RE: [math-fun] Lost in fraction
Waow, so quick, thanks, Christoph ! (and forgive my poor math skills !) This fraction has a nice self-referential smell (in french) : 13847/1111 = 12.4635 4635 4635 4635 (a) ... take any digit of (a); when written in french, this digit has as much letters as described by the next digit : UN (2 letters) DEUX (4 letters) QUATRE (6 letters) SIX (3 letters) TROIS (5 letters) CINQ (4 letters) QUATRE (6 letters) ... What could be a similar english fraction ? Best and thanks, É. -----Message d'origine----- De : math-fun-bounces+eric.angelini=kntv.be@mailman.xmission.com [mailto:math-fun-bounces+eric.angelini=kntv.be@mailman.xmission.com] De la part de Pacher Christoph Envoyé : mardi 5 septembre 2006 18:00 À : math-fun Objet : RE: [math-fun] Lost in fraction Hello,
is there a (simple) fraction whose result is : 12.4635 4635 4635 4635 ... ?
12+4635/9999= 13847/1111, isn't it? since it is unique (after canceling), simpler is not possible... best wishes Christoph
This fraction has a nice self-referential smell (in french) : ... ... take any digit of (a); when written in french, this digit has as much letters as described by the next digit : ... What could be a similar english fraction ?
The English-language number -> length mapping has a unique fixed point, where all iterations end up. So we'd get, e.g., 1354444444444. Put the decimal point at a place of your choosing and follow the same procedure as before; so, e.g., x = 1.35444444444... 10x = 13.54444444444... 9x = 12.19000000000... = 1219/100 so x = 1219/900. -- g
So that's kind of interesting. If you start with any English phrase, or any piece of literature for that matter, and repeatedly write the number of letters in the phrase in English, you end up at "four". I wonder what the smallest lengths of text are that survive n iterations... ----- Original Message ----- From: "Gareth McCaughan" <gareth.mccaughan@pobox.com> To: <math-fun@mailman.xmission.com> Cc: "Eric Angelini" <Eric.Angelini@kntv.be> Sent: Tuesday, September 05, 2006 12:29 PM Subject: Re: [math-fun] Lost in fraction
This fraction has a nice self-referential smell (in french) : ... ... take any digit of (a); when written in french, this digit has as much letters as described by the next digit : ... What could be a similar english fraction ?
The English-language number -> length mapping has a unique fixed point, where all iterations end up. So we'd get, e.g., 1354444444444. Put the decimal point at a place of your choosing and follow the same procedure as before; so, e.g.,
x = 1.35444444444... 10x = 13.54444444444... 9x = 12.19000000000... = 1219/100
so x = 1219/900.
-- g
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Eric Angelini wrote:
... take any digit of (a); when written in french, this digit has as much letters as described by the next digit :
UN (2 letters) DEUX (4 letters) QUATRE (6 letters) SIX (3 letters) TROIS (5 letters) CINQ (4 letters) QUATRE (6 letters) ...
In English, the map (n -> # letters in the name of n) always ends up at its unique fixed point: FOUR has 4 letters. One of my early Intelligencer Entertainments columns riffed on the slighly more interesting point of view where you say that the map is, instead, (n -> product of lengths of words in the name of n). So 24 -> "twenty four" -> 6*4 = 24. If you find this amusing, the column is here: http://people.brandeis.edu/~kleber/Papers/twenty-four.pdf --Michael -- It is very dark and after 2000. If you continue you are likely to be eaten by a bleen.
To wrap up this thread, I am adding this to the OEIS. Neil %I A119383 %S A119383 1,2,4,6,3,5,4,6,3,5,4,6,3,5,4,6,3,5,4,6,3,5,4,6,3,5,4,6,3,5,4,6,3, %T A119383 5,4,6,3,5,4,6,3,5,4,6,3,5,4,6,3,5,4,6,3,5,4,6,3,5,4,6,3,5,4,6,3,5, %U A119383 4,6,3,5,4,6,3,5,4,6,3,5,4,6,3,5,4,6,3,5,4,6,3,5,4,6,3,5,4,6,3,5,4 %N A119383 a(1) = 1; for n>1, a(n) = numbers of letters in French name for a(n-1). %C A119383 Memo: 13847/1111 = 12.4635463546354635... %C A119383 The analogue in English is not so interesting: all starting values eventually lead to ...,4,4,4,4,4,... - Gareth McCaughan. %O A119383 1,2 %e A119383 UN (2 letters), DEUX (4 letters), QUATRE (6 letters), SIX (3 letters), TROIS (5 letters), CINQ (4 letters), QUATRE (6 letters), ... %K A119383 nonn,word %A A119383 Eric Angelini (Eric.Angelini(AT)kntv.be), Sep 17 2006
participants (5)
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David Wilson -
Eric Angelini -
Gareth McCaughan -
Michael Kleber -
N. J. A. Sloane