prove that is irrational number.


Asked by Topperlearning User | 12th Dec, 2013, 01:40: AM

Expert Answer:

 Let us assume, to the contrary that is rational



                 --- (1)

         Therefore 2 divides  and 2 divides p.  So we can   

         write  p = 2r for some integer r


   Substituting this value of p = 2r in (1)


This means 2 divides to  or 2 divides q also.  Therefore, p and q have at least 2 as a common factor.  But this contradict the fact that p and q have no common factor other than 1.


This contradiction arises because of our wrong assumption that  is a rational number.  So we conclude that   is irrational number.

Answered by  | 12th Dec, 2013, 03:40: AM