logo www.randallfly.net

Fibonacci Numbers

The Fibonacci Numbers - Click here to open an Excel spreadsheet containing the Fibonacci Series.

Fib(3) means the third Fibonacci Number in the series.  So, 144 is the 13th number in the series, or Fib(13), or F13.
F1 0 Reduction ad Infinitum means you add all the digits in each fibonacci number and repeat until the result is a single-digit number.

Now, the first 24 fib numbers create a pattern that repeats up through the Fibonaccia Series to infinity!
The numbers are
0, 1, 1, 2, 3, 4, 8, 4, 3, 7, 1, 8,
9, 8, 8, 7, 6, 4, 1, 5, 6, 2, 8, 1


For example, the 24th and final number is 1, arrived at by applying Reduction ad Infinitum to the 24th Fib Number.
ie, 28,657 = 2 + 8 + 6 + 5 + 7 = 10 + 11 + 7 = 21 + 7 = 28 = 2 + 8 = 10 = 1 + 0 = 1   whew!

So, if you apply Reduction ad Infinitum to Fib Numbers 25 thru 48, you will get the exact same series as above.


Further, if you take the final digit of the first 60 Fib #'s, a patter emerges here, too, that repeats up thru infinity. We say the final digit has a cycle of 60.  The final 2 digits have a cycle of 150.  The final 3 digits have a cycle of 1,500.  The final 4 digits have a cycle of 15,000; and they all repeat forever into infinity!
F2 1
F3 1
F4 2
F5 3
F6 5
F7 8
F8 13
F9 21
F10 34
F11 55
F12 89
F13 144
F14 233
F15 377
F16 610
F17 987
F18 1597
F19 2584
F20 4181
F21 6765
F22 10946
F23 17711
F24 28657
Ok, let's get a little deeper.  By some freakish chance, the fraction 1/89 is intimately related to the Fibonacci Series.  It is a repeating decimal fraction with 44 digits.   Here it is:

1/89 = .01123595505617977528089887640449438202247191

You can already see the Fib Series emerging in the first 6 digits.  Now, here's the insane part: take the first number in the Fib Series, 0, and divide it by 10 to the power of 1.
Take the 2nd #, 1, and divide it by 10 to the power of 2.
Divide the 3rd #, 1, by 10 to 3rd power.  etc,

0/E+1 is how you write Zero divided by 10 to the first power.
Continue thru Fib(44) ...
Now add all these results together, and you get 1/89!!!!!!!!

(0/E+1) + (1/E+2) + (1/E+3) + (2/E+4) + (3/E+5) + (5/E+6) + (8/E+7) + ...

0.0 + 0.01 + 0.001 + 0.0002 + 0.00003 + 0.000005 + 0.0000008 +...
= 1/89 = .01123595505617977528089887640449438202247191

0
1
1
2
3
5
8
13
21
34
55
89
144
233
377
610
987
1597
2584
4181
6765
10946
17711
28657 - Fib(24)
46368
75025
121393
196418
317811
514229
832040
1346269
2178309
3524578
5702887
9227465
14930352
24157817
39088169
63245986
102334155
165580141
267914296
433494437
701408733
1134903170
1836311903
2971215073
4807526976
7778742049
12586269025
20365011074
32951280099
53316291173
86267571272 is Fib(55)
139,583,862,445
225,851,433,717
365,435,296,162
591,286,729,879
956,722,026,041
1,548,008,755,920
2,504,730,781,961
4,052,739,537,881
6,557,470,319,842
10,610,209,857,723
17,167,680,177,565
27,777,890,035,288
44,945,570,212,853
72,723,460,248,141
117,669,030,460,994
190,392,490,709,135
308,061,521,170,129
498,454,011,879,264
806,515,533,049,393 is Fib(74)
Here are the first 1,000 digits of Phi.  Phi is the ratio of two neighboring Fibonacci Numbers.   The higher you go in the Series, the closer you get to Phi.  This fraction goes on forever without repeating, just like Pi.

1.6180339887498948482045868343656381177 2030917980576286213544862270526046281890 2449707207204189391137484754088075386891 75212663386222353693179318006076672635 4433389086595939582905638322661319928290 26788067520876689250171169620703222104 32162695486262963136144381497587012203 4080588795445474924618569536486444924104 4320771344947049565846788509874339442212 54487706647809158846074998871240076521 70575179788341662562494075890697040002812 1042762177111777805315317141011704666 59914669798731761356006708748071013179 5236894275219484353056783002287856997829 7783478458782289110976250030269615617002 50464338243776486102838312683303724292 675263116533924731671112115881863851331 620384005222165791286675294654906811317 159934323597349498509040947621322298101 726107059611645629909816290555208524790 352406020172799747175342777592778625619 432082750513121815628551222480939471234 145170223735805772786160086883829523045 926478780178899219902707769038953219681 986151437803149974110692608867429622675 7560523172777520353613936