Proof that √2 is Irrational

Imagine that √2 is rational, then it can be written as the ratio of integers, p and q. 

(1) √2 = p/q

It may be presumed that p and q have no common factors, and thus cannot be reduced. Square both sides to get:

(2) 2 = p²/q²

which leads to

(3) p² = 2q²

Therefore, p² is an even number. It can only be even if p is an even number. Then p² is divisible by 4. Hence q² and (ergo) q must also be even. Since p and q are both even, the assumption that p and q have no common factors is contradicted. Therefore,  √2 must be irrational. Q.E.D. (quod erat demonstrandum). 

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