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

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 √p =

(√p)² = ( )²

P =

a² = pb²

Since a² is 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.