 Hey folks, welcome again. So in this session, we are going to take up one theorem related to certs and understand very critical and important theorem says Square root of a rational number. So when you find out a square root of a rational number It would either be a purely rational number or it would be purely irrationals You will never get square root of a rational number M food M that is where M is what I'm saying M is a rational number M is a Rational Number you will never get root M as a plus root B Okay, we'll never be equal to a plus root B where a is rational and This is irrational root B irrational you will never get Yeah, yeah, the square root of M as a plus root B square root of M will be let us say square root of N Or square root of M will either not be possible that is square root of M will be simply square root of M Or if M is a perfect square, then you will get n is equal to n such that n square is equal to M Okay, so if you see Either you will not have the square root only or if you have you will have as a Rational number so hence root M will never be equal to let us say a plus root B form It will never happen Right, it will never happen. So how to prove this? If possible, let us say So we'll use the process of let's say method of contradiction Contradiction so I am saying let us say root M can be expressed as a plus root B. So you are getting a Let us say this is possible root M is equal to a plus root B where a belongs to Rational number set that is a is a rational number and root B is a is an irrational number root B is an irrational Number, let us say it is possible for for for the time being so squaring both sides squaring Squaring both sides Both sides. What will you get? You will get M root M square is equal to a plus root B whole square that is M is equal to a Plus root B square plus to a root B That means M is equal to a Sorry a square. It will be a square plus B plus to a root B That means what does it mean? M minus a square minus B is Equal to 2 a root B and hence root B, which is an irrational number can be equated to M minus a square minus B upon 2 a Now if you closely look at this equation here, this is a this is a irrational number irrational Irrational number and it is on the right hand side. It is a rational number Rational why it is a rational because a B M all are all our rationals is it all our Rational numbers that means RHS is a rational number. So how can an irrational number be equal to a rational number? So hence there is a contradiction contradiction you You arrive at a conclusion that an irrational is equal to rational and that is happening because You assumed this to be true. This was you you assumed that root M can be expressed as Root M can be expressed as a plus root B. So hence clearly root M cannot be expressed as a plus root B. So hence hence root M will either be Root of M will be root M itself. That means it is not Possible to find the square root or if M is a perfect square M is a perfect square That means root of M is equal to n such that n square is equal to m where n is a Rational Number right example root of three not possible. So root three is an purely irrational purely Irrational root of four is two. So you see purely rational Purely rational root of five again not possible. So irrational Root of six is irrational Root of seven also is irrational Root of eight can be written as two root two, which is again irrational and Root of nine if you see is three, which is purely rational So hence either it will be purely rational or it will be purely Irrational so square root of a rational number will be either a purely rational or irrational irrational It cannot be a mix of rational and irrational