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Below is a simple proof of irrationality of square-roots that I discovered as a teenager (inspired by a proof of Dedekind). It employs the Bezout identity for the gcd, i.e. $\rm\,\gcd(a,b)\,$ is an integral linear combination of the integers $\rm\,a,b,\,$ i.e. $\rm\,\gcd(a,b)\, =\, a d - b c\,$ f...