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.
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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.
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:
.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.
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Note the Note the |
…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.
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Note the Fibonacci series in green Note the sequence number of the Fibonacci series inred |
1 / 109 = 0 / (10 ^ 1 ) + . . . |
0.00917431…= 0.0 + . . . |
{ 2 comments… read them below or add one }
square 98 twice is 3.14 …but then again shown on this website just about any number squared over and over would usually almost equal the 1.6 phi or usually would almost equal the pi also…which 39+63+96 is 198 and the 36+93+69 is 198 making double 99 or (which also is the 33 66 and 99 also being 198) (I personally think its the Shield of David if made into triangles on a graph which I think has in common with the Fibonacci by being somewhat on a graph but showing dimension or a picture of depth (though the numbers them-self are different as there is an end to the Shield of David number series) which they also lead to the number 144 as 39+36+69 make 144 and also 1/69 is .0144 (which of course is square of 12 and is 1+1+2+3+5)
I found an interesting phenomenon, but I don’t have a proof for it, just a conjecture and I don’t know why it works. I also don’t know what post to post this on so I am posting it on this. I conjecture that if you take any random number, subtract the nearest Fibonacci number from it, and then do it again with the difference of that equation, and then again, so on and so forth, you eventually get a Fibonacci number. For example take the random number 40 for example. The closest Fibonacci number is 34. So take 40 – 34 to get 6. Then take the closest Fibonacci number to the number 6 which is 5. 6-5 which equals 1, which is Fibonacci!