89, 109 and the Fibonacci Series

The reciprocal of 89, a Fibonacci number, is based on the Fibonacci series

This is a little curiousity involving the number 89, one of the Fibonacci series numbers.

1/89 is a repeating decimal fraction with 44 characters:

.01123595505617977528089887640449438202247191

You can see the beginning of the Fibonacci sequence in the first 6 digits of the decimal equivalent of 1/89. (i.e., 0,1,1,2,3,5 appears as 0.011235..)

If you take each Fibonacci number, divide it by 10 raised to the power of its position in the Fibonacci sequence and add them all together, you get 0.011235955..., the same number as the reciprocal of 89.

Note the Fibonacci series in green

Note the sequence number of the Fibonacci series in red

1 / 89 =

0 / (10 ^ 1 ) +
1 / (10 ^ 2 ) +
1 / (10 ^ 3 ) +
2 / (10 ^ 4 ) +
3 / (10 ^ 5 ) +
5 / (10 ^ 6 ) +
8 / (10 ^ 7 ) +
13 / (10 ^ 8 ) +

. . .

0.011235955... =

0.0 +
0.01 +
0.001 +
0.0002 +
0.00003 +
0.000005 +
0.0000008 +
0.00000013 +

. . .

This relationship was discovered in the fall of 1994 by Cody Birsner, a student at the University of Oklahoma, while doing a term paper on the Fibonacci series.

The reciprocal of 109 is also based on the Fibonacci series, forwards and backwards

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:

.009174311926605504587155963302752293577981651376146788990825688073394495412844036697247706422018348623853211

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
Fibonacci series
in green

Note the
sequence
number of the
Fibonacci series
in red

1 / 109 =

0 / (10 ^ 109 ) +
1 / (10 ^ 108 ) +
1 / (10 ^ 107 ) +
2 / (10 ^ 106 ) +
3 / (10 ^ 105 ) +
5 / (10 ^ 104 ) +
8 / (10 ^ 103 ) +
13 / (10 ^ 102 ) +
21 / (10 ^ 101 ) +
34 / (10 ^ 100 ) +
55 / (10 ^ 99 ) +
89 / (10 ^ 98 ) +
144 / (10 ^ 97 ) +
233 / (10 ^ 97 ) +
377 / (10 ^ 97 ) +

. . .

...18348623853211=

...000000000000000 +
...00000000000001 +
...0000000000001 +
...000000000002 +
...00000000003 +
...0000000005 +
...000000008 +
...00000013 +
...0000021 +
...000034 +
...00055 +
...0089 +
...144 +
...33 +
...7 +

. . .

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 ) -
1 / (10 ^ 2 ) +
1 / (10 ^ 3 ) -
2 / (10 ^ 4 ) +
3 / (10 ^ 5 ) -
5 / (10 ^ 6 ) +
8 / (10 ^ 7 ) -
13 / (10 ^ 8 ) +
21 / (10 ^ 9 ) -

. . .

0.00917431...=

0.0 -
0.01 +
0.001 -
0.0002 +
0.00003 -
0.000005 +
0.0000008 -
0.00000013 +
0.000000021 -

. . .