 Hi folks once again welcome to this session on the number system in this session We are taking up another question and the question is to that root 2 minus root 2 is an irrational number Okay, so to prove that root 3 minus root 2 is an irrational number So in the previous sessions if you have gone through them, you would have noticed that we have done proving on how how root 2 is an irrational number we have proved that and And we discussed a method of proving in that particular session and that method is called method of contradiction, isn't it? So we discussed method of contradiction and using this method of contradiction when we know that Only two possibilities of outcomes are there for an event Then we can use the method and prove the given theorem. So hence We start with an assertion in this case also that let root 3 minus root 2 be an irrational number instead of an irrational number And we will eventually Contradict our assertion, right? So we will come up with some result where it contradicts our assertion that it is an it is a rational number Hence automatically it will become an irrational number. So we start with an assertion saying let us say Let us assume let us assume root 3 minus root 2 to be to be a Rational number to be a rational number. Okay Why we are doing it so that you know, uh, we will eventually contradict this assertion and then By logic it will automatically become an irrational number. So the moment we say It is a rational number. We know that a rational number will be having three criteria One is it must be expressed in the form of p by q where p and q are integers And q must not be equal to zero. That's condition number two and condition number three is Gcd or hcf of p and q is equal to 1 Right. So hence since root 3 minus root 2 is uh, is a rational number. So hence we can say p upon q Is equal to root 3 minus root 2 So since root 3 minus root 2 is a rational number what we are asserting Then therefore There must exist p and q two integers such that p by q is 3 minus root 3 minus root 2 So squaring both sides squaring Both sides and when we square We get rid of the root signs, isn't it? So let us square. So we'll get p square by q square Is equal to root 3 Minus root 2 whole square Which is equal to if you know the identity a plus or sorry in this case a minus b whole square is a square That is root 3 square Plus b square. That is root 2 square minus twice of ab. So root 3 into root 2 correct Simplifying what will you get you will get You will get p square upon q square is equal to 3 plus 2 minus 2 into under root 3 times 2 that is 6 correct Simplifying further you will get p square by q square Minus 5 is equal to minus root 2 by root 6. That means this implies simply 5 Minus p square by q square is equal to 2 root 6 So I just removed the negative sign from the right hand side So on the left hand side the order of the terms will change And hence now I can say 5 minus p square upon q square by 2 Is equal to root 6 Isn't it after simplification now? Let us let us see LHS. What is LHS? In LHS my dear friends if you see what is LHS LHS is 5 minus p square by q square by 2 Okay, now p and q are integers Are integers So if p and q are integers that mean that means p by q Is a p square by q square is a rational number Is a rational number isn't it that means 5 minus a rational number Will be a rational number is a Rational number Why because rational plus rational or rational minus rational is always a rational number that means 5 minus p square by q square upon 2 Is a rational number why because I am multiplying by 1 by 2 Rational number So by this logic LHS is a LHS is a Rational number but RHS my dear friends if you see RHS RHS is Root 6 and it doesn't take much time to prove that Root 6 is a irrational irrational Number that means we are trying to equate a rational number into in Rational number to irrational number that is not correct. Hence LHS will never be equal to RHS hence we contradict our own assertion What is our own our assertion was that that Root 3 minus root 2 is a Rational number that cannot be possible because it is leading to a result Which is not valid right this particular assumption led to this result that An irrational number is equal to rational number. Hence we say hence Hence we declare that root 3 minus root 2 is not a Rational not a Rational number and something is not a rational number and it is real then root 3 minus root 2 is An Irrational number guys irrational number Understood. So what did we learn in this process again an example of method of contradiction? We start with an assertion and eventually Eventually prove that it's not going to be True and hence whatever we have started as an assertion was wrong Hence the other outcome of the assertion will be true, right? So in this case again root 3 minus root 2 is an irrational number. Thank you