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| GoldenNumber.net |
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89, 109 and the Fibonacci Series
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Note the Fibonacci series in green Note the sequence number of the Fibonacci series in red |
1 / 89 = 0 / (10 ^ 1 ) + . . . |
0.011235955... = 0.0 + . . . |
John Hulbert, who has published on similar sequences in the Bulletin of the Institute of Mathematics and its Applications, notes that this relationship was presented in in the book "The Spell of Mathematics" by W.J.Reichmann, published by Pelican books in 1972.
Here's another curiousity involving the number 109, discovered and contributed (10/20/2003) by Rick Toews.
1/109 is a repeating decimal fraction with 108 characters:
.009174311926605504587155963302752293577981651376146788
990825688073394495412844036697247706422018348623853211
You can see the beginning of the Fibonacci sequence in the LAST 6 digits of the decimal equivalent of 1/109, appearing in REVERSE order starting from the END of the decimal. (i.e., 0,1,1,2,3,5, 8 appears as ...853211)
If you take each Fibonacci number, divide it by 10 raised to the power of 109 MINUS its position in the Fibonacci sequence (starting with 0) and add them all together, you get the reciprocal of 109.
Note the Note the |
1 / 109 =
0 / (10 ^
109
) + . . . |
...18348623853211=
...000000000000000 + . . . |
Lastly, here's one more curiosity involving the number 109.
If you take each Fibonacci number, divide it by 10 raised to the power of its position in the Fibonacci sequence and add and subtract each alternate term together, you get .00917431... again, the reciprocal of 109.
Note the Fibonacci series in green Note the sequence number of the Fibonacci series in red |
1 / 109 = 0 / (10 ^ 1
) + . . . |
0.00917431...= 0.0
+ . . . |
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