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# 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 Ø.  After the 40th number in the series, the ratio is accurate to 15 decimal places.

### Compute any number in the Fibonacci Series easily!

You can use phi to compute the nth number in the Fibonacci series (fn):

fn =  Øn / 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 Ø) Difference from Ø 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

Phi - The Golden Number
A source to some of Net's "phi-nest" information on the
Golden Section / Mean / Proportion / Ratio / Number,
Divine Proportion, Fibonacci Series and Phi (1.6180339887...)