One to Nine by Andrew Hodges

Author:Andrew Hodges [Hodges, Andrew]
Language: eng
Format: epub
ISBN: 978-0-385-67266-5
Publisher: Doubleday Canada
Published: 2007-11-28T16:00:00+00:00

which is useful, but does not make it obvious that each Fn is an integer! This is typical in the aesthetics of mathematics: a formula or picture illuminates one aspect of a structure, yet disguises another one. The mind needs many different pictures to build up understanding, piecing them together into a manifold of insight.

As integers, the Fn have Diophantine relationships similar to those of the numbers used to calculate in Chapter 4. The square of Fn differs by 1 from the product of its neighbours. F2n-1 is the sum of the squares of Fn and Fn-1, and F2n = Fn × (Fn-1 + Fn+1); these are analogous to the jumping-ahead rule for approximations to . In fact the fractions (Fn-1 + Fn+1)/Fn give better and better approximations to , and satisfy

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