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NEW EXTREMELY SIMPLE ARITHMETICAL ROOT-SOLVING METHODS
http://mipagina.cantv.net/arithmetic/rmdef.htm
A brief summary on extremely simple higher-order root-solving
algorithms established by agency of the simple arithmetical operation:
'Rational Mean' which is a general and unifying principle of
Quantity that rules not only Bernoulli's, Newton's, Halley's,
Householder's methods and continued fractions, but many other new
higher-order iterating functions, as well as all known means:
Arithmetic, Harmonic, Geometric, Golden, Mediant, etc.
It is a real SHAME to realize that from all the evidence at hand, none
of these extremely simple high-order arithmetical methods based on the
Rational Mean appear in neither any Chinese, nor Hindu, nor Arabic, nor
European, nor Inca, nor Mayan text on numbers -- since Babylonian
times up to now--. Including, of course, modern scholar texts on
numbers, as well as all kind of modern encyclopedias (including many
new web-electronic ones).
A real SHAME, indeed. It is very difficult to assimilate the arrogance
of some people who consider themselves as "experts" on numerical
algorithms but they do not know that all those
Cartesian-Infinitesimal-Decimal methods could have been trivially
stated -since ancient times-- by means of the most simple
Arithmetic. A real shame, indeed. No need of any Cartesian system,
decimal numbers, infinitesimals, fluxions nor any other primitive
artifices.
Dear reader, just notice that if I were wrong then any "experts" on
either root-solving or History of Mathematics could easily reply to
this post of mine by bringing out to the audience of this Math-Forum a
plethora of precedents on these GENERAL AND TRIVIAL HIGH-ORDER
ARITHMETICAL METHODS. Also, the same "experts" could show you that
by agency of just the "Rational Mean" (Generalized Mediant) you
cannot at once generate Newton's, Bernoulli's, Halley's, and
Householder's methods.
HOWEVER, NONE OF THEM WILL BE ABLE TO DO THAT (specially the most
egotist ones), because there are no precedents on these general
arithmetical methods, and also because the Rational Mean is certainly a
General and unifying principle of Quantity embracing all those
well-known methods among many other new ones, as well as all known
Means (Arithmetic, Harmonic, Geometric, Golden, etc.) and the Continued
Fractions.
A True Natural Principle of Quantity which surprisingly DO NOT APPEAR
IN ANY TEXT ON NUMBERS SINCE ANCIENT TIMES UP TO NOW.
If any doubts, I invite you to revisit the whole history of
root-solving:
The whole history of root-solving algorithms is plenty on Geometrical
and Trial-&-Error methods for dealing with roots. Indeed, there is no
trace of any Natural arithmetical methods based on Number itself but
just geometrical artifices (Cartesian system, fluxions, etc.) and/or
Trial-&-Error checking. The reader can check by himself that all those
ancient and re-known root-solving methods are based exclusively on
Geometry and Trial-&-Error checking.
Moreover, it was almost impossible for ancients to bring out any
numerical approximations on a simple cube root, not to say anything
about higher-degree roots.
Ancient Greeks failed to find any Natural Arithmetical Method (any
Natural Order in root-solving algorithms), for solving problems related
to that we call nowadays "algebraic equations of any degree".
Notwithstanding, their noble ideas on some "Natural Order"
pre-established by the mind of God remained there for so long, but it
was so hard to hide such a shocking arithmetical failure: 'No Natural
Order in there'. There were no Natural Methods based only on
Arithmetic (The Science of Quantity) exempt from Geometry and any
Trial-&-Error checking.
The direct consequence of such ancient arithmetical flop was: The
decimal system, the Cartesian system, the infinitesimals and fluxions
and consequently the modern concept of Number and many other modern
bizarre ideas and opinions who only contribute to distort the real
image of Quantity. Apparently, there was no choice for mathematicians
of past times, they needed to produce root-solving algorithms and
surprisingly the only way they found to do so was by using geometrical
artifices. Moreover, a plethora of papers from many scholar journals
have stated (say, imposed) that the Cartesian System was the only tool
mathematicians needed in order to achieve all those modern high-order
root-solving algorithms (Newton's, Halley's, Householder, etc.).
After all, Descartes as well as many other erudite mathematicians from
past times used to talk to God even when dreaming, so it seems one must
follow his commandments.
However, for all those who don't care about any self-called Demigods,
the following question comes up:
Question:
Is there any chance of generating natural root-solving methods --and
consequently generating irrational numbers-- just by means of Number
itself (Arithmetic) instead of using extrinsic
'Geometrical-Trial-&-Error-Cartesian-Infinitesimal-Fluxion'
elements?
Answer: Despite the fact that up to these days there are no precedents
on Natural Arithmetical root-solving methods, the answer to that
question is a sound 'Yes', all the irrational numbers and their
arithmetical operations can be defined and approximated by agency of
simplest Arithmetic.
In these web pages:
mipagina.cant.net/arithmetic/rmdef.htm">http://mipagina.cant.net/arithmetic/rmdef.htm
there is a brief introduction to some of the new arithmetical
root-solving algorithms which -from all the evidences at hand-- have
never appeared neither in any Greek, nor Chinese, nor Arab, nor Hindu,
nor Maya, nor Inca, nor European text on numbers since Babylonian times
up to now. More important, all these new arithmetical methods are all
ruled by the same general and unifying arithmetical operation: "The
Rational Mean" leading the way to fully define and establish not only
Bernoulli's, Newton's, Halley's and Householder's methods but
many other new higher-order iterating functions.
It is very important to notice that in order to generate Newton's
method it was mandatory to create not only the decimal-number concept
but also the Cartesian System, infinitesimals, fluxions, etc.. A lot of
work for computing roots, indeed.
So considering the origins of Newton's, Halley's, Householder's
and Bernoulli's method and the way they were established, it is
clear they cannot be considered as Natural Arithmetical Methods (See:
'Secondary Arithmonic Process'), at all, for not to mention as well
all those old and well-known root-solving algorithms based on
logarithmic computations. All those methods do not constitute any
single piece of a true Natural Philosophy.
Fortunately, there is a true Natural way for dealing with Number, a
true Natural Philosophy which some arrogant-egotist people have tried
to distort and as a direct consequence denying any "Natural Order"
conception and promoting just a primitive Technocratic System instead
of a true Natural Science, that is, a true Natural Philosophy.
This is the time to change all that mechanistic arrogance and egotism.
Some experts as well as some friends of mine have constantly suggested
me to try to publish all those new trivial high-order methods in any
peer-review journal, and I always answer back them that some time ago
I published a sample on a linear-order method (related to Bernoulli's
method) in an excellent journal in Berlin, Germany, and would be happy
to do so again mainly because its owner and Chief-Editor is an
excellent human being, however, I am not interested in sending any of
these new high-order methods to any scholar peer-review journal,
because if such trivial methods inexplicably have no precedents on any
of those peer-review journals --all through the history of root solving
-- then those journals do not deserve to get these Natural high-order
algorithms imprinted in any of their pages. Moreover, I would beg all
those people who in a near future could find any related advances on
these new methods not to publish them in any scholar peer-review
journal.
Domingo Gomez Morin
Structural Engineer. Civil Engineer.
djesusg@xxxxxxxxx
mipagina.cant.net/arithmetic">http://mipagina.cant.net/arithmetic
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