From the book of Martzloff (of which I have a digital copy), the chapter on Planimetry starts at page 277 and probably gives the formulas which are used to enclose pi between two fractions.
R.E. Boss > -----Original Message----- > From: Chat [mailto:[email protected]] On Behalf Of > Martin Kreuzer > Sent: woensdag 31 januari 2018 13:47 > To: [email protected] > Subject: Re: [Jchat] weak | strong fraction ... > > *** 2nd try *** (hopefully getting the formatting right this time) ... > > Seems that up til now I have been totally unaware of the existence of the > Stern-Brocot tree of fractions or the Mediant ... Thanks for opening up that > door for me. -- > > After some reading I now accept that going through the tree I could > approximate any real number with fractions (taking mediants on the way). > But doesn't that mean I have to know the target in the first place..? > > In the case of pi approximation (in Ancient Chinese Math) the real number > (target) isn't known ... > > If I started out with the interval [3;4] written as fractions 3/1 and > 4/1 (or 'coefficients' 3 1 and 4 1 as they called it, stating circumference- > diameter pairs) > > piacm 3 1;4 1;0 > 3 4 > 1 1 > > and repeatedly take the mediant of the lower interval boundary and the > upper (replacing the upper with the mediant), after 7 steps I'll be past the > (famous) first reasonable approximation 22/7 > > piacm 3 1;4 1;7 > 3 4 7 10 13 16 19 22 25 > 1 1 2 3 4 5 6 7 8 > > Taking the last two thus obtained values as interval boundaries and continue > the process for another 15 steps > > piacm 22 7;25 8;15 > 22 25 47 69 91 113 135 157 179 201 223 245 267 289 311 333 355 > 7 8 15 22 29 36 43 50 57 64 71 78 85 92 99 106 113 > > I'll come up with the (famous) better approximation 355/113, which coincides > with the partial sum of the continued fraction [3;7,15,1] > > ecf 3 7 15 1 > 3.14159 > x: ecf 3 7 15 1 > 355r113 > > [Almost all of these values are shown in the animated picture contained in > the Wikipedia article.] > > Now my question is: > In that algorithm, what would be the indicator (or break-off > condition) for having hit a reasonable result, and which new 'summand' to > use in improving it..? > > -M > > ps: > Citing from the Martzloff book (p. 281): > --- > "... Zu Chongzhi: > '... [Hence] the "close models" mi lu, 113 for the diameter and 355 for the > circumference, and the "rough models" yue lu, 7 for the diameter and 22 for > the circumference [respectively].' ..." > --- > So the terms weak | strong fraction hasn't its origin in that text ... > > > > At 2018-01-29 21:01, you wrote: > > >Guess you are right. > >Brocot (from the Stern-Brocot tree of fractions) came up with the algo > >to produce from two fractions p/q and r/s the fraction (p+r)/(q+s), > >which is in between the two parent fractions. Starting with 0/1 and 1/0 > >you can converge to any positive real number x, where (p+r)/(q+s) > >becomes the new 'weak' or 'strong' fraction dependent of its position > >w.r.t. x. > > > > > >R.E. Boss > > > > > > > -----Original Message----- > > > From: Chat [mailto:[email protected]] On Behalf Of > > > Roger Hui > > > Sent: maandag 29 januari 2018 18:50 > > > To: Chat Forum <[email protected]> > > > Subject: Re: [Jchat] weak | strong fraction ... > > > > > > Guess: p<x and x<q. p and q are rational. p is a "weak > > fraction" and q is a > > > "strong fraction". If you have a process where at each step the > > > weak becomes less weak and the strong becomes less strong, then you > > > are in business. > > > > > > > > > > > > On Mon, Jan 29, 2018 at 9:05 AM, Martin Kreuzer > > > <[email protected]> > > > wrote: > > > > > > > Hi - > > > > > > > > This question goes to you Math people: > > > > > > > > Looking at this article > > > > https://en.wikipedia.org/wiki/Mil%C3%BC > > > > I found (in the last paragraph) the terms "weak" and "strong" fractions. > > > > > > > > I've never before seen this mentioned anywhere and I'm in doubt > > > > whether it is a valid concept. > > > > [nb: I do own the Martzloff (1996) book.] > > > > > > > > Any enlightenment welcome ... > > > > > > > > -M > > > > > > > > ------------------------------------------------------------------ > > > > ---- 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 ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
