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Topic: cube root of a given number
Replies: 112   Last Post: Jan 10, 2013 1:39 PM

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 arithmetic Posts: 106 From: venezuela Registered: 1/23/06
Re: cube root of a given number
Posted: Jul 21, 2007 4:38 PM

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.

yours, because I cannot realize if your remarks are the consequence
of your ignorance on the methods shown in my webpages, or you just
want to disturb people and cause confusion by making false
statements.

You mentioned "Farey Fractions".
Do you know what you are talking about?

I have never used any "Farey Fractions" in any of my methods, so I
cannot understand why you are mentioning them as the center of the
issue. Farey Fractions have been restricted only to reduced
fractions,
and they operate only between TWO FRACTIONS (even when you can compute
many Mediants at the same time it always operate between TWO
FRACTIONS). I cannot believe you dare
methods based on
the Rational Mean which is not restricted to reduced fractions and
operate on any set
of fractions.

> Hurwitz wrote a paper "Ueber die Irrationalzahlen"
> in the 1890s which describes a "mediant" method based on Farey
> fractions that produces best rational approximations.

The paper you are mentioning deals only with best approximations to
any given number, but that is far from being a ROOT-SOLVING ALGORITHM.
You are not bringing any reference to a ROOT-SOLVING ALGORITHM by far
similar to those shown in my web pages, worst, you are not bringing,
at all, any reference to any ROOT-SOLVING ALGORITHM. So, it is clear
that you don't know what you are talking about.

The issue on "best approximations" is well explained on my web pages.
It is well known that the fundamental law for generating the
convergents of continued fractions
is the "Mediant" which is the operation which rules Farey Fractions.
The fact that the Mediant is a particular case of the Rational Mean
does not mean nothing. All those well-known methods for finding best
rational approximations that you metined are basically "Continued
Fractions of the second degree" (all this is fully defined and
explained in my webpages, but it is clear that you have not read any
single bit of it)

Indeed, I cannot believe you dare to talk about all this without
taking care of what you read and say.
Indeed, it seems you even have not read any single bit of my webpages.
The only reason I can find for your making false statements and
story really hurt you. I'm so sorry for that, but I have to continue
doing so, because all that is the crude Truth: It is strinking to
realize that such simple and trivial methods do not appear in any book
on numbers since Babylonian times up to now. I know that can really
hurt some people so much, sorry for that.

> Monkmeyer and Mahler have examined generalizations
> of Farey fractions, essentially a higher order
> "mediant" method, intended to produce best rational
> simultaneous approximations to a set of irrationals.

Again: You are referring to best approximations to any ginven number,
those mathematicians you mentioned didn't make any ROOT-SOLVING
METHOD. To find best approximations to any given number is far from
being similar to a GENERAL ROOT-SOLVING METHOD.

> Can Morin find best rational approximations
> to cubrt(2), cubrt(4) with his methods ?

YES, by means of these so TRIVIAL methods everybody CAN, even young
students at secondary schools.

It is finally clear that you have never read any single bit of my web
pages.
These new methods based on the Rational Means embrace --apart from
many other new algorithms--
all the well known Bernoulli's, Newton's, Halley's, Householder's
methods.
Many examples are shown in my web pages, you can compute any root of
any number with any convergence speed.
The examples are shown in my webpages, do not ask me to explain all

Your statements are the main reason for the title of my posting:
"DEDICATED TO YOUNG MATH STUDENTS..."
mainly because young people are usually eager to find new things and
try to read before talking about any issue, young people do not
pretend to be "experts" but just to learn and to think and that is
really wonderful, that is a virtue that many others have lost trough
the years that have them passed by.

> He has not been able to do so in the past.- Ocultar texto de la cita -
>

That's not true. Again it is finally clear that you have never read
any single bit of my web pages.

The only reason I can find for your making false statements and
story really hurt you. I'm so sorry for that, but I have to continue
doing so, because all that is the crude Truth: It is strinking to
realize that such simple and trivial methods do not appear in any book
on numbers since Babylonian times up to now. I know that can really
hurt some people so much, sorry for that.

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