Prove that √2is irrational?

Let us assume on the contrary that √ 2 is a rational number. Then, there exist positive integers a and b such that √ 2 = a b where, a and b , are co-prime i.e. their H C F is 1 ⇒ ( √ 2 ) 2 = ( a b ) 2 ⇒ 2 = a 2 b 2 ⇒ 2 b 2 = a 2 ⇒ 2 | a 2 [ ∵ 2 | 2 b 2 a n d 2 b 2 = a 2 ] ⇒ 2 | a . . . ( i ) ⇒ a = 2 c for some integer c ⇒ a 2 = 4 c 2 ⇒ 2 b 2 = 4 c 2 [ ∵ 2 b 2 = a 2 ] ⇒ b 2 = 2 c 2 ⇒ 2 | b 2 [ ∵ 2 | 2 c 2 ] ⇒ 2 | b . . . ( i i ) From ( i ) and ( i i ) , we obtain that 2 is a common factor of a and b . But, this contradicts the fact that a and b have no common factor other than 1 . This means that our supposition is wrong. Hence, √ 2 is an irrational number.