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 ) + . . . | 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.
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.
Note the Fibonacci series in green Note the sequence number of the Fibonacci series inred | 1 / 109 = 0 / (10 ^ 1 ) + . . . | 0.00917431…= 0.0 + . . . |
j says
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)
Jamie says
It is because of the fact that .98 + .01 is .99 but .99 + .01 is the whole number 1, causing change from 9 to 0 in both places where the previous addition of the same unit only caused change in one of them. Pi can be explained by this as well… basically pi makes it so that 1 is instead .99 thus a natural perfect third is inherent in the unit and instead of a continuous nonsensical error that contaminates complex calculations, you end up with a normalized change in the relevant decimal places (10000) and apply the expanding and contracting decimal remained we see when multiplying pi by steadily increasing numbers. This is because the ancients were brilliant and recognized the initial flaw I described, realized the implications and abstracted a way to take these uniquely occurring extra decimal digits and make them floating digits in an expanding and contracting decimal remainder. It is all abstraction. Remember these symbols are simply ways to keep track of things. It is not something we discovered it is something we invented out of an overwhelming need for some kind of structured way to track it.
Kilogram says
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!
Ian1/89 says
Look up Kaprekar’s constant and his process its kind of similar but not related to the fib sequence.
Random Name says
if you are always subtracting smaller Fibonacci numbers, then wouldn’t the difference either make a Fibonacci number >1 or 1 which are both Fibonacci numbers. This would be true because a number-a smaller number is positive so you will never reach a negative number.
Michael says
Found a counter example. Try 284
Me says
This is a consequence of Zeckendorf’s theorem: https://en.wikipedia.org/wiki/Zeckendorf%27s_theorem
Anonymous says
This is always trivially true. Any time you subtract the nearest fibonacci number, you get a smaller number (In amplitude). So you must get down to d0wn to 0. To get 0, you must subtract a number from itself. So the previous step was a fibonacci number.
Jess Tauber says
Hi. If you look at: http://scienceforums.com/topic/25035-the-189-lemma/ you will see that 1/109 relates to the base 11 equivalent of the Fib series. In the past week I discovered that ALL base 10 generalized Fib series when decimalized like 1/89 are also based on multiples of 1/89. For ex. the Lucas number decimalization is 19/89. But if we go up higher hierarchically to the generalized Metallic Means (of which the Golden Mean is just one, relating to ratios of contiguous generalized Fib), the sequences giving these are just as regular, in base 10 anyway. For the Silver Mean 1+sqrt2 the basis decimalization fraction is 1/79, for generalized Pell numbers, where TWO copies of any later term are added to ONE copy of the previous term (for gen. Fib ONE for ONE) to generate the next. For the next Copper/Bronze Mean THREE or FOUR copies for ONE previous, they get 1/69 and 1/59 as basis fractions for decimalization. And so it goes. I’ve been unable to find much of anything online about all this and am still waiting to hear back from professional mathematicians specializing in Phi, Fib, and Metallics. The next question is whether this can be generalized to other bases than 10, since the Fiblike fraction can, and also even higher up hierarchically to equations of Pisot-Vijayaraghavan numbers related to the Metallic Means.
Max Bazovsky says
Wow. The wonders of mathematical relationships and interelationships are dazzelingly beautiful and very inspiring..
W. van Zeeland says
@ Kilogram
It seams it ends with 1
Two examples:
134 – 89 = 45, 45 – 34 = 11, 11 – 8 = 3. 3 – 2 = 1
61 – 55 = 6, 6 – 5 = 1
So it’s kind of a hailstone-series!
My english is a little rusty, so what do you mean by : moderation?
“Moderation” means that comments are awaiting review and approve by the site moderator, which per http://www.merriam-webster.com/dictionary/moderator is “ someone who leads a discussion in a group and tells each person when to speak : someone who moderates a meeting or discussion.” All comments on this site are reviewed to assure that they are respectful, relevant and not outright spam.
If I meet with a Fib.number more than 1 (cf. 11 – 8) then I’m sure to end with the number 1, of course.
It’s also possible to work your way up. If you want to arrive at a certain Fib. number you can start the other way around.
P. e. : Say you want to arrive at Fib. 89, than you add a different Fib. p.e. 233, go 89 + 233, them add 377, 610 and so on.Of course you can make bigger steps.
So all we want now is a decent (preferably short) proof of the conjecture!
From a limit(less) point of view, 1/109 = the infinite sum of 110^-n
So 1/110 + 1/12100 + 1/1331000… etc. = 1/109
This is directly related to Pascal’triangle, since the numerators ad up right to left:
….1331000 + 12100 + 110 + 1 (/110^n)
The decimal expansion of 1/89 can also be viewed as: 90^n/1/90^-n
…..8100 + 90 + 1 + 1/90 + 1\8100….
Notice that :
11/89 = ,11235…+ ,011235…= ,1235…
21/89 = ,235,,,
32/89 = ,35…
53/89 = ,5…
10-1=9
20-3=17
50-8=42
130-21=109
The product of 1/89 and 1/109 I.e. 1/9701 is also very interesting. 10/9701= 0,010308215… (every second Fibonacci number).
99/9701=0,0102051334…
(correction: 10/9701= 0,0010308215….)