Rational numbers have always fascinated me. I wanted to build a gear train for a science fair where the gears form a loop where the gear train does not mesh. But only after many thousands of revolutions. Then a sign on the front asking people to break the gears by turning the crank.
Long ago the Hammond organ used a gear train to approximate the twelfth root of two and powers of it to build the notes in the octave. Now electronics do that more effectively. The ratio Roger showed in a previous post are too large to be pleasing. Smaller numbers in the ratio could still be useful and still be accurate enough for some purposes. I realize that this request is a little vague, but this is just an extention to the original challenge. On Mar 9, 2016 9:38 AM, "Raul Miller" <rauldmil...@gmail.com> wrote: > That seems a bit underspecified, or open-ended, at the moment. > > For example, pi could be 1p1 (or o.1) or pi could be any of a number > of algorithms: https://en.wikipedia.org/wiki/Category:Pi_algorithms > > Meanwhile, there's also the precision aspect - that could also be > specified in a variety of ways. > > In other words, you might be asking for something like this: > 7%~x:<.0.5+7*1p1 > 22r7 > > Or you might have something different in mind, like perhaps this: > ~.@(#~ (= <./)@:|@:-&1p1),%/~i.100x > 22r7 > > Or, significantly slower, but almost tolerable: > ~.@(#~ (= <./)@:|@:-&1p1),%/~i.1000x > 355r113 > > Or, ... any of a variety of other mechanisms... (and even brute force > search techniques could be made to be significantly more efficient - a > moment's thought would show that incorporating the rounding technique, > for example, could have increased the speed of that last search by > something near a factor of 1000). > > Anyways, if you can say a bit more about what you were aiming for, > maybe we could do a better job of getting there? > > Thanks, > > -- > Raul > > On Wed, Mar 9, 2016 at 11:18 AM, Don Guinn <dongu...@gmail.com> wrote: > > How about rounding to a rational of some precision like pi rounded to > 22r7 ? > > On Mar 9, 2016 8:52 AM, "Kip Murray" <thekipmur...@gmail.com> wrote: > > > >> Thanks, Roger and Raul. Not understanding GCD +. I had a Rube > Goldberg > >> solution involving Format ": and Do ". --Kip > >> > >> On Wednesday, March 9, 2016, Roger Hui <rogerhui.can...@gmail.com> > wrote: > >> > >> > den and num illustrate different ways of computing the same thing, > >> > hopefully in so doing improves understanding. If you want speed 2&x: > is > >> > going to be faster. > >> > > >> > Also, watch this: > >> > > >> > 2 x: o. 1 > >> > 1285290289249 409120605684 > >> > x: o. 1 > >> > 1285290289249r409120605684 > >> > > >> > 2 x:!.0 o. 1 > >> > 884279719003555 281474976710656 > >> > x:!.0 o. 1 > >> > 884279719003555r281474976710656 > >> > > >> > > >> > > >> > > >> > On Wed, Mar 9, 2016 at 7:20 AM, Mike Day <mike_liz....@tiscali.co.uk > >> > <javascript:;>> wrote: > >> > > >> > > Roger, are these (among) recommended and preferred methods for > >> > > recovering num & den, or do they just show an elegant way of > >> > > avoiding 2 x: ? > >> > > > >> > > Thanks, > >> > > > >> > > Mike > >> > > > >> > > > >> > > On 09/03/2016 04:32, Roger Hui wrote: > >> > > > >> > >> x=: %/?2$10^8x > >> > >> x > >> > >> 69904549r40669028 > >> > >> > >> > >> den=: [: % 1 +. ] > >> > >> num=: * den > >> > >> > >> > >> den x > >> > >> 40669028 > >> > >> num x > >> > >> 69904549 > >> > >> > >> > >> > >> > >> > >> > >> > >> > >> On Tue, Mar 8, 2016 at 7:28 PM, Kip Murray <thekipmur...@gmail.com > >> > <javascript:;>> > >> > >> wrote: > >> > >> > >> > >> That's great! It's still a nice puzzle to write your own. --Kip > >> > >>> > >> > >>> On Tuesday, March 8, 2016, Raul Miller <rauldmil...@gmail.com > >> > <javascript:;>> wrote: > >> > >>> > >> > >>> 2 x: 6r4 > >> > >>>> 3 2 > >> > >>>> > >> > >>>> -- > >> > >>>> Raul > >> > >>>> > >> > >>>> > >> > >>>> On Tue, Mar 8, 2016 at 10:04 PM, Kip Murray < > thekipmur...@gmail.com > >> > <javascript:;> > >> > >>>> <javascript:;>> wrote: > >> > >>>> > >> > >>>>> How do you find the numerator and denominator in lowest terms > of a > >> > >>>>> > >> > >>>> rational > >> > >>>> > >> > >>>>> fraction? For example, > >> > >>>>> > >> > >>>>> nd 6r4 > >> > >>>>> 3 2 > >> > >>>>> > >> > >>>>> --Kip Murray > >> > >>>>> > >> > >>>>> > >> > >>>>> > >> > >>>>> -- > >> > >>>>> Sent from Gmail Mobile > >> > >>>>> > >> > ---------------------------------------------------------------------- > >> > >>>>> For information about J forums see > >> > http://www.jsoftware.com/forums.htm > >> > >>>>> > >> > >>>> > >> ---------------------------------------------------------------------- > >> > >>>> For information about J forums see > >> > http://www.jsoftware.com/forums.htm > >> > >>>> > >> > >>> > >> > >>> > >> > >>> -- > >> > >>> Sent from Gmail Mobile > >> > >>> > >> ---------------------------------------------------------------------- > >> > >>> For information about J forums see > >> http://www.jsoftware.com/forums.htm > >> > >>> > >> > >>> > >> ---------------------------------------------------------------------- > >> > >> For information about J forums see > >> http://www.jsoftware.com/forums.htm > >> > >> > >> > > > >> > > > >> > > --- > >> > > This email has been checked for viruses by Avast antivirus software. > >> > > https://www.avast.com/antivirus > >> > > > >> > > > ---------------------------------------------------------------------- > >> > > For information about J forums see > http://www.jsoftware.com/forums.htm > >> > ---------------------------------------------------------------------- > >> > For information about J forums see > http://www.jsoftware.com/forums.htm > >> > >> > >> > >> -- > >> Sent from Gmail Mobile > >> ---------------------------------------------------------------------- > >> For information about J forums see http://www.jsoftware.com/forums.htm > > ---------------------------------------------------------------------- > > For information about J forums see http://www.jsoftware.com/forums.htm > ---------------------------------------------------------------------- > For information about J forums see http://www.jsoftware.com/forums.htm ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
