[math-fun] Fracfac expansions (fractions in factorial base)
Since high school I've thought factorial base is an interesting way to express real numbers in the unit interval (0,1). That is, since 1/2! + 2/3! + 3/4! + ... = (d/dx((e^x-1)/x))(1) = 1, every x in (0,1) has a unique representation of the form x = Sum_{n=1..oo} c_n/(n+1)! if the integers c_n satisfy 0 <= c_n <= n. E.g., e = 2 + (1,1,1,...) and pi = 3 + (0,0,3,1,5,6,5,0,1,4,7,8,0,...). But I've found very little written on the subject. If anyone can direct me to articles on this subject I'd appreciate it. In particular: Is anything known about fracfac expansions of pi ? Of algebraic numbers like quadratic irrationals? Even rational numbers' fracfac expansions, which clearly terminate, are interesting to look at, especially reciprocal primes. Is there a theorem like the one about Khinchin's constant K for continued fractions? (To wit: For almost every real number, the convergents of its continued fraction have the same constant K = 2.685... as the limit of their geometric mean; see < http://en.wikipedia.org/wiki/Khinchin%27s_constant >.) (Actually, it seems clear that for almost every real number, the coefficients c_n of the fracfac expansion must approach the uniform distribution on {0,1,2,...,n}. More precisely, as n -> oo, the distribution of the ratios c_n/n must approach the uniform distribution on (0,1).) ((( Note: It's a pleasant problem, if you haven't seen it, to calculate f(K) := Sum{n=0..oo} 1/(nK)! ))) --Dan ________________________________________________________________________________________ It goes without saying that .
* Dan Asimov <dasimov@earthlink.net> [Jan 01. 2012 17:16]:
Since high school I've thought factorial base is an interesting way to express real numbers in the unit interval (0,1).
That is, since 1/2! + 2/3! + 3/4! + ... = (d/dx((e^x-1)/x))(1) = 1, every x in (0,1) has a unique representation of the form
x = Sum_{n=1..oo} c_n/(n+1)!
if the integers c_n satisfy 0 <= c_n <= n.
I do not think so: 1/2 = 0.1 = sum( n>=2, n/(n+1)! ) The problem (same as 1.0 = 0.999.. ) stays with all mixed radix representations. For uniqueness I'd should check non-adjacent forms (no two adjacent digits nonzero), e.g., digits 0,+1,-1 in binary.
[...]
Related to this (but not the same) is the rather interesting paper "Profinite Fibonacci Numbers" by Hendrik Lenstra: http://www.math.leidenuniv.nl/~hwl/papers/fibo.pdf The idea is that expressing integers in the factorial base is a good way to describe "Z-hat" the profinite completion of Z. The Fibonacci numbers can be uniquely continued to a piecewise analytic function on "Z-hat". And the paper has some neat pictures. Victor On Sat, Dec 31, 2011 at 7:04 PM, Dan Asimov <dasimov@earthlink.net> wrote:
Since high school I've thought factorial base is an interesting way to express real numbers in the unit interval (0,1).
That is, since 1/2! + 2/3! + 3/4! + ... = (d/dx((e^x-1)/x))(1) = 1, every x in (0,1) has a unique representation of the form
x = Sum_{n=1..oo} c_n/(n+1)!
if the integers c_n satisfy 0 <= c_n <= n.
E.g., e = 2 + (1,1,1,...) and pi = 3 + (0,0,3,1,5,6,5,0,1,4,7,8,0,...).
But I've found very little written on the subject. If anyone can direct me to articles on this subject I'd appreciate it.
In particular: Is anything known about fracfac expansions of pi ? Of algebraic numbers like quadratic irrationals?
Even rational numbers' fracfac expansions, which clearly terminate, are interesting to look at, especially reciprocal primes.
Is there a theorem like the one about Khinchin's constant K for continued fractions? (To wit: For almost every real number, the convergents of its continued fraction have the same constant K = 2.685... as the limit of their geometric mean; see < http://en.wikipedia.org/wiki/Khinchin%27s_constant >.)
(Actually, it seems clear that for almost every real number, the coefficients c_n of the fracfac expansion must approach the uniform distribution on {0,1,2,...,n}. More precisely, as n -> oo, the distribution of the ratios c_n/n must approach the uniform distribution on (0,1).)
((( Note: It's a pleasant problem, if you haven't seen it, to calculate
f(K) := Sum{n=0..oo} 1/(nK)! )))
--Dan
________________________________________________________________________________________ It goes without saying that .
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participants (3)
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Dan Asimov -
Joerg Arndt -
Victor Miller