Wondering about Domingo's Rational Mean book"

[paraguana at my-deja.com]

In article <8i5vsn$llr$71 at epos.tesco.net>,
  "Iain Davidson" <Sttscitrans at tesco.net> wrote:
>
> > "Iain Davidson" <Sttscitrans at tesco.net> replied:
> >
>
> It just seems to be a variant on the method used in schools to find
square

That _very particular_ example of rational process for approximating
the square root has been analyzed by using the Mediant, by many
mathematicians as I previously pointed out, i.e: Chuquet, J. Ortega,
Heath, etc.

> The same idea can obviously be extended to
> higher degree polynomials but conditions for
> convergence may be complicated.

Of course!, it is pretty obvious how to extend such very particular
case of the new general and fundamental rational process concept.
WORST!,  it is pretty obvious how to find Bernoulli´s, Newton and
Halley´s method (among many other new algorithms) by agency of
this fundamental and general principle of Arithmetic: The Rational Mean.
No Geometry, no decimals, no derivatives, no Cartesian system.

Of course, that´s pretty obvious!

So,
If you have some doubts, something to say,  about the rational mean and
the rational process, then 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 as has been trivially and
obviously stated in my book and summarized in my home page:
www.etheron.net/usuarios/dgomez/

All of them are just rational processes ruled by the rational mean.

Of course it is fairly obvious!. That´s why I find of so much
importance to consider the striking fact that western culture needed
more than two or three thousands years to find Bernoulli´s, Newton´s
and Halley´s methods by means of 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.

Now, considering that this not a one-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 sequence of best approximations for the
cube root of any number, 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 about the other topics
I have mentioned on the precedents and history of developing
Bernoulli´s, Newton´s, and specially Halley´s method
by means of simple arithmetic, if I´m wrong on this then
there must be, at least, two or three thousands years, 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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