Another cute proof: square root of 2 is irrational.

Reference: Elementary Number Theory, David M. Burton, Sixth Edition, Tata McGraw-Hill.

(We are all aware of the proof we learn in high school that \sqrt{2} is irrational. (due Pythagoras)). But, there is an interesting variation of that proof.

Let \sqrt{2}=\frac{a}{b} with gcd(a,b)=1, there must exist integers r and s such that ar+bs=1. As a result, \sqrt{2}=\sqrt{2}(ar+bs)=(\sqrt{2}a)r+(\sqrt{2}b)s=2br+2bs. This representation leads us to conclude that \sqrt{2} is an integer, an obvious impossibility. QED.

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