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:
“The thing with Catholicism, the same as all religions, is that it teaches what should be, which seems rather incorrect. This is “what should be.” Now, if you’re taught to live up to a “what should be” that never existed—only an occult superstition, no proof of this “should be”Mthen you can sit on a jury and indict easily, you can cast the first stone, you can burn Adolf Eichmann, like that!”
—Lenny Bruce (1925–1966)
“Sculpture and painting are very justly called liberal arts; a lively and strong imagination, together with a just observation, being absolutely necessary to excel in either; which, in my opinion, is by no means the case of music, though called a liberal art, and now in Italy placed even above the other two—a proof of the decline of that country.”
—Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)