arithmetic
Posts:
106
From:
venezuela
Registered:
1/23/06
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Re: cube root of a given number
Posted:
Jul 21, 2007 11:28 PM
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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 ROOT-SOLVING 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 ROOT-SOLVING ALGORITHM.
>
> If you approximate sqrt(2) you are solving x^2-2 =0
No, you are wrong, Hurtwitz DID NOT find any NEW GENERAL ROOT-SOLVING
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 ROOT-SOLVING
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 ROOT-SOLVING 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 cube-version 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
ROOT-SOLVING 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.
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