question_answer

  Prove that is irrational, where p, q are primes. OR What can you say about the prime factorization of the denominator of .

Answer:

  Let us suppose that  be rational. Letwhere r is rational. Therefore, Squaring both sides, we get  Therefore Now R.H.S. is a rational number, since r is rational, q, p are both rationals [Since p, q are primes] But is irrational, since q is prime, we arrive at a contradiction.  our supposition is wrong. Hence  is irrational. OR Since  is a terminating decimal number  denominator must be of the form  where m, n are natural numbers.