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assume sqrt(2) = a/b where a & b are both coprime integers
2=(a^2)/(b^2)
2b^2=a^2; a^2 is even, thus a is even
thus: a=2k where k is an integer
2b^2=(2k)^2
2b^2=4k^2
b^2=2k^2; b^2 is even, thus b is even
both a and b are divisible by 2, thus they are not coprime, sqrt(2) is irrational
