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Approximating the Golden Ration by Dividing Fibonacci Numbers

The golden ratio is an irrational number, approximately 1.6180339887498948482045868. Dividing one Fibonacci number by the previous gives better and better approximations to the golden ratio the bigger the numbers get. One division gets a little under, then the next a little over, as the table below shows. 144/89 is probably good enough for most practical purposes.

F(n)/F(n - 1)Decimal ValueApproximation Error
1/11-0.6180339887498948482045868343656381
2/120.3819660112501051517954131656343618
3/21.5-0.1180339887498948482045868343656381
5/31.6~0.0486326779167718184620798323010285
8/51.6-0.0180339887498948482045868343656381
13/81.6250.0069660112501051517954131656343618
21/131.6153846153846153846153846153846154-0.00264937336527946358920221898102273
34/211.61904761904761904761904761904761900.001013630297724199414460784681980
55/341.6176470588235294117647058823529411-0.0003869299263654364398809520126969
89/551.618~0.000147829431923333613594983816180064097
144/891.6179775280898876404494382022471910-0.000056460660007207755148632118447106484
233/1441.61805~0.000021566805660707350968721189917437835
377/2331.6180257510729613733905579399141630-0.000008237676933474814028894451475027591
610/3771.61803713527851458885941644562334210.000003146528619740654829611257704057346
987/6101.6180327868852459016393442622950819-1.201864648946565242572070556150507
1597/9871.61803444782168186423505572441742650.000000459071787016030468890051788427365
2584/15971.6180338134001252348152786474639949-0.000000175349769613389308186901643127112
4181/25841.618034055727554179566563467492260.00000066977659331361976633126621944199
6765/41811.6180339631667065295383879454675914-0.000000025583188318666198888898046632429
10946/67651.618033998521803399852180339985210.000000009771908551647593505619579916278
17711/109461.618033985017357938973140873378403-0.00000000373253690923144596098723504810583812
28657/177111.61803399017559708655637739258088190.00000000142570223835179055821524382005847236
46368/286571.6180339882053250514708448197648044-0.00000000054456979673374201460083370693062428
75025/463681.6180339889579020013802622498274670.00000000020800715317567541546182910105125741

Values calculated with Mathematica 4.