Wondering about Domingo's Rational Mean book"

[dgomezm at etheron.net]

You insist to rise differences between Cfs and the Rational
Process, however, there is no difference, at all. You were not aware of
that,
however by this time you certainly  know that Continued fractions are
just a particular case of the rational process ruled by the rational
mean.

I clearly stated that point, moreover, at the beginning of this thread
Kirby Urner´s remark was very much to the point, he got the point and
cleverly told you that this was not a problem of CFs vs. Rational
process.
Continued fractions are ruled by the rational mean as well as
Bernoulli´s, Newton´s and Halley´s, among many other new methods, all
of them are just a rational process ruled by the rational mean (as
stated in my web page and book). That´s why the rational process is a
new fundamental and general concept.

If you need to see some best approximations then go to my web page,
enjoy some of them and do some homework by yourself, there you have
many rational processes i.e.: Bernoulli´s, Newton´s, Halley´s and many
others, I hope you enjoy them and please any question on best
approximations ask to Bernoulli, Newton and Halley, the problem is
that  they were unfortunately  unaware of the __trivial and
obvious__    fact  that their methods can be __trivially and
obviously__   found  by means of simple ARITHMETIC: The Rational
process).
No Geometry, no decimals, no derivatives, no Cartesian system.
JUST the most simple ARITHMETIC.
WORST!, there are no precedents on this matter, at all


Of course, you are free to  continue your self-talk on CFs vs. Rational
Process. You insist to ignore all kind of questions and remarks, and it
is clear that even by this time you haven´t read my web page, so I
shall reply _your_ messages by repeating my previous one.


Again:
If you have some doubts, something to say,  about the rational mean and
the rational process, then all your arguments are straightaway directed
to the CFs, Bernoulli´s, Newton and Halley´s method, because all of
them are ruled by the rational mean (They are just rational processes)
as has been trivially and obviously stated in my book and summarized in
my home page:
www.etheron.net/usuarios/dgomez/default.htm

That´s why it is just striking to realize that western culture needed
more than two or three thousand years to find Bernoulli´s, Newton´s and
Halley´s methods by means of previously inventing the infinitesimal
calculus and the cartesian system, also that all  their most
outstanding mathematicians never realized that such methods could have
been  easily developed by using simple arithmetic.

Just striking, indeed, specially for a humble purchaser
of "rigurous" math books containing so much rigorous talk about best
approximations and roots solving.

Finnally, considering that this must be a two-way discussion I´m sure
you will be so kind and take some time to formulate here those CFs
method you mentioned for finding a "fundamental and general" sequence
of best approximations for some cube roots, I´m sure it will be amusing
and very useful not only for me but for the whole audience of this
newsgroup.
 Also I would like to get any response from you about the other issues
I rised on any  precedents on developing
Bernoulli´s, Newton´s, and specially Halley´s method
by means of very simple arithmetic, if I´m wrong on this then
YOU SHOULD BRING OUT, at least, two or three thousands years,
Oooops...sorry,
I meant to say "two of three thousands references" (books, articles,
etc) on such a very simple arithmetical matter.


Domingo





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