Prove that square root 2 is an irrational number.

Answers (1)

Solution: Let us assume on the contrary that sqrt2 is

       a rational number . there exist positive integers

       a   and  b  such that.

       sqrt2=fracab    Where  a and b are co-prime .

    \ Rightarrow (sqrt2)^2=(fracab)^2\ \Rightarrow 2=(fraca^2b^2)\ \Rightarrow 2b^2=a^2\ \Rightarrow 2|a^2  [ecause 2|2b^2hspace0.2cmand hspace0.2cm2b^2=a^2]

  \ Rightarrow a=2c   for some integer c

 \Rightarrow a^2=4c^2\ \Rightarrow 2b^2=4c^2\ \Rightarrow b^2=2c^2\ \Rightarrow 2|b^2\ \Rightarrow 2|b

 We obtan that 2 is a common factor of a and b .

But this contradiccts the fact that a and b have

no common factor other than 1.

Hence ,sqrt2 is an irrational number.

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