According to Excursions in Mathematics by Ogilvie (Dover) Joseph Louis LaGrange (1736-1813) expressed the proof of the irrationality of the square root of 2, in a handy sentence:
It (square root of 2) cannot be found in fractions, for if you take a fraction reduced to its lowest terms, the square of that fraction will again be a fraction reduced to its lowest terms and consequently cannot be equal to the whole number 2.
I think that is interesting and related to the other proofs presented.