Question

Prove that √2 is an irrational number​

Answer 1

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Answer 2

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Here is u answer.!!

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• 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 , co prime therefore H.C.F is 1. ) -----------  ( 1 )

(√2 )²  = ( a/b)²

2 = a²/b²

2b²  = a²

2 Divides a²

                    ------------------------------------------------------------------- ( 2 )

2 Divide a

a = 2c for some integer c

a² = 4c²

2b² = 4c²

b² = 2c²

2 Divides b²

                       

                     ----------------------------------------------------------------------- ( 3 )

2 Divide b

From ( i ), ( ii ) and ( iii) 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.  √2 is not a rational number.

Hence , √2 is an irrational number​.

Hence proved...!!!

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