arithmetic
Posts:
106
From:
venezuela
Registered:
1/23/06


Re: cube root of a given number
Posted:
Jul 21, 2007 11:28 PM


On 21 jul, 17:39, "sttscitr...@tesco.net" <sttscitr...@tesco.net> wrote: > On 21 Jul, 21:38, arithmonic <djes...@gmail.com> wrote: > > > On 16 jul, 04:39, "sttscitr...@tesco.net" <sttscitr...@tesco.net> > > wrote: > > > > I don't think the claim that these methods are in any way > > > new stands up to scrutiny. > > > The idea of Farey dissections is clearly not new. > > > It is mentioned in Hardy and Wright for example. > > . > > > You mentioned "Farey Fractions". > > Do you know what you are talking about? > > and they operate only between TWO FRACTIONS (even when you can compute > > many Mediants at the same time it always operate between TWO > > FRACTIONS). > > Again you reveal your profound ignorance in these > matters. If you had read Hurwitz's paper you would > realize that you can generalize Farey fractions to > simultaneously approximate two irrationals, this > involves either double or triple mediants.
AGAIN: My webpages and book deal with methods for APPROXIMATING ROOTS OF ANY DEGREE OF ANY POSITIVE NUMBER WITH ANY DESIRED CONVERGENCE SPEED which certainly produce best approximations, But you do not want to hear that, you just want to talk about simultaneous approximations of two irrational numbers by means Farey Fractions.
You are talking about methods of finding best rational approximations to any given number, in the same way as anyone could do by using Continued Fractions. All those methods on best approximations using Farey Fractions are basically just the same thing than computing Continued Fractions of second order (as explained in my web pages).
Let me put it clear to you and the sci.math audience:
I challenge you to show to the sci.math audience a very simple arithmetical example on your alleged Hurwitz's method for computing, say, THE FIFHT ROOT OF 2.
Just the FIFHT ROOT OF 2.
Come on, show to the audience such a simple example on Hurtwitz method. Come on. Do not forget not to mention anything related to Continued Fractions.
Hurtwitz DID NOT FIND ANY GENERAL ROOTSOLVING METHOD AND YOU KNOW THAT, but you just want to cause confusion.
Every body knows that so many people have worked with Farey Fractions that way, not only Hurtwitz but many others, the math journals are plenty of articles on working with Farey Fractions.
HOWEVER, to your disgrace, I DO NOT work with Farey Fractions. I do not work with the Mediant and I got not only Newton's, Bernoulli's, Halley's, Householder's methods but many other new algorithms with high convergence speed which can be extende to algebraic equations. YOU CERTAINLY KNOW that all that is far beyond the limited scope of all your statements, some numerical examples are in my webpages and to your disgrace you CAN NOT PREVENT PEOPLE FROM READING THEM. ALL THOSE NEW METHODS HAVE NO PRECEDENTS, AT ALL, and you cannot deny that.
> > The paper you are mentioning deals only with best approximations to > > any given number, but that is far from being a ROOTSOLVING ALGORITHM. > > If you approximate sqrt(2) you are solving x^22 =0
No, you are wrong, Hurtwitz DID NOT find any NEW GENERAL ROOTSOLVING METHODS he was basically working the same thing that has been known since long time ago as CONTINUED FRACTIONS, that is, those things I use to call Continued Fractions of Second Degree, because there are Continued Fractions of Higher degrees.
I challenge you to show to the sci.math audience a very simple arithmetical example on your alleged Hurwitz's method for computing, say, THE FIFHT ROOT OF 2.
Just the FIFHT ROOT OF 2.
Come on, show to the audience such a simple example on Hurtwitz method. Come on. Do not forget not to mention anything related to Continued Fractions.
> > > The issue on "best approximations" is well explained on my web pages. > > Then find the best simultaneous approximations to > cubrt(25), cubrt(625). > > I've made this simple challenge many times in the past > and you were never able to meet it.
I really do not remember to have discussed with you any single line, if you have addressed any message to me in the past be sure it did not arrive at my end. Anyway, I can only say is: read my webpages, and compute all the roots you want and find all the best approximations you desire. My webpages deal exclusively with EXTREMELY SIMPLE ROOTSOLVING METHODS WHICH DO NOT APPEAR IN NEITHER ANY CHINESE, NOR EUROPEAN, NOR ARAB, NOR HINDU, NOR AMERICAN BOOK ON NUMBERS, since Babylonian times up to now, and you are trying to cause confusion by challenging me to solve the cube version of Pell's equation. No, you are wrong, what you see in my postings, my webpages and book is what you get. You are the only one who insists to talk about any simultaneous approximations by agency of Farey Fractions, but my work is about: NEW EXTREMELY SIMPLE ROOTSOLVING METHODS WHICH SURPRISINGLY DO NOT APPEAR IN NEITHER ANY CHINESE, NOR EUROPEAN, NOR ARAB, NOR HINDU, NOR AMERICAN BOOK ON NUMBERS, since Babylonian times up to now. My point has been stated very clear, and be sure that I will not allow any strategy from yours to divert that.
NOW, you are trying to introduce much more confusion by challenging me to solve the cubeversion of Pell's equation.
YOU ARE WRONG, I challenged to you and your friend Grover Hughes to ask the following issues:
1. I challenge you to show such Eshbach's method in this thread, because both of you are trying to state that my methods based on the Rational Mean are the same as the one you read in Eshbach's work ("Handbook of Engineering Fundamentals).
2. I challenged you in this posting to show to the sci.math audience a very simple numerical example on your alleged GENERAL Hurwitz's ROOTSOLVING METHOD for computing, say, THE FIFHT ROOT OF 2.
You have made very specific statements about my methods and I am challenging you to prove them by means of very concrete evidence. Notice that I am challenging you and your friend Grover Hughes with two very simple inquires.
If none of you is able to ask such simple challenges then both of you will have to face the conseequences of your negligence.
> > How can anyone believe what you say, if your > methods cam't produce what you claim.

