A Magic Trick from Fibonacci
Short summary:
A Magic Trick from Fibonacci. James Smoak (jsmoak@worldnet.att.net) ... What is behind this magic trick? A check of several additional fractions with a ...
Long summary:
A Magic Trick from FibonacciPage 1by using increments and limits instead of differentials but the present form helps todemonstrate the utility of the differential calculus in addressing such problems. Thisalternate approach will hopefully provide a better link between thought processes inboth calculus and physics.Reference1. G. B. Thomas and R. L. Finney Calculus 9th ed. AddisonWesley 1996.A Magic Trick from FibonacciJames Smoak (jsmoak@worldnet.att.net) 12140 E. Iowa Drive Aurora CO 80012Thomas J. Osler (Osler@rowan.edu) Mathematics Department Rowan UniversityGlassboro NJ 08028A mathematical magician displays the following remarkable fractions and asks usto think Fibonacci.10089= 1. 1 2 3 5 955056 100009899= 1. 01 02 03 05 08 13 21 34 55 9046368 1000000998999= 1. 001 002 003 005 008 013 021 034 055 089144 233 377 610 98859958818777596 The decimal expansion of our first fraction generates the first five Fibonacci numbersbefore blurring into other digits. Our second fraction generates the first ten and thethird fraction generates the first fifteen. Notice that the successive fractions change byappending two 0s to the numerator and a 9 to the front and back of the denominator.Will the next fraction10000000099989999generate the first twenty Fibonacci numbers in a similarway? Does this pattern continue forever? What is behind this magic trick?A check of several additional fractions with a computing aide like Mathematicashows that the pattern does appear to continue. (Roberts 2] mentions the fraction10000/9899.)We now reveal the machinery of the magician. We use the familiar notation F1= 1F2= 1 F3= 2... for the Fibonacci numbers with the recurrence relation Fn=Fn1+ Fn2. The generating function 1] for these numbers is the key item11 x x2= F1+ F2x + F3x2+ F4x3+ .(1)Note that(F1+ F2x + F3x2+ )(1 x x2) = F1+ x(F2 F1)+n=2xn(Fn+1 Fn Fn1) = 1since F1= 1 F2 F1= 0 and Fn+1 Fn Fn1= 0 for all n 2.58c THE MATHEMATICAL ASSOCIATION OF AMERICAPage 2Now letting x = 0.1 we get11 0 ...
Source: www.rowan.edu
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