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The golden ratio $\phi$ 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 divided by 89 is probably good enough for most practical purposes.
| $F_n$ |
$F_{n - 1}$ |
$\frac{F_n}{F_{n - 1}}$ to 20 decimal places |
Approximation error |
| 1 |
1 |
1.00000000000000000000 |
-0.61803398874989484820 |
| 2 |
1 |
2.00000000000000000000 |
0.38196601125010515179 |
| 3 |
2 |
1.50000000000000000000 |
-0.11803398874989484820 |
| 5 |
3 |
1.66666666666666666666 |
0.04863267791677181846 |
| 8 |
5 |
1.60000000000000000000 |
-0.01803398874989484820 |
| 13 |
8 |
1.62500000000000000000 |
0.00696601125010515179 |
| 21 |
13 |
1.61538461538461538461 |
-0.00264937336527946358 |
| 34 |
21 |
1.61904761904761904761 |
0.00101363029772419941 |
| 55 |
34 |
1.61764705882352941176 |
-0.00038692992636543643 |
| 89 |
55 |
1.61818181818181818181 |
0.00014782943192333361 |
| 144 |
89 |
1.61797752808988764044 |
-0.00005646066000720775 |
| 233 |
144 |
1.61805555555555555555 |
0.00002156680566070735 |
| 377 |
233 |
1.61802575107296137339 |
-0.00000823767693347481 |
| 610 |
377 |
1.61803713527851458885 |
0.00000314652861974065 |
| 987 |
610 |
1.61803278688524590163 |
-1.20186464894656524257 |
| 1597 |
987 |
1.61803444782168186423 |
0.00000045907178701603 |
| 2584 |
1597 |
1.61803381340012523481 |
-0.00000017534976961338 |
| 4181 |
2584 |
1.61803405572755417956 |
0.00000066977659331361 |
| 6765 |
4181 |
1.61803396316670652953 |
-0.00000002558318831866 |
| 10946 |
6765 |
1.61803399852180339985 |
0.00000000977190855164 |
| 17711 |
10946 |
1.61803398501735793897 |
-0.00000000373253690923 |
| 28657 |
17711 |
1.61803399017559708655 |
0.00000000142570223835 |
| 46368 |
28657 |
1.61803398820532505147 |
-0.00000000054456979673 |
| 75025 |
46368 |
1.61803398895790200138 |
0.00000000020800715317 |
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(view preamble | get metadata)
Cross-references: decimal places, division, numbers, approximations, Fibonacci number, irrational number, golden ratio
This is version 1 of values of  for  , born on 2007-07-14.
Object id is 9767, canonical name is ValuesOfFracF_nF_n1For1N26.
Accessed 749 times total.
Classification:
| AMS MSC: | 11B39 (Number theory :: Sequences and sets :: Fibonacci and Lucas numbers and polynomials and generalizations) | | | 40A05 (Sequences, series, summability :: Convergence and divergence of infinite limiting processes :: Convergence and divergence of series and sequences) |
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Pending Errata and Addenda
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