Phi and the Fibonacci Series
Leonardo Fibonacci discovered the series which converges on
phi
 In
the 12th century, Leonardo Fibonacci discovered a simple numerical series that is the
foundation for an incredible mathematical relationship behind phi.
Starting with 0 and 1, each new number in the series is simply the sum of
the two before it.
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, . . .
The ratio of each successive pair of numbers in the series approximates
phi (1.618. . .) , as 5 divided by
3 is 1.666..., and 8 divided by 5 is 1.60.
The table below shows how the ratios of the successive numbers in the
Fibonacci series quickly converge on Phi or Φ. After the 40th number in the series,
the ratio is accurate to 15 decimal places.
1.618033988749895 . . .
Compute any number in the Fibonacci Series easily!
You can use phi to compute the nth number in the Fibonacci series (fn):
fn
= Φn / 5½
(This provides an estimate which always rounds to the correct Fibonacci number.)
The ratio of successive Fibonacci numbers converges on phi
Sequence
in the
series |
Resulting
Fibonacci
number
(the sum
of the two
numbers
before it) |
Ratio of each
number to the
one before it
(this estimates
phi or Φ) |
Difference
from
Φ |
|
0 |
0 |
|
|
| 1 |
1 |
|
|
| 2 |
1 |
1.000000000000000 |
+0.618033988749895 |
| 3 |
2 |
2.000000000000000 |
-0.381966011250105 |
| 4 |
3 |
1.500000000000000 |
+0.118033988749895 |
| 5 |
5 |
1.666666666666667 |
-0.048632677916772 |
| 6 |
8 |
1.600000000000000 |
+0.018033988749895 |
| 7 |
13 |
1.625000000000000 |
-0.006966011250105 |
| 8 |
21 |
1.615384615384615 |
+0.002649373365279 |
| 9 |
34 |
1.619047619047619 |
-0.001013630297724 |
| 10 |
55 |
1.617647058823529 |
+0.000386929926365 |
| 11 |
89 |
1.618181818181818 |
-0.000147829431923 |
| 12 |
144 |
1.617977528089888 |
+0.000056460660007 |
| 13 |
233 |
1.618055555555556 |
-0.000021566805661 |
| 14 |
377 |
1.618025751072961 |
+0.000008237676933 |
| 15 |
610 |
1.618037135278515 |
-0.000003146528620 |
| 16 |
987 |
1.618032786885246 |
+0.000001201864649 |
| 17 |
1,597 |
1.618034447821682 |
-0.000000459071787 |
| 18 |
2,584 |
1.618033813400125 |
+0.000000175349770 |
| 19 |
4,181 |
1.618034055727554 |
-0.000000066977659 |
| 20 |
6,765 |
1.618033963166707 |
+0.000000025583188 |
| 21 |
10,946 |
1.618033998521803 |
-0.000000009771909 |
| 22 |
17,711 |
1.618033985017358 |
+0.000000003732537 |
| 23 |
28,657 |
1.618033990175597 |
-0.000000001425702 |
| 24 |
46,368 |
1.618033988205325 |
+0.000000000544570 |
| 25 |
75,025 |
1.618033988957902 |
-0.000000000208007 |
| 26 |
121,393 |
1.618033988670443 |
+0.000000000079452 |
| 27 |
196,418 |
1.618033988780243 |
-0.000000000030348 |
| 28 |
317,811 |
1.618033988738303 |
+0.000000000011592 |
| 29 |
514,229 |
1.618033988754323 |
-0.000000000004428 |
| 30 |
832,040 |
1.618033988748204 |
+0.000000000001691 |
| 31 |
1,346,269 |
1.618033988750541 |
-0.000000000000646 |
| 32 |
2,178,309 |
1.618033988749648 |
+0.000000000000247 |
| 33 |
3,524,578 |
1.618033988749989 |
-0.000000000000094 |
| 34 |
5,702,887 |
1.618033988749859 |
+0.000000000000036 |
| 35 |
9,227,465 |
1.618033988749909 |
-0.000000000000014 |
| 36 |
14,930,352 |
1.618033988749890 |
+0.000000000000005 |
| 37 |
24,157,817 |
1.618033988749897 |
-0.000000000000002 |
| 38 |
39,088,169 |
1.618033988749894 |
+0.000000000000001 |
| 39 |
63,245,986 |
1.618033988749895 |
-0.000000000000000 |
| 40 |
102,334,155 |
1.618033988749895 |
+0.000000000000000 |
|
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