06 Apr If p is a prime then √p is irrational
To prove √p is irrational, where p is a prime, we will need the following theorem:
Theorem: If is prime then then or
Proof: If we are done. So let us assume that Therefore, Hence, by Euclid’s lemma, [Euclid’s lemma: If with then ]
Now to prove √p is irrational (where p is a prime):
Assume that √p is rational.
Then there exists two integers a, b such that where gcd(a,b)=1 and
Then Which implies
divides (By the above theorem.)
So there exists an integer such that
So, from (1), we get
(Dividing both sides by p.)
This implies p divides b. This is a contradiction since gcd(a,b)=1.
Thus, when p is a prime, √p is irrational.
Particular Case:- √2 is irrational.