
Re: cube root of a given number
Posted:
Jul 26, 2007 5:33 AM


On 26 Jul, 03:14, arithmonic <djes...@gmail.com> wrote: > On 25 jul, 19:10, "sttscitr...@tesco.net" <sttscitr...@tesco.net> > wrote:
> *************************************************************************** > I have never asserted in any posting from mine nor in my webpages, nor > in my book, nor in any paper that "The methods shown in my webpages > yield ALL THE BEST APPROXIMATIONS". There is nothing by far similar to > such assertion in my webpages and postings.
Not true see below: "The point here is that a Rational Proccess which allow us to trivially develop:
1. Traditional and Generalized Continued Fractions"
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Domingo Gomez Morin
Posts: 215 Registered: 12/3/04 Re: Wondering about Domingo's Rational Mean book" Posted: Jun 9, 2000 12:10 AM Plain Text Reply
In article <8hp1g8$47o$1@barcode.tesco.net>, "Iain Davidson" <Sttscitrans@tesco.net> wrote: > > I don't think anyone would say that Farey dissections are less fundamental > than CFs, they are equivalent and both have optimal properties.
ThatÃÂÃÂ´s right, all that has been said here is that CFs are just a particular case of the rational process, as can be easily seen by means of the following example:
Given a set of two initial fractions: [3/2, 4/3]
According to my definition and notation (see home page), the following very simple iteration algorithm: (the symbol _ means subscript, and "Rm": Rational Mean)
Rm[{3/2, 4/3},{2*3/2*2, 4/3}]=[7/5,10/7] Rm[{7/5, 10/7},{2*7/5*2, 10/7}]=[17/12,24/17] Rm[{17/12, 24/17},{2*17/12*2, 24/17}]=[41/29,58/41] Rm[{41/29, 58/41},{2*41/29*2, 58/41}]=[99/70,140/99] and so on....
is the most simple example of a Rational Process for aproximating the square root of 2 (yielding sets of two fractions whose product is always 2). Thus, as everyone can see, this very simple rational process yields two column of values, the first one (3/2, 7/5, 17/12, 41/29, 99/70,...) and the second one (4/3, 10/7, 24/17, 58/41, 140/99,...)
The first column (best approximations) corresponds to the simple CF representation of the square root of 2.
All this applies for all CFs, that is, not only for simple continued fractions and the particular case Mediant,
all CFs are just a column of values within a rational process of second order, however, there are other rational processes of higher order as can be seen in my home page.
Thus, noone will ever say there is any diference between traditional CFs and the Rational Process. You mentioned above "Farey dissections" (Mediant) and IÃÂÃÂ´m forced again to remark that the Mediant is just a special case of the Rational Mean, so the Rational Process embraces _ALL_ CFs.
The problem on finding all the best approximations for irrational of higher degrees is another matter, in this way you said:
> > However CFs have other useful feature like finding quadratic units, > characterising rational and quadratic surds etc.
Again: the rational process embraces _all_ CFs.
> > If your rational mean method is fundamental and general then it should be > capable > of finding all solutions to the cubic analogue of the Pell Equation > > X^3 + kY^3 + (k^2)Z^3  3kXYZ = 1, k>1 > > by finding best approximations to cubt(k^2):cubrt(k):1 > as CFs do for sqrt(k):1 to solve X^2 kY^2 = 1
I could tell you a very similar statement: "If your CFs of second order is fundamental and general then it should be capable of finding all solutions to the cubic analogue of the Pell Equation". Although the point on best approximations is really amusing and useful, I donÃÂÃÂ´t feel that your particular definition of "fundamental and general" could be taken by far as fundamental and general :). If someone feel compelled to ask me for a higher root method which yields all the best approximations then I could ask him to do the same thing, thatÃÂÃÂ´s just fair.
The point here is that a Rational Proccess which allow us to trivially develop:
1. Traditional and Generalized Continued Fractions 2. BernoulliÃÂÃÂ´s method 3. NewtonÃÂÃÂ´s method 4. HalleyÃÂÃÂ´s method 5. Power series expansions 5. Many other new methods 6. A new point of view on means definition 7. A new definition of irrational numbers and their arithmetical operations
is certainly a __general and fundamental__ concept, moreover, when considering that we are talking about a very simple __ARITHMETICAL__ method (No derivatives, no decimals, no cartesian system) which could have been easily implemented since __ancient times__, unfortunatedly, "from all the evidences", ancient mathematicians were inexplicabily unaware of this general and fundamental concept.
The most general and fundamental news are that from now on our children of scholar age will be able to easily handle, by means of simple sums (simple arithmetic), all those "extremely advanced" methods (NewtonÃÂÃÂ´s and HalleyÃÂÃÂ´s) which have been sold to us as exclusive, exquisite and sophisticated creations of the "divine" cartesian system. That is the mean point here, and _I guess_ that was one of the reasons Proff. Kirby Urner asked for any precedent on the rational process at this newsgroup. In the case he couldnÃÂÃÂ´t get any answer from sci.math on this very specific topic, I would suggest him to look at the HistoriaMatematica mailing list.
I know that all what I say could sound as harshly talk (worst Via internet), however, be sure this not directed against any individual but to the whole actual math scientificsystem and its terrible social consequences. Indeed, IÃÂÃÂ´m very grateful to all of you for all your comments, no matter what they could be. Finnally I must publicy say that IÃÂÃÂ´m greatly impressed, indeed, by the wonderful and amusing KirbyÃÂÃÂ´s web pages at:
http://www.inetarena.com/~pdx4d/ocn/numeracy0.html
This means there is still people who really care for our young people.
Greetings, Domingo Gomez Morin
Date Subject Author 5/30/00 Wondering about Domingo's Rational Mean book Kirby Urner 5/30/00 Re: Wondering about Domingo's Rational Mean book Kirby Urner 5/30/00 Re: Wondering about Domingo's Rational Mean book Iain Davidson 5/31/00 Re: Wondering about Domingo's Rational Mean book Kirby Urner 5/31/00 Re: Wondering about Domingo's Rational Mean book Iain Davidson 6/1/00 Re: Wondering about Domingo's Rational Mean book Kirby Urner 6/7/00 "Re: Wondering about Domingo's Rational Mean book" Domingo Gomez Morin 6/8/00 Re: Wondering about Domingo's Rational Mean book" Iain Davidson 6/9/00 Re: Wondering about Domingo's Rational Mean book" Domingo Gomez Morin 6/10/00 Re: Wondering about Domingo's Rational Mean book" Iain Davidson 6/12/00 Re: Wondering about Domingo's Rational Mean book Domingo Gomez Morin 6/12/00 Re: Wondering about Domingo's Rational Mean book" Domingo Gomez Morin 6/13/00 Re: Wondering about Domingo's Rational Mean book" Iain Davidson 6/14/00 Re: Wondering about Domingo's Rational Mean book" Domingo Gomez Morin 6/14/00 Re: Wondering about Domingo's Rational Mean book" Iain Davidson 6/14/00 Re: Wondering about Domingo's Rational Mean book" Domingo Gomez Morin 6/14/00 Re: Wondering about Domingo's Rational Mean book" Huaiyu Zhu 6/17/00 Re: Wondering about Domingo's Rational Mean book" Domingo Gomez Morin 6/16/00 Re: Wondering about Domingo's Rational Mean book" Iain Davidson 6/17/00 Re: Wondering about Domingo's Rational Mean book" Domingo Gomez Morin 6/18/00 Re: Wondering about Domingo's Rational Mean book" david_ullrich@mydeja.com 6/19/00 Re: Wondering about Domingo's Rational Mean book" Domingo Gomez Morin 6/20/00 Is this new, rational means? Milo Gardner 6/21/00 Re: Is this new, rational means? Domingo Gomez Morin 6/22/00 inverse golden proportion Milo Gardner 6/22/00 Re: inverse golden proportion Domingo Gomez Morin 6/23/00 Archimedes' finite numeration system, and calculus Milo Gardner 6/29/00 Wondering about DomingoÃÂ´s rational mean book Domingo Gomez Morin 6/30/00 Archimedes used Egyptian finite arithmetic to solve 4A/3 and n/pq conversion Milo Gardner 7/1/00 Re: Archimedes used Egyptian finite arithmetic to solve 4A/3 and n/pq conversion Domingo Gomez Morin 7/1/00 Re: Is this new, rational means? Domingo Gomez Morin 5/31/00 Re: Wondering about Domingo's Rational Mean book Kirby Urner 6/7/00 "Re: Wondering about Domingo's Rational Mean book" Domingo Gomez Morin
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