Proof of Irrationality
This irrationality proof for the square root of 5 uses Fermat's method of infinite descent:
Suppose that √5 is rational, and express it in lowest possible terms (i.e., as a fully reduced fraction) as for natural numbers m and n. Then √5 can be expressed in lower terms as, which is a contradiction. (The two fractional expressions are equal because equating them, cross-multiplying, and canceling like additive terms gives and hence, which is true by the premise. The second fractional expression for √5 is in lower terms since, comparing denominators, since since since . And both the numerator and the denominator of the second fractional expression are positive since and .)
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