 I am Professor K. Rajaram Gandhi. Deport from Department of Mathematics, Bitsu Vizag. So today I would like to discuss about how to analyze a mathematical problem regularly and more regressively. Because generally what will happen today, maybe I am working in engineering college. I started my career from a school, 3 years I am a school teacher. Generally I am addressing these kind of questions. When I used to teach mathematics for school children, these kind of questions many I have seen. So these are the questions actually very important and need. Why it is need? Because we are not answering these kind of questions, we cannot motivate the students towards research. What I feel personally. For example this question is actually there in 70 class. When I used to teach at that time, maybe probably now also available. And the same question I have seen in the Cali class and as well as in higher mathematics. The thing is is 0.39 dot dot I mean and so on is really equal to 1 or not. So now I will give you different proofs and then I will give some remarks also. And also welcome if any faculty member would like to comment any question on particular way. On the proofs or any what that suggestions. Generally if you divide 1 with 3, so you will see that 0.333 that means 3 is repeating. So now I would like to multiply both sides with 3. So that 3 into 1 by 3 1 and this side we will see that 0.99 and so on. In this way we can convince the children 0.99 and so on is really equal to 1. And another way is if you take x is equal to 0.99 and so on if you consider as x. By multiplying it with 10 on both sides you see 10x equal to 9.99 and so on. Again now I will separate these 2 equations 1 from other either 1 minus 2 or 2 minus 1. First equation for second or second equation from first does not matter. You see that 9x equal to 9 and x is equal to 1. So this is another kind of convention that 0.99 and really equal to 1 or not. Now if you go for the third proof 0.39 and so on that is nothing but 9 by 10 plus 9 by 100 plus 9 by 1000 plus and so on. We can see this is a geometrical progression series that first term and common rest very clearly we can find out. And we already have a formula for finding the test infinity in the GP series. So by using a by 1 minus r definitely we see that 1. So again by this proof also we can conclude 0.99 and so on is definitely equal to 1. And then if you take a is equal to 0.99 let us consider that 0.39 and so on is a. And I am taking b is equal to 1. Now I would like to add a and b we can see 1.39 and so on that means 9 is actually repeating. Now I would like to divide both sides equal to 2. So we can see that 0.39 and so on that is nothing but equal to a. So because we already considered that 0.39 equal to a previously. Now a plus b by 2 equal to a we will get by cross multiplication we can see that b and a are both are same. So 1 is equal to 0.39 and so on. 0.39 and so on means this is very clearly by this proof so you can understand it exactly equal to 1. Now I am going in a different direction so now and so on 999. So now what I am going to do I will consider this term as a. So now I will multiply with the both sides with 10. We can see that the third equation again by subtraction 9 and I am actually just subtracting at left side 9. And that is nothing but a minus 9 is equal to 10 a we see a is equal to minus 1. So why I given this slide because 3 minus 2 is 1, 2 minus 3 is minus 1. So now I am actually going in reverse order so now we can conclude by this slide. If 0.39 is really equal to 1 and so on that 0.39 definitely equal to minus 1. This is one kind of convention to prove 0.39 and so on is equal to 1. So why these many proofs, one may get the doubt. So the reason for giving these many proofs what secret actually there. Because generally in mathematics one proof is sufficient but actually in the morning session that what are told. We can prove Pythagoras theorem in many ways. In the same way this 0.39 is real and so on equal to 1. Like that we can prove in many ways but still here I don't want to show that number of proofs. Here the main intention of this what are the present patient is. So sometimes we feel 0.99 is some slightly less than 1. When it is slightly less than 1 but then these proofs of saying very clearly it is exactly 1. So it is clearly without mathematical argument we can understand it is close to 1. Why we are saying it is exactly equal to 1. Is there any flaw in the proof or is there any flaw there in our understanding. So it is very clearly we can go for little higher mathematics we can understand. If you take two consecutive numbers in between there is a number existing. So if you want to make it any particular number in between 0.39 and so on and 1. Some particular some number according to this property this rule definitely that number more than this and less than 1. So easy to possible. So here we will what are that contradictory and all things will come. What we can say 0.39 and so on really equal to 1 or not. So very clearly we can understand by this argument by convincing the children. So there is some what are that number is existing for every two consecutive numbers. That is not possible for this case therefore definitely 0.0001 is that is not we won't consider as a number. Therefore definitely that variation is very minute that minute generally we won't count. Because how less is it is very for that infinitely less number is existing therefore we won't bother about that numbers between what are that existing number. So we can conclude very clearly 0.39 and so on is really equal to 1. Now I am going in other way so if you take capital X is equal to 1 by 1 minus X. Now we would like to plug in the place of X by 0.9 in first case so that we can see capital X is equal to 10. And then if you replace 0.99 and then if you replace 0.0001 like that if we increase 119 and you can see that the capital X gives very big number. Now here if you give small X is equal to 1. If you give small X equal to 1 naturally we will get what are that infinity. So now we can't say that infinity we can say very simply it is undefined not exactly infinity I am sorry it is undefined. If we give small X equal to 1 we see capital X is undefined. So here for 0.39 and so on we got capital X very huge whereas for small X if you give 1 we will see that capital X is undefined. That means if that 1 followed by many 0s are equal to undefined then it is clearly not equal. 1 followed by many 0s is definitely not equal to undefined very clearly 0.39 and so on not equal to 1. But here there is a flaw we have to observe. So by this proof we can understand 0.39 and so on is really not equal to 1. Previously by 3 or 4 proofs I showed from that we can understand 0.39 and so on is exactly equal to 1. Now where is the flaw there? Here people can put little more concentration if they can think more vigorously we will produce that quality ideas quality research papers. Here the flaw there in the function. So what kind of flaw there in the function? Why we are not supposed to take X value is 1. If we take what is the flaw actually happening graphical representation what is saying. So it is all the things it is just matter of proof and what is that some higher level of calculus is required. So in the point of more ideas because people are listening from different remote centers. Maybe some of the people are not interested about mathematics they come from different backgrounds. So I do not want to test their patience here lot of calculus is advanced calculus is required. By using pre-calculus I showed the flaw and everything why we are not supposed to take X is equal to 1 and what is happening in the graph these are all the things if any faculty member can discuss what is that what is the video conference I am very happy. Thank you very much for giving this opportunity sir. Thank you, thank you so much. Okay, thank you very much Rajaram Gandhi. Okay, so we had some mathematics though there were some simple equations. The thing was interesting I am just saying that you know we did not have the integral sign the differential sign and so on. So I am looking at it from the point of your presentation. But it was interesting in the sense that while the other two presentations were completely devoid of mathematics here we have some mathematical orientation. Okay, let me get a couple of comments from some centers to the presentations and then I will proceed with my own part. So MES, Pillai, New Panvel, any comments or questions? Yes, yes we can hear you. So I wanted to make a comment on the third presentation by professor Rajaram Gandhi. So there I thought that the color contrast was very poor. So he has the background color and the lateral color he should have used a different color combination. So that is one and on the first I think the first presentation was very good and it was very crisp and she brought it very clearly. She had some message to convey and I think she did very effectively. Thank you, over to you. Yes, thank you for your comments. Let us have just one more. Yes, BDT College of Engineering, Dhawan Giri, you wanted us to connect to you. Sir that first presentation was very good and the flow was tailor made and the second presentation I think it has gone out of the limits and it was not up to standard I think. And the third one was okay but the slides could have been little better but the third one was 75% okay but the complete continuity was not there. Over to you sir. Can you please elaborate on what do you mean by saying continuity was not there? Subject continuity was not there. The starting was very good but the concluding was not complete. Concluding was not complete. Starting was very good and in the middle part also was okay but what he was able to conclude so that was not clear. Okay and what do you think was where the problems with the second presentation? See we need to be very specific with our comments. If you say that some presentation was good or not good you know it is not clear why you feel so. So you should mention about specific aspects. That is what we want to do in this communication skills. Sir I got one comment on the first presentation. The contribution she didn't mention what exactly the contribution of the work. The first one only she mentioned replace recognition which algorithm what is the contribution of the work she didn't mention. That is very important I think in technical presentation. Just one more and then I think I will start with my part. Angadhi institute you are connected. Good afternoon sir. I think this presentation was basically not to comment on the content because all are from the different background. So I would like to speak about the strengths and weaknesses about the presentation. The first presentation the strengths are clarity of voice and language. The slides were precise and clear body language was there. Timely completed and technical details well contained. The scope of improvement could be need some voice modulation and more technical depth and eye contact was required. Regarding the second presentation the audio was not really clear to many of our centres. So whatever we could get it we have done with it. So I feel the presenter when giving a presentation should face the audience and that was really missing because there is no communication through eye contact between the presenter and the audience. So that was one there is no link between the slides that were shown to us and what he was speaking and lot of theory related to the content. There was no figures there were no some mathematical equations involved. Otherwise I mean there is no outline of the presentation no conclusion as such. The third participant he was confident in his talk whatever he was but it was somewhat like a class than a presentation. He was bit fast then again the same thing he was not facing the audience the presenter should need to be that. And I feel PPP didn't have the outline. There was no flow because the audience need to know what is to be covered next. So there was no outline and there was no conclusion references but otherwise overall presentation was fine. Thank you. How to you. Okay. Okay. So I think we had an interactive session there.