 Hello friends, so welcome to new session on real numbers. Today, carrying on with where we had left in the last session, we will discuss a new theorem and this theorem is called and this term is called fundamental theorem of arithmetic. Okay, so we are going to discuss fundamental, fundamental, fundamental theorem, theorem of arithmetic. So you can see as the name itself suggests, it is a fundamental theorem. So very basic theorem in number theory or real numbers as you know it in your grade 10 mathematics book, right? So the very important, very, very important, important theorem, theorem and this theorem you will be surprised to know was given by Euclid. So Euclid was the first person who actually coded this in or codified this in his book called The Elements and he was before Christ. We all know that he existed before Christ, correct? Later on, Karls-Fedrich Gross, very famous European mathematician, also tried and gave the proof of this theorem. So let us see and you know what does the web says about fundamental theorem of arithmetic. So I already made a quick search on Wikipedia, so it's always a good habit to search on net about the topic which you are studying and try to understand what whatever has been said about that particular topic. Not all the topics are authentic, but yes, Wikipedia to a large extent because it is peer reviewed and verified by many people on the web. So you can reasonably expect that the data given here or the facts given here are correct. So now let's see what does it say. So you can see there is a picture here of a book cover it seems and it says the unique factorization theorem. So if you see here they are talking about unique factorization theorem. So this is fundamental theorem of arithmetic they are saying and it is also called unique factorization theorem or the unique prime factorization theorem, okay? Now what does it say? It states that every integer greater than one either is a prime number. So if you take a positive number either it will be a prime number or it can be represented as the product of prime numbers. So we will first see what is the prime number and then try to explain this and they go on to say that moreover this representation is unique. So one is they can be represented as products of prime numbers and moreover they are saying this representation is unique that means you cannot express any composite number in terms of products of prime numbers which are you know or you cannot express in two different ways, okay? So let us understand now what is fundamental theorem of arithmetic which was proposed by Euclid given by Euclid he tried proving it also and then later on Gauss also proved it in 19th century. Now if you remember we were talking about few terms so there was one term called composite number there was another term called prime numbers, right? Now prime numbers what are these types of numbers? So we know and to tell you we are going to discuss only non-negative or rather only positive, positive, positive integers, positive integers we are going to discuss only about positive integers, okay? So what we are saying so what is a composite number? A composite number is a number or a positive integer a positive integer it is positive integer, integer, integer which has, which has, which has more than, more than two factors, okay? So any integer which has more than two factors will be called as composite numbers example four, four why four is a composite number because the factors are one, two and four there are three factors similarly six, six has factors one, two, three and six these are the factors of six similarly eight if you see this is one, two, four and eight there are four factors now what is a prime number then? So prime numbers as you already know prime numbers are positive integers, positive, positive integers they are positive integers the integers which have, which have, which have only, only two factors you already know this, know this what all factors one is the number it's one and the number itself and the number, number itself and we know the smallest prime number is what is the smallest prime number? Smallest prime number we know is two smallest prime number is two because it has factors one and two then we have three it has factors one and three similarly five similarly seven eleven thirteen so on and so forth okay all these are what are these numbers these are prime numbers right now what does fundamental theorem of arithmetic say definition written over here is every composite number now we know what is a composite number can be expressed as a product as a product of prime numbers so this is what the fundamental arithmetic says and it goes on to say that this factorization is unique also right unique so whatever factors you will be getting it will be unique you cannot get different sorts of factorizations for one composite number except for the order in which the prime factors occur let's take an example our composite number will be let's say ten so ten is a composite number clearly why because it has many factors one two five ten so it can be clearly written as two to the power one into five to the power one yeah now in no way you can express ten in terms of three seven or even different powers of two right so hence it is unique let us take twelve so twelve is two into two so two to the power two into three okay so twelve is two square four into three twelve let us take another example eighteen so eighteen is two into three square twenty is equal to two square into five right so basically what we are seeing that every composite number can be expressed as powers of prime factors and this particular representation is also called canonical form of canonical form canonical form of representation representation representation representation of any composite number any composite okay friends so in this session we understood what fundamental theorem of arithmetic is all about and we also saw that these representation is also called canonical form of representation of any composite number in the next session we are going to discuss a few more examples of fundamental theorem of arithmetic in terms of real express few bigger composite numbers in in their prime factors and we'll also learn how to find out or let's say find this expression or how to represent any composite number in its canonical form thanks for watching this video