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
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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Bill Gosper: "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." Albers & Alexanderson put together "More Mathematical People" (1990) which I'm certain I have but can't at the moment locate. Therein are the first few paragraphs. Google Books search results and the annoying snippet views within the book itself sometimes generates sufficient overlap to find the next snippet/text. Toggling thus, I have: Continued fractions are hard to like. People who like continued fractions eat pickled okra and drive Citroens. Books on the subject are filled with dull proofs of dull properties, and recent papers relating continued fractions to computers have bordered on libel. But the literature is not the real problem. Let's face it; a continued fraction is a very awkward object for our intuitions to grasp. Just to estimate the size of a purely numerical continued fraction would seem, at first, to require discarding all but the first few terms, followed by converting to improper fractions in a bottom-to-top repetition. Since it isn't immediately clear how much error we committed by discarding the "tail" we have been penalized for asking even a simple question about size. Do continued fractions suffer from the "observer effect"? Why, if they are so intractable, are we about to attempt arithmetic with them? In fact, modern mathematical writers have denied the feasibility of the idea! Of course, chopsticks are, at first, very awkward objects for our fingers to grasp, and many Chinatown tourists have doubted the feasibility of eating with them. With practice and the proper technique, however, we eventually learn to pity those poor Europeans who must stab their salad greens with a sour-tasting, bent metal object with no moving parts. Such is the pity I feel for everyone who must crunch his numbers decimally, or cast his points to float among electronic registers. I will admit that continued fraction techniques are not the best way to handle everything, but then neither are chopsticks.
participants (3)
-
Allan Wechsler -
Bill Gosper -
Hans Havermann