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.