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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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Iain Davidson

Posts: 1,173
Registered: 12/12/04
Re: cube root of a given number
Posted: Jul 26, 2007 5:33 AM
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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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Math Forum » Discussions » sci.math.* » sci.math
Topic: Wondering about Domingo's Rational Mean book
Replies: 32 Last Post: Jul 1, 2000 10:28 PM
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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.

That´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 I´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 don´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, that´s just fair.

The point here is that a Rational Proccess which
allow us
to trivially develop:

1.- Traditional and Generalized Continued Fractions
2.- Bernoulli´s method
3.- Newton´s method
4.- Halley´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 (Newton´s and Halley´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 couldn´t get any answer from
sci.math on
this very
specific topic, I would suggest him to look at the
Historia-Matematica 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-
scientific-system
and its terrible social consequences.
Indeed, I´m very grateful to all of you for
all your
comments, no matter
what they could be.
Finnally I must publicy say that I´m greatly
impressed,
indeed,
by the wonderful and amusing Kirby´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@my-deja.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 Domingo´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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