Prove √ 2 is irrational
Let us assume √2 is rational number.a rational number can be written into he form of p/q√2=p/qp=√2qSquaring on both sidesp²=2q²__________(1).·.2 divides p² then 2 also divides p.·.p is an even numberLet p=2a (definition of even number,\’a\’ is positive integer)Put p=2a in eq (1)p²=2q²(2a)²=2q²4a²=2q²q²=2a².·.2 divides q² then 2 also divides qBoth p and q have 2 as common factor.But this contradicts the fact that p and q are co primes or integers.Our supposition is false.·.√2 is an irrational number.
Let , us assume that √2 is rational number then it can be written in the form of a/b where b≠0 and (a,b are co prime numbers). So, √2=a/b, now squaring on both sides gives 2=a²/b²… 2b²=a² so we can say thay a² is divisible by 2 , therefore a is also divisible by 2. Then take a=2c, now putting value ofa and squaring on both sides giv gives us …… 2b²=4c² => b²=2c² so b² is divisible by 2 therefore b is also divisible by 2. Hence our assumption is wrong , they have other common factor as 2. There fore √2 is irrational.hence, proved