simont | (no subject)

(no subject)

Mathematical curiosity of the day:

   100/9899     = 0 . 01 01 02 03 05 08 13 21 ...
  1000/998999   = 0 . 001 001 002 003 005 008 013 021 ...
 10000/99989999 = 0 . 0001 0001 0002 0003 0005 0008 0013 0021 ...

And in general, 10^n / (10^2n-10^n-1) displays the Fibonacci numbers in n-digit blocks of its decimal expansion.

Sorry, I just thought that was outstandingly cute.

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http://mathworld.wolfram.com/FibonacciNumber.html

is a worthwhile read on this and associated subjects.

-m-

in particular look at the section on the "curious addition tree" (10/89). i'd guess that your series are going to do the same when you get to fibonacci numbers that are more than n digits long.

the bit i find fascinating about that page is the bit about generating fibonacci numbers in terms of the golden ratio and its lesser-known sibling (does (1-sqrt(5))/2 have its own name, or is it just the discarded root of 1-x-x^2 ?)

-m-

Yes, I'm sure you're right; being rationals, those things have to have repeating decimal expansions, so once they start carrying the carries will somehow arrange to wrap around and turn into a periodic digit sequence (though of course it needn't go right back to the beginning, and neither need it have a period that's a multiple of n). That's magical and cool in itself of course.

Actually, looking at that page (thanks for the link!) I'm far more interested in the generating function bit that appears just above the mention of 10/89; I'd already noticed that the denominators of my fractions (10^2n-10^n-1) looked suspiciously similar to the difference equation satisfied by the Fibonacci numbers, and that generating function looks as if it explains the whole lot rather better than I had.

I don't think (1-sqrt(5))/2 has a big impressive name comparable to "the golden ratio", but when the golden ratio is expressed as phi I usually see the other root expressed as psi. (Of course it's -1/phi or 1-phi or probably several other things like that too, so one might reasonably question the need for a separate symbol at all :-)

Oddly, the expansion of the Fibonacci sequence as a sum of basis solutions to the difference equation (i.e. powers of phi and psi) was the way I arrived at those fractions in the first place. I've known since playing with a calculator when I was at school that dividing 1 by 10^n-k (for small k) gives you a display of the geometric progression of powers of k; for example 1/98 = 0.010204081632... And the other day it occurred to me to apply this fact to the Fibonacci sequence by expressing it as the weighted sum of two GPs and thereby derive an expression for a real number which displayed it in the same way. I was mostly curious to see if all the sqrt(5)s would cancel out of my construction and leave me with a rational; and indeed they did. It was only later that I noticed the curious resemblance between the denominator of the fraction and the difference equation I first thought of, which was rather cool in itself.

Nice to see someone else gives a hoot about this sort of thing at all :-)

agreed on cuteness.
Hmm. must go noseying on that one.
Thanks!

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