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)
“It comes to pass oft that a terrible oath, with a swaggering accent sharply twanged off, gives manhood more approbation than ever proof itself would have earned him.”
—William Shakespeare (1564–1616)