show that √2 is irrational
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let us assume to the contrary that root 2 is rational
so,we can fin integer r and s such that root 2 =r/s
suppose r and s common factor other than 1 .then we duvided by the common factor to get root 2 =a/b,where a and b are coprime so,b root 2=a
squaring on both side and rearranging we get 2b2=a2.therefore 2 divides a2
now,by theorem1.3, it follows that 2 divides a so,we can write a=2c for some integer c
subsituting for a we get 2b2=4c2 that id b2=2c2
but thus contradicts fact that a and b have no common factor othar than 1 this contradiction has arisen because of our incorrect assumption that root2 is rational
so,we conclude that root 2 is irrational
so,we can fin integer r and s such that root 2 =r/s
suppose r and s common factor other than 1 .then we duvided by the common factor to get root 2 =a/b,where a and b are coprime so,b root 2=a
squaring on both side and rearranging we get 2b2=a2.therefore 2 divides a2
now,by theorem1.3, it follows that 2 divides a so,we can write a=2c for some integer c
subsituting for a we get 2b2=4c2 that id b2=2c2
but thus contradicts fact that a and b have no common factor othar than 1 this contradiction has arisen because of our incorrect assumption that root2 is rational
so,we conclude that root 2 is irrational
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