 Hello friends, so welcome to the session of real numbers and today we are going to discuss a new topic, which is Euclid's Euclid's Division Division Lemma Okay, so this is a very you know popular topic or very useful There are topics also With regards to number theory Or number systems, which you are studying in grade 10 now before we plunge into this First thing which will do is as we have been doing let's do a recap of whatever we have done so far So if you see we have done we just talked about real numbers in the first session What they are who coined or let's say who was the first person to use the term real numbers in Mathematics, then we talked about divisibility What is meant by divisibility and Then we talked about Some results related to divisibility or let's say properties properties properties related to or properties of Divisibility Divisibility now these are the topics which we have discussed so far now Now we are dealing with I'm going to deal with a new topic, which is Euclid's division lemma Now the first thing whenever we are the first time when I studied this topic, you know the question which came to my mind Was what these first of all? Who was this guy? Euclid who was he and you know when did he exist and why is that we are studying his or was this lemma given by him or someone else gave it or How did we come to know about his work? Second is I understand the word division because we have been you know doing division 30 divided by 5 is 6 and all that all this while so division we kind of know But what is this new term called lemma now? This is surprising why all of a sudden these you know terminologies are coming in my way So these were the thoughts which were there in my mind when we used to study all at 70 I was in class 10 now Um Then during our time it was very difficult to actually do a research on these things and you have to sit for hours together in Library to mine all this information But today we you know just a click of your mouse and you can find everything out So let's first find about find out who was Euclid so We know that this guy now was Was there in 4th century? 4th century our 4th century or 2 3rd century BC 3rd century before Christ Yeah, this was a time when he existed this he was a Greek philosopher or Greek mathematician that time You know every subject used to originate from philosophy. So he was a Greek scholar. Let us use the word scholar and He is also He's also famous for writing a book called he elements Okay, okay elements where he has talked about number theory Number theory and geometry Number theory and primarily geometry Geometry actually this is not one book This is a set of 13 or you can say that there are 13 volumes to this book you can always go to euclidia and Find out about find out more information. So you can see this is Euclid This is how he looked like I'm not sure Who drew his picture or how did they come to know about it? But anyway, so this is Euclid and it says can you see Euclid Euclid? Yeah, or he was Euclid of Alexandria. So this was in you know, a Greek kingdom of Greece So this guy was during the reign of Ptolemy I so this is the information and he has done a lot of work And specially in conic sections spherical geometry number theory and drug So these are the things which he has done and he has written this book the elements, right? So this book as you can see reduced the theorem support is now called Euclidean geometry So we are studying whatever geometry are studying today You can either praise this guy or if you do not like geometry you can even you know curse him But I would suggest don't curse him because mathematics has gained a lot from this guy So this he he's the guy who has done a lot of work and today we are going to study his lemma now having known this this About this guy now it's pertinent to know what is lemma. So again, you know what to do go to you can always go back to Your Wikipedia and you can always search. So let me make a search. So this is a lot of lemma So if you can see lemma and what is so lemma mathematics, that's what we are going to understand So if you see what is lemma? So lemma is what is lemma a lemma is a helping theorem So helping them right is a or is a proven proposition which is used as a stepping stone to larger result Rather than as a statement of interest by itself So it's something which supports some other theorem. So it is used to let's approve other theorems and It's it's derived from an ancient Greek word lemma again So this is how lemma is pronounced to there and which in Greek means anything which is received such as a gift Prophet or a bribe. So if you can see this is nothing but You can say this is used to achieve something else Used to so this particular gift or prophet is used to do more business or get something done bribe as you all know bribe is done Bribe is made to get some job done easily. So which is not legal by the way, but yeah This is this is what it means. So it is used to get something done get something Something done. So this is what This is what lemma means. So hence in in Euclid's lemma also would be important to prove many other theorems and hence the word lemma so if you see there are many other well-known lemmas and You know for fewer, you know gosses lemma is there Can you see you can see that you'll study about all these later on if you pick up mathematics as your career option? Then there is point cares lemma. There are so many other You know lemmas which will come across if you happen to pick up mathematics as your career option So now let's go back. So enough of drilling down. We discussed about Euclid. We discussed about Lemma and division all you know. So in divisibility also we understood what is division process So and we also talked about obelisk Symbol in the previous sessions if you remember, so now it's clear who was Euclid He is because this guy was a Greek fellow Greek scholar He was there in 4th and or 3rd century BC And he wrote a book called elements in which he has described all these things and then he has talked about Division, which is if you know studied under number 3 which in your 10th grade syllabus, it's it's coined as number system now And then we talked about lemma, which is nothing but a supportive supporting supporting Supporting proposition isn't it supporting proposition Proposition right. So what we'll do is so after having this understanding, let's go and understand What is Euclid's division lemma?