Prove that √5 is irrational
•We shall start by assuming √5 as rational. In other words, we need to find integers x and y such that √5 = x/y.
•Let x and y have a common factor other than 1, and so we can divide by that common factor and assume that x and y are coprime. So, y√5 = x.
•Squaring both side, we get, 5 y^2 = x^2.
•Thus, x^2 is divisible by 5, and by theorem we can say that x is also divisible by 5.
•Hence, x = 5z for some integer z.
•Substituting x, we get, 5x^2 = 25 z^2 i.e. y^2 = 25 z^2; which means y^2 is divisible by 5, and so y will also be divisible by 5.
•Now, from theorem, x and y will have 5 as a common factor. But, it is opposite to fact that x and y are co-prime.
•Hence, we can conclude √5 is irrational.