 Hello friends welcome to this session on number system and in the last session I promised you that I will be telling you a trick on how to find out whether a particular fractions Decimal representation will be terminating or not. So here is the trick the trick is and it's very easy to see also You can do a little bit of mathematical manipulation and you will be able to Derive it yourself. So the thing is fractions with denominators of the form 2 to the power m and 5 to the power n will always have terminating decimal representation and What are m and n? m and n are non-negative integers. What are non-negative integers folks? Non-negative means something which is not negative. So something which is not negative is 0 1 2 and Any number in any positive number, right? So let us understand this first. So Let us take an example of 2 to kind of you know explain this. So, let us say we have 21 upon 200 Okay, 21 upon 200. Okay, so let us try to find out It's decimal representation. So it's decimal representation if you see If you divide 21 by 2 is 10.5. So it is nothing but 0.1 1 1 5 5 I'm sorry my bad So the representation is 21 by it is 105. Sorry. So 105 you divide 21 by 2 it is 10.5 and there are two zeroes. So it will be simply 100 10.5 by 100. So which is equal to 0 point 1 0 5 I Hope, you know, I'm not making a mistake. Otherwise, you know how to divide 21 by 200 and find the so in any case You will see that, you know, the this is what is observation. This is terminating Terminating decimal representation, isn't it? Terminating decimal, isn't it? Now Let's take another example. So another example could be let's say 7 upon 25 Okay, so 7 upon 25, let us do the long division 25 and then 7 so 0 Then you take the symbol and then from 70 so 25 2 times is 50 Is it it? So remainder is 20 and then again 1 0 comes so hence 28 Okay, so 25 times 8 is 200 So hence this goes and hence I get what 7 upon 25 is 0.28 Right, let us take one more a simpler one. Maybe 3 by let us say 3 upon 80 Okay, so what is it? So let us first find out 8 by 3 by 8. So hence, let me do this. So or you can take it directly 80 So this is 3 so you put 0 point and then becomes 0, but it's not sufficient again put a 0 So it becomes 300 now 83 times 3 times if you do you'll get 240. Okay, so what is the remainder here? Reminder is 60 put another 0. So what next right 8 7s of 56 so 80 80 times 7 is 560, okay, so what is left? 40 right put a 0 now 5 times 400 Wow again, what do I see? I see that 3 by 80 again We got a terminating terminating decimal representation So what is the observation if you look closely? 200 can be written as 2 into 100 or 4 in our 8 into 25 isn't it? So it is simply 2 to the power 3 into 5 squared Okay, what is here 25 so 25 if you see denominated 25 can be written as 2 to the power 0 which is 1 to the time times 5 squared All right, similarly 80 is 16 times 5 which is nothing but 2 to the power 4 into 5 to the power 1 so all these Denominators if you see can be reduced to what form 2 to the power m and 2 n 5 to the power n So if you see here, it is m is equal to 3 and n is equal to 2 in this case M is equal to 0 and n equals to 2 and here it is m equals to 4 and n equals 1 guys Okay, so this is a very important Observation right and why is it happening? Let us now see why it is happening or for that matter before We go to the proof part of it. Let us take some other example where only 2 and 5 are not there So for example, let us say we do 7 upon 15 Okay, what is 7 upon 15 so 7 upon 15 if you see 7 upon 3 into 5 right? It is not of the form of so 15 is not equal to 2 to the power m and 5 to the power n any mn for any mn it is not True, so you can say sir 2 to the power 0 into 5, but then you also have a 3 which is extra So hence we don't require 3 hence This is not satisfying our criteria. So let us see whether we really get our terminating decimal or not so 15 and This is 7 so 0 and then a decimal gives you 70 Then 15 times 4 is 60 So this is 10 then you another put another 0 so 15 times 6 is 90 and then you put you get 10 you get another 0 so 6 again So 90 and then you now know it is going to repeat isn't it? So hence I can see it is non-terminating so 7 upon 15 is 0.46666 and so on and so forth Which is equal to 0.46 bar in our methodology, isn't it so clearly it satisfies the rule which we just you know we were discussing so far now Let us see how and why it is happening Now, let us say we have a fraction p upon q Okay, general fraction where q where q is equal to 2 to the power m and 5 to the power n Okay, now, let us take case one case one Case wise why because let us say m is equal to n because there could be three possibilities what all? m could be more than n m could be equal to n or M could be less than n for any two integers m and n, isn't it? So let us first take case where m is equal to n in that case. What will this be? Q will be 2 to the power m into 5 to the power m now Don't get confused when I use this point this point represents multiplication as well Okay, so now if you know the rules of indices, it is nothing but 2 times 5 It's not in 2. So let me write cross only so 2 into 5 to the power m Which is nothing but 10 to the power m right now guys if you divide any number p by 10 to the power m That means one followed by some zeros. You know that it will be a Terminating decimal only you can just count the number of zeros from the left-hand left-hand side of p You can just count the number of digits and put a decimal for example if 23 by 2 to the power 3 into 5 to the power 3 is there. So what is this basically nothing but 23 upon 2 into 5 to the power 3 right which is 23 upon thousand Which is nothing but 0.0 to 3 which is a Terminating decimal. Okay. So case one was well understood now Let us take case 2 case 2 is m is greater than n Okay, so if m is greater than n guys So I can find a small r such that m is equal to m is equal to n plus r We can say that right now if m is greater than n there can be a Small r which when added to n will give you m. Isn't it now? Let us say p upon q is nothing but p upon 2 to the power m and Q sorry 5 to the power 5 to the power n So what does this mean? For example, let us say if m was 5 and n is 3 so you can get r is equal to 2 Right, so hence 5 is equal to 3 plus 2 right now So hence p divided by 2 to the power m 5 to our n which is now can be written as p divided by 2 to the power n Plus r into 5 to the power n n Right now the same thing can be written as p 2 to the power n into 5 to the power n Into 2 to the power r why because I know a to the power m plus sorry into a to the power n Is equal to a to the power m plus n so the reverse of this I Know now 2 to the power n plus r can be written as 2 to the power n into 2 to the power r Okay, so hence now what do I get from here? So I get P upon again I can take the exponent to be common 2 into 5 to the power n Into 2 to the power r. Is it it? So hence this is p divided by 10 to the power n Into 2 to the power r Correct now. Can I do a trick? I am doing this trick what I am multiplying and dividing the denominator and numerator by 5 to the power r can I do that a fraction doesn't change if you multiply and divide or If you multiply both numerator and denominator by the same Number why do why did I do this? I'll tell you a little while Later so in this is p into 5 to the power r divided by 10 to the power n into 10 to the power r Why because 2 to the power r into 5 to the power r can be written as 2 into 5 to the power R which is nothing but 10 to the power r. So that's what I have written here That means this becomes 5 p into 5 to the power r divided by 10 to the power n plus r Is it it? 10 to the power n plus r That is you are dividing a new number in the numerator by a power of 10 so hence again, this will be a terminating terminating Decimal is it it? Terminating decimal representation Likewise you can again see case 3 case 3 Case 3 so what is case 3 then m is less than n Now the same logic will happen. So what will happen? It will be now we can find out small r which is equal to n So now what can we do? We can express the same e by q can be written now as p upon q q is 2 to the power m and 5 to the power m plus r instead of n I am writing like that and the same mechanism can be done now instead of multiplying by Power of 5 here. You have to multiply by power of 2. So if you see P divided by 2 to the power m into 5 to the power m into 5 to the power r Which is equal to p upon 2 into 5 to the power m Into 5 to the power r Correct. I am just repeating what I did above So you'll have to be a little patient and go to go through this video a little slowly You will be able to get this so 2 into 5 to the power m and now what I'm doing I am multiplying and dividing by 2 to the power r both numerator and The denominator, right? Why did I do that? So that I get powers of 10 in the denominator So it is 10 to the power m into 10 to the power r So hence it is p into 2 to the power r divided by 10 to the power M plus r you must be a little, you know Aware of the indices and the laws of exponents then you then this becomes very easier proof But if you don't know that it at the time at this time So you can just to be You know familiar with the trick and the trick is if it is of the form of 2 to the power 5 and Sorry 2 to the power r and 5 to the power or 2 to the power m and 5 to the power m Then you know it is going to be terminating, right? So again, you see here. This is terminating Terminating decimal. Why because it is being divided by a power of 10 So what is the learning from this? The learning is simply this that if you have fractions with the denominators of the form 2 to the power m and 5 to the power n then it will always have terminating decimal representation Hope you understood the logic. Thank you