 Hello friends welcome to another session on number systems now today in this session we are going to take up a very common problem which is there in the syllabus of a number system and this is nothing but proving that root 2 is an irrational number now those who are in cbc curriculum they will know that this is a very common question and it has been asked time and again in various you know fora now proving that root 2 is an irrational number is also important because in this proof we learn a very important method of proving in mathematics and that is called method of contradiction method of contradiction now this is one of the methods which is deployed which is deployed to prove multiple theorems and other results in mathematics okay so while you learn how to prove root 2 in irrational number you also learn a very important method and that is called method of contradictions now so what do we use and how do we use this method of contradiction is sim something like this let us say there are only two possibilities of any event okay so there are only two possibilities either possibility one possibility number one or possibility number two let us say for any event there are only two outcomes for any event there are only two outcomes that means in a let's say a soccer game or any any sports let's say there is no draw there are only two possibilities either the either the team will win or it will lose win or lose right there is no possibility of a draw or a tie let us say okay so if win is not happening then definitely lose will happen right lose some if someone is not winning then definitely it will lose and if someone is not losing that means definitely it is going to win so this is this is what we are using going to use in the method of contradiction so hence what we say is if root 2 is if root 2 is not rational is not a rational rational then root 2 is definitely definitely root 2 is definitely irrational if i'm saying if root 2 is not rational root 2 is definitely irrational okay so we will use this philosophy that if root 2 is not rational then root 2 is definitely irrational so instead of proving root 2 is an irrational number we can use or you can you can go this way that is prove that root 2 is not rational okay and hence we can conclude that root 2 is irrational this is happening because root 2 can can be either rational can be either rational or irrational there is no third possibility if there is a third possibility then we cannot use this method right only two methods two possibilities are there either root 2 will be a rational number or it will be a irrational number now if we somehow prove that root 2 is not rational then definitely it is going to be irrational number so now let us try and prove root 2 to be a rational number okay so why am i doing this because to prove that irrational number it is much more difficult and why is it difficult because we let us go by the definition of irrational number so what are irrational number irrational numbers are irrational numbers are irrational numbers are those numbers who have which has got non terminating non terminating and non repeating non terminating and non repeating what decimal decimal representation representation right so it will be very difficult to prove that someone some number has non terminating and non repeating for example in the case of root 2 itself last session we saw that we kept on finding the root 2 value but we never got a repeating pattern so we are not sure whether after 1 billion digits also let us say it will repeat or not so it will be a humongous task to prove this way so hence what method do we adopt we adopt that if somehow we prove that root 2 is not rational then definitely root 2 is irrational and why is that so because there is only two possibilities or two possibilities for root 2 to be in any of these groups either it will be in the group of irrational number or it will be in the set of rational numbers now i'm using the words group and sets loosely here to not take you know you know in fact in mathematics groups and sets all have different meanings but in your standard let us let us talk about sets so if root 2 doesn't fall into rational number set then definitely it is falling in irrational number set so to prove that root 2 is rational let us say then what do we have to do we have to abide by or we have to prove that all the properties of rational numbers are met so what are the properties of rational number so there are three criteria 1 it must be expressed in the form of p by q 2 q must not be equal to 0 and yes please note that p and q are integers right and 3 gcd of p and q is equal to 1 right this is what we have defined rational numbers as and any one of these rules are violated then we know that root 2 is not a rational number right so let us start out with our with an assertion let root 2 be a rational number right this is my assertion i am saying let this be true okay so hence all the three criteria all the three criteria must be met so hence i can write root 2 as p by q correct where p and q are some integers and q is not equal to 0 isn't it and also gcd of p and q gcd is nothing but hcf of p and q is 1 hcf of p and q is 1 okay so now p and q b by q is root 2 then squaring let us say let us now do some mathematical manipulation squaring 1 squaring 1 what do we get you get p by q whole squared is equal to root 2 squared isn't it this implies p square by q square is equal to 2 right root 2 square is 2 so this implies p square is equal to 2 q squared correct p square is equal to 2 q square that means p square is a multiple of multiple of 2 that means 2 divides p square isn't it because something p square is equal to 2 times something now q square is also an integer so 2 times an integer is another integer that means p square is a multiple of 2 if p square is a multiple of 2 guys then p is also a multiple of 2 p is also a multiple of 2 meaning if there is a perfect square which is a multiple of 2 let us say if 36 36 is a multiple of 2 so square root of 36 is 6 so 6 also is a multiple of 2 so 8 64 is a multiple of 2 so square root of 64 8 is also a multiple of 2 isn't it so you can take so basically even even numbers will have will have square even squares also will have even squares meaning any even number is there let us say 8 so if you square it you will get 64 which is also an even number another example let us say 10 if you square it you will get 100 which is an even number let us say 12 is 144 which is also a even number so hence if you square root 64 you will get even 8 if you square root 100 you will get 10 if you square root 144 you will get 12 that means if p square is multiple of 2 p is also a multiple of 2 that means a multiple of 2 can always be written as 2 times k isn't it where k is an integer again k is an integer what do i mean any even number you take let us say 18 so 18 can be written as 2 times 9 isn't it 22 is nothing but 2 times 11 you take any integer you will always find another integer k any even integer p you will always find another integer k such that 2 times k is p so p is 2k form p is 2k form now that means i'm writing now here p is 2k that means i can write p square is equal to 2q square from here p square is 2q square so writing p as 2k i can write 2k square is equal to 2q square that means i can write 4k square is equal to 2q square that means i can write q square is equal to 2k square so if you see q square is again is a multiple of multiple of 2 similarly you can say q is also a multiple of 2 multiple of so q can be written as let us say 2m right multiple of 2 2m where m is a integer isn't it so clearly if you see both both p and q are multiples multiples of 2 that means what does it mean this means they have they have a common factor factor as 2 if both p and q are multiples of 2 then they will have a common factor as 2 so the 2 divides 2 is a factor of p as well as 2 is a factor of q so there is a common factor 2 for both which is greater than 1 so hence we can say that gcd of p and q is not equal to 1 why because there is a factor 2 already right so the greatest common divisor of pq is not 1 there is a another greater factor 2 which is common to both hence it violates it violates it violates the condition the condition that that gcd of p and q is 1 and because of this the one of the conditions for rationality is lost hence and this is coming why because our assertion our assumption or assertion was wrong assertion is contradicted contradicted right it is proven wrong hence we declare hence hence root 2 is not a rational number and we know that if something is not a rational number it will definitely be a irrational number hence we say hence root 2 is an irrational number okay so we we just contradicted one of these you know we just prove that one of the condition is not valid is not getting met hence it contradicts our assertion and hence whatever we assumed we had assumed that root 2 is a rational number is wrong and hence root 2 has been proven as a rational number the same kind of methodology can be used to prove any root 3 root 7 root 11 whatever likewise to you know that to be irrational number I hope you understood the methodology