 7 is 12, so it's 2 and 1 is there, 15, 16, and 1 goes to the next position. That's what's very important. I'm overflowing the period and 1 should be added to the next position. So now instead of 7, 5, 12, I will have 13. But then everything else will be exactly the same. So it will be also 6 and 1 here. Next will be exactly the same thing. 3, 6, and 1 here. And next will be 1. So as you see, 63 now becomes a period. If I start at somewhere much further, this also will be 63. So if we imagine that the whole thing is infinite towards right, then all will be 63s. So it seems to be the same story. The period length is basically repeated. And the only difference is that I might have some overflow to the very left digits. If it used to be 0, then it would be 1. Okay, that's actually true. I mean, if you will consider the most general case when you have any period here and any period there that the lengths will be the same, what happens in the worst case? In the worst case, you will have a number 1 transferred from one period to another one, which just increases the last digit in the period. So I can actually say that in the general case, the worst which can happen is that two different numbers, two different periods, and together, might actually increase by 1. And basically, that's the only thing. Is it a proof, actually, that everything really will be periodic with the same lengths of period, etc.? Well, you might consider this not exactly a proof because I was using the real numbers instead of some kind of a generalized, algebraic, symbolical notation. But, you know, let's try to do it in a more general case and I will probably get exactly the same result. So the result would be that 1 should be transferred to the next position. Well, how can we do it? Well, let's just consider it this way. If you have number, let's say 0.p in a period and you have another number, which is 0.q in a period. p and q having the same lengths. Like in this case, p is age 7 and q is 75. Now, what actually happens in this case? Let's consider, for definitiveness, that the lengths of the period is n digits. What does it mean? It means that p contains, p is an integer number, like age 7, which has n digits. In this case, it's 2. q also is a different integer number, which is 75 in this case, and it also has 2 digits. What it means is that this is a representation, if you remember from the lectures, which preceded that thing, of a rational number which can be obtained from summing up geometrical progression. Remember, if you have something like this, sum of geometrical progression, which starts with a and multipliers q, then it's actually equal to a divided by 1 minus q. I do remember this formula, that if you don't know the formula, it's very easy to derive it by multiplying this by q and subtracting the results. Alright, so in this particular case, what this is, is the following. Let's just wipe out this, we'll leave this example. So we can say that this particular number is p times 10 to the minus n divided by 1 minus 10 to the minus n. Why did they write it down? Well, first of all, you can always return back to the lecture, which preceded this particular problem, where I derive actually this particular expression. But in any case, you understand that this is exactly what it is. Because what this represents, it represents the first member of this geometrical sequence. Let's go back to the numbers. It's 0.87. The next one will be 0.0087. The next one will be 000087. And sum of all these up to infinity will give this particular number. So the beginning is 0.87, because in this case, it's 87 times 10 to the minus 2, the power of minus 2. And every next one is 100s of the previous. So 10 to the minus 2 is actually the Q in this particular formula. And obviously, this one can be written as this, which has exactly the same period length. That's why we can write it this way. Now, if I will summarize them together, what I will have is obviously p plus Q times 10 to the minus n divided by 1 minus 10 to the n. It would be great if p plus Q is less than 10 to the nth degree, like 87 and 75, unfortunately, is greater than 100. So I can't really say that this is exactly the representation of this type of, the sum of this type of geometrical progression. So if p is this, and Q is 0.75 plus 0075, etc. So 75 and 87 exceeds 100. So I can't really say that I can summarize it. That's exactly the overflow which we're talking about. However, what I can always say that if p plus Q is greater than 10 to the nth degree, in this case, 10 square, which is 100, I can always say that this is the same as p plus Q minus 10 to the nth degree times 10 to the minus n divided by 1 minus 10 to the minus n. Plus 10 to the nth degree times 10 to the minus n divided by 1 minus 10 to the minus n. So I subtracted 10 to the nth and added 10 to the nth. Now, if 87 plus 75 is greater than 100, like in this particular case where n is equal to, I can always subtract that 100. So what will be, it's 15162, right? So it will be 62. So this actually represents a nice periodic number, which is 0.62 in this particular case, right? Because p is 87, Q is 75 minus 100. So it's 162 minus 100, so it's 62. So this represents a nice periodic decimal number. And what is this? As you see, this is exactly 1 over 1 minus 10 to the minus n degree. That's what it is, right? Because this actually is reducible. Now, what is this particular expression? Okay, let's do this. It's just quite obvious that 1 over 1 minus 10 to the minus n represents 1 over 10 to the minus n plus 1 over 10 to the minus 2n, plus, etc., etc. It's also a somewhat geometrical progression with the first member 1 over 10 to the minus n and the every other member factors or multiplied by 10 to the minus 2nd degree. Now, so what we have right now is 1 periodic number, which is 0.62 in this particular example, and another periodic number, which is 0.010101, etc., etc. for our example when n is equal to 2, right? This is a period. This is a period, this is a period, this is a period. So what happens if I add periodic number 62 with a periodic number 01? Again, in a general case, what happens if I add some kind of a periodic number which has n digits in the period r? The number which is 0.001 where the number of digits is exactly n. In our case, it was 0.62 in period and this one was 0.01 in period. So the lengths of these periods is exactly the same and all I'm doing is I'm adding 1 to the very last digit. What have I just done? I have reduced, again, a more complicated problem. What happens if there is an overflow when I'm adding two periodic numbers? I have reduced this to a slightly simpler problem. What happens if I add two periodic numbers where the second one is just 1 in the very last position of the period? So what happens is very easy to understand in this particular case because all I'm adding this right now is just 1 to the very last digit, which doesn't cause any overflow under any circumstances, no matter what r actually is, except 1. What if r is 9, 9, etc. 9? End positions. Then if I add 0.001 also end positions, then I will have an overflow. So that's the only difficult point which remains in this particular problem because if it's 8, for instance, and I add 1, I will still be within the period. I don't have an overflow. Okay, fine. So all these more difficult cases were reduced to only 1 when I'm adding 0.001 end positions to a number which is a decimal fraction which has 9 repeated end times as a period. Okay, the whole big deal of different rational numbers represented in different ways and with different periods with overflow, without overflow is basically reduced to one small problem. What happens if I add this to this? And this is relatively trivial case because of the following. These numbers which have 9 as certain number of 9s actually in the period they are really very special kind of numbers and here is why. It's actually the same thing to write 0.9 in the period and 1.0. These are exactly the same numbers. Why? Well, first of all, we can use the formula for a sum of geometrical progression in which case, in this case, it will be 0.9 as the first member of this sequence and it's 1.10 as a multiplier. Right? Which basically means 1. Because this is 0.9 and this is 9.10. So, using the formula, this thing actually means exactly 1. Now, just using the common sense it's also kind of obvious because look at these numbers 0.9, 0.99, 0.999, etc. Each one of them gets closer and closer to the number 1. This by 1.10, this by 1.100, this by 1.000. But if you have an infinite number of minds it's infinitely close to 1 and infinitely close to 1 means equal to 1. So that's why these numbers are really special and they introduce certain level of how should I say I don't like the fact that the number 1 can be represented in two different ways using decimal fractions. But, well, I guess I don't have a choice to allow these numbers or alternatively I can say that any number any decimal fraction which has 9 as a period should actually be prohibited completely from the usage and replaced with a different number where all these 9s are completely cut off and the digit which is preceding these 9s is increased by 1. Like in this case I can cut off the tail all 9s and increase 0 by 1. If it's a different number let's say 0.57 and then all 9s then I can say that this is really 58. So by basically changing this type this is actually 1. 1.0 and then if I add it to this number I will have 1 and then this type of thing as a period if I add these two together which is a periodic decimal number which we actually wanted to prove in all different cases. So, again a smooth presentation of somebody who rehearsed this type of thing in front of the audience I was trying to present it in exactly the same way as I was thinking about this problem myself and I think it's very important so again you consider your problem in certain particular cases then you understand the deficiency when it's not really complete how can you generalize it what kind of more general cases you should consider and finally you exhaust all these cases one by one so the proof is relatively complete. Well, hope you enjoyed it. Thank you very much. That's it for problem 1.