I'm pretty sure I have a hardcopy of Gosper's CF paper. Perhaps I will rummage for it tomorrow, though I have already promised Neil Sloane some work. On Sat, Mar 17, 2018 at 7:38 PM, Bill Gosper <billgosper@gmail.com> wrote:
Also, from http://www.tweedledum.com/rwg/cfup.htm
Similarly, if you want 100/2.54, the number of inches per meter, you will find
39 2 1 2 2 1 4
which is much nicer than
39.(370078740157480314960629921259842519685039)
where the part in parentheses repeats forever. (Incidentally, this repeating decimal is easily computed since the remainder of 2 after the quotient digits 3937 ensures that, starting with 7874..., the answer will be just twice the original one, ignoring the decimal point. Thus you just double the quotient, starting from the left (using one digit lookahead for carries), placing the answer digits on the right, so as to make the number chase its tail. This trick is even easier in expansions of ratios where some remainder is exactly 1/nth of an earlier one, for small n. You forget about the divisor and simply start shortdividing by n at the quotient digit corresponding to the earlier remainder. If this seems confusing, forget it--it has nothing to do with continued fractions.)
I just remembered that this writeup used to have a preface that began "Continued fractions are hard to like. People who like continued fractions drive Citroens and eat pickled okra." Followed by some disparagement of people who eat with bent metal objects instead of chopsticks.
I can't Google this anywhere. I bet somebody has a hardcopy. --rwg On 2018-03-17 14:13, Simon Plouffe wrote:
Hello,
we can remark that 142857 is symmetrical, 142 + 857 = 999
so one just has to remember half of the period in order to know all of the digits, 1/13 = 076923 , 076+923 = 999 too.
this phenomena is true whenever p <> 2 or p <> 11 and the period is even.
best regards, Simon Plouffe
Le 2018-03-17 à 21:59, Cris Moore a écrit :
well,
1/13 = 0769/9997 = 0.0769 (1 + 0.0003 + 0.00000009 + …) = 0.0769 2307 …
although this doesn’t quite break up the repetend 076923. You could use the clumsier
1/13 = 076/988 = 0.076 (1 + 0.012 + 0.000144) = 0.076 + 0.000912 + 0.0000109… = 0.076923…
but this involves a bunch of carrying.
Cris
On Mar 17, 2018, at 2:36 PM, James Propp <jamespropp@gmail.com> wrote:
I just figured out for myself a probably well-known trick for deriving/remembering the decimal expansion of 1/7: 1/7 = 14/98 = 14/(100-2) = .14/(1-.02) = .14 + .0028 + .000056 + .00000112 + ... = .142857...
Are there other examples where the repetend of the decimal expansion of 1/n in splits into blocks that are related to this sort of fashion?
Jim Propp
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