Let √7 is not rational number ⇒√7 is rational number ⇒√7=
p
q
p2
q2
=7 ⇒p2=7q2 ⇒ 7 is a factor of p2 ⇒ 7 is a factor of p Let p = 7m for some natural number m Then p=7m p2=72m2 ⇒7q2=72m2 ⇒q2=7m2;[∵p2=7q2] ⇒ 7 is a factor of q2 ⇒ 7 is a factor of q But, 7 is a factor of p and 7 is a factor of q means that 7 is a factor of both p and q. This contradicts the assumption that p and q have no common factor. Hence, our supposition is wrong. Hence, √7 can not be a rational number.