Question

It 5 is a prime number ,than prove √5 is irrational

Answer 1

Let us assume, to the contrary, that √5 is rational.

That is, we can find integers a & b (b ≠ 0)

Such that √5 = a/b

Suppose a & b have a common factor other than 1 , then we can divide by tha Common factor , and assume that a and b are co prime.

So , b√5 = a

Squaring both sides:-

we get 5b² = a²

∴ a² is divisible by 5 , and a is also divisible by 5.

So, we can say a = 5c

=> B² = 5c²

=> b² is divisible by 5, so b is also divisible by 5

∴ a & b at least have 5 Common Factor

Hence this contradiction has arisen because is our incorrect assumption √5 is rational

∴ √5 is irrational number Hence proved