RE: [math-fun] Decimal question
Christian writes: << With a = 17537 = prime number, And b = 12800(...0) = 2^(9+k) * 5^(2+k), my last family produce always reduced fractions a/b, i.e. on (0, 1): 17537/128000 = 0.1370078125
Very nice example. Now I wonder if there's an example p/q in lowest terms, with 1/10 < p/q < 1, so that the most compact way of writing the decimal uses the same digits (with the same multiplicity). (By compact I mean no leading or trailing zeroes.) Okay, we can start with binary. How about a binary fraction p/q in lowest terms, 1/2 < p/q < 1, with its compact binary representation using the same number of 0's & 1's as there are in p_2 & q_2 (with multiplicity) ? --Dan
--- Dan wrote: Now I wonder if there's an example p/q in lowest terms, with 1/10 < p/q < 1, so that the most compact way of writing the decimal uses the same digits (with the same multiplicity). (By compact I mean no leading or trailing zeroes.) 322673/512000 = .630220703125 Then: 322673/5120000 = .0630220703125 322673/51200000 = .00630220703125 ... In France, we never use this way to write decimal numbers. We need always digits before our decimal point... in fact a comma. Christian.
You know that the most compact solution in base 10 is: 5/2 = 2.5 But in other bases? I have now the most compact solutions of all (non-prime) bases <= 16. For example, in base 6, the most compact solution is: 35/4 = 5.43 Hmmm... difficult to read with our "decimal eyes"... The same irreducible fraction written in base 10: 23/4 = 5.75 In fact, only ONE exception: in the small base 4, I have not yet found any solution. Is the problem impossible in base 4, for an unknown reason????? Christian.
Base 4 is not impossible, most compact solutions found. Completely incredible, they have 14 digits!!! So strange. If somebody can check, I hope to have not missed easiest solutions in base 4. Including the crazy results on base 4, including the sudoku-like solution 124,983/576 = 216.984 375 all the main numerical results on this decimal question are now in this page: http://cboyer.club.fr/FractionsDigits.htm And an Excel file can be downloaded from this page, including the full list of solutions from 2 to 8 digits, starting by the solution 5/2 = 2.5 (using our classical base 10). David, thanks for your interesting question. Christian. -----Message d'origine----- De : math-fun-bounces+cboyer=club-internet.fr@mailman.xmission.com [mailto:math-fun-bounces+cboyer=club-internet.fr@mailman.xmission.com] De la part de Christian Boyer Envoyé : lundi 25 septembre 2006 21:57 À : 'math-fun' Objet : RE: [math-fun] Decimal question You know that the most compact solution in base 10 is: 5/2 = 2.5 But in other bases? I have now the most compact solutions of all (non-prime) bases <= 16. For example, in base 6, the most compact solution is: 35/4 = 5.43 Hmmm... difficult to read with our "decimal eyes"... The same irreducible fraction written in base 10: 23/4 = 5.75 In fact, only ONE exception: in the small base 4, I have not yet found any solution. Is the problem impossible in base 4, for an unknown reason????? Christian. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
I don't see how the base 4 solutions you've posted can work. The denominators have prime factors other than 2 that are not in the numerators, so how can their quaternals (?) terminate? --ms Christian Boyer wrote:
Base 4 is not impossible, most compact solutions found. Completely incredible, they have 14 digits!!! So strange. If somebody can check, I hope to have not missed easiest solutions in base 4.
Including the crazy results on base 4, including the sudoku-like solution 124,983/576 = 216.984 375 all the main numerical results on this decimal question are now in this page:
http://cboyer.club.fr/FractionsDigits.htm
And an Excel file can be downloaded from this page, including the full list of solutions from 2 to 8 digits, starting by the solution 5/2 = 2.5 (using our classical base 10).
David, thanks for your interesting question.
Christian.
-----Message d'origine----- De : math-fun-bounces+cboyer=club-internet.fr@mailman.xmission.com [mailto:math-fun-bounces+cboyer=club-internet.fr@mailman.xmission.com] De la part de Christian Boyer Envoyé : lundi 25 septembre 2006 21:57 À : 'math-fun' Objet : RE: [math-fun] Decimal question
You know that the most compact solution in base 10 is:
5/2 = 2.5
But in other bases? I have now the most compact solutions of all (non-prime) bases <= 16. For example, in base 6, the most compact solution is:
35/4 = 5.43 Hmmm... difficult to read with our "decimal eyes"... The same irreducible fraction written in base 10: 23/4 = 5.75
In fact, only ONE exception: in the small base 4, I have not yet found any solution. Is the problem impossible in base 4, for an unknown reason?????
Christian.
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Oooops, you are perfectly right. Not checked this result directly given by the computer, I was too happy to have seen something on the screen this morning... This problem on base 4 is a precision problem, 14 digits were too long for my program. Sorry. I have already deleted the result on the base 4 in http://cboyer.club.fr/FractionsDigits.htm Who can provide at least one solution in base 4? Christian. -----Message d'origine----- De : math-fun-bounces+cboyer=club-internet.fr@mailman.xmission.com [mailto:math-fun-bounces+cboyer=club-internet.fr@mailman.xmission.com] De la part de Mike Speciner Envoyé : mardi 26 septembre 2006 17:23 À : math-fun Objet : Re: [math-fun] Decimal question I don't see how the base 4 solutions you've posted can work. The denominators have prime factors other than 2 that are not in the numerators, so how can their quaternals (?) terminate? --ms Christian Boyer wrote:
Base 4 is not impossible, most compact solutions found. Completely incredible, they have 14 digits!!! So strange. If somebody can check, I hope to have not missed easiest solutions in base 4.
Including the crazy results on base 4, including the sudoku-like solution 124,983/576 = 216.984 375 all the main numerical results on this decimal question are now in this page:
http://cboyer.club.fr/FractionsDigits.htm
And an Excel file can be downloaded from this page, including the full list of solutions from 2 to 8 digits, starting by the solution 5/2 = 2.5 (using our classical base 10).
David, thanks for your interesting question.
Christian.
-----Message d'origine----- De : math-fun-bounces+cboyer=club-internet.fr@mailman.xmission.com [mailto:math-fun-bounces+cboyer=club-internet.fr@mailman.xmission.com] De la part de Christian Boyer Envoyé : lundi 25 septembre 2006 21:57 À : 'math-fun' Objet : RE: [math-fun] Decimal question
You know that the most compact solution in base 10 is:
5/2 = 2.5
But in other bases? I have now the most compact solutions of all (non-prime) bases <= 16. For example, in base 6, the most compact solution is:
35/4 = 5.43 Hmmm... difficult to read with our "decimal eyes"... The same irreducible fraction written in base 10: 23/4 = 5.75
In fact, only ONE exception: in the small base 4, I have not yet found any solution. Is the problem impossible in base 4, for an unknown reason?????
Christian.
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Christian Boyer -
Daniel Asimov -
Mike Speciner