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Theorem: the square root x of 2 is irrational. Proof: x=n/m with gcd(n,m)=1 implies 2=n^2/m^2 which is 2 m^2=n^2 so that n must be even and n^2 a multiple of 4. Therefore m is even. This contradicts gcd(n,m)=1. 
19992000, www.dynamicalsystems.org 