Prove that for any prime positive integer p, is an irrational number.

Given: Any prime positive integer p.

To find: is an irrational number.

Concept used: Assume p to be rational number and prove it is irrational by contradiction.

Explanation:

_{Let assume that √p is rational}

_{Therefore, it can be expressed in the form of} , where a and b are integers and b≠0

Therefore, we can write

a² = pb²

Since a²is divided by b², therefore a is divisible by b.

Let a = kc

(kc)²= pb²

K²c²= pb²

Here also b is divided by c, therefore b² is divisible by c². This contradicts that a and b are co-primes. Hence is an irrational number.