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 .)
Read more about this topic: Square Root Of 5
Famous quotes containing the words proof of and/or proof:
“There is no better proof of a man’s being truly good than his desiring to be constantly under the observation of good men.”
—François, Duc De La Rochefoucauld (1613–1680)
“If some books are deemed most baneful and their sale forbid, how, then, with deadlier facts, not dreams of doting men? Those whom books will hurt will not be proof against events. Events, not books, should be forbid.”
—Herman Melville (1819–1891)