Prove that √2 is an irrational no

1.	Prove that √2 is an irrational no
Answer» Let assume root 2 is rationalRoot 2=a/b(where a and b are co -prime and b is not equal to 0)Root2b=a--(1)Squaring on both sides(Root 2b)^2=a^2--(2)2 divide a^2}2 divide a}a=2c(where c is some integer)Put a in (2)(Root 2b)^2=(2c)^22b^2=4c^2b^2=4c^2/2b^2=2c^2So a and b have atleast 2 as common factor.But this condradictss that a and b have no,common factor other than 1 and this aris due to incorrect assumption that root 2 is rational.Therefore root 2 is irrational. Hence proved Let is assume that root 2 is rational.Let a and b is a positive integer. Root 2= a/bSo root 2 is rational number But this contradicts the fact that root 2 is rationalBut this contradiction our assumption is wrong so root 2 is rational number.

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