prove that is irrational number.
Let us assume, to the contrary that is rational
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.
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