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 Value | Approximation Error |
|---|---|---|
| 1/1 | 1 | -0.6180339887498948482045868343656381 |
| 2/1 | 2 | 0.3819660112501051517954131656343618 |
| 3/2 | 1.5 | -0.1180339887498948482045868343656381 |
| 5/3 | 1.6~ | 0.0486326779167718184620798323010285 |
| 8/5 | 1.6 | -0.0180339887498948482045868343656381 |
| 13/8 | 1.625 | 0.0069660112501051517954131656343618 |
| 21/13 | 1.6153846153846153846153846153846154 | -0.00264937336527946358920221898102273 |
| 34/21 | 1.6190476190476190476190476190476190 | 0.001013630297724199414460784681980 |
| 55/34 | 1.6176470588235294117647058823529411 | -0.0003869299263654364398809520126969 |
| 89/55 | 1.618~ | 0.000147829431923333613594983816180064097 |
| 144/89 | 1.6179775280898876404494382022471910 | -0.000056460660007207755148632118447106484 |
| 233/144 | 1.61805~ | 0.000021566805660707350968721189917437835 |
| 377/233 | 1.6180257510729613733905579399141630 | -0.000008237676933474814028894451475027591 |
| 610/377 | 1.6180371352785145888594164456233421 | 0.000003146528619740654829611257704057346 |
| 987/610 | 1.6180327868852459016393442622950819 | -1.201864648946565242572070556150507 |
| 1597/987 | 1.6180344478216818642350557244174265 | 0.000000459071787016030468890051788427365 |
| 2584/1597 | 1.6180338134001252348152786474639949 | -0.000000175349769613389308186901643127112 |
| 4181/2584 | 1.61803405572755417956656346749226 | 0.00000066977659331361976633126621944199 |
| 6765/4181 | 1.6180339631667065295383879454675914 | -0.000000025583188318666198888898046632429 |
| 10946/6765 | 1.61803399852180339985218033998521 | 0.000000009771908551647593505619579916278 |
| 17711/10946 | 1.618033985017357938973140873378403 | -0.00000000373253690923144596098723504810583812 |
| 28657/17711 | 1.6180339901755970865563773925808819 | 0.00000000142570223835179055821524382005847236 |
| 46368/28657 | 1.6180339882053250514708448197648044 | -0.00000000054456979673374201460083370693062428 |
| 75025/46368 | 1.618033988957902001380262249827467 | 0.00000000020800715317567541546182910105125741 |
Values calculated with Mathematica 4.