number roots solving householder newton halley bernoulli� mean arithmetic harmonic mean fraction ratio continued generalized raiz media armonica golden numero algorithm new method metodo convergence raices fifth operation algoritmo nuevo nueva rational racional irrational surd cubic square fourth polynome polinomio fractal generalizada math theory���������������������������������������������������

THE FIFTH ARITHMETICAL OPERATION

 

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��������� The Rational Mean: The fifth arithmetical operation. All means as particular cases of the Rational Mean��������������������� (Generalized Mediant). The new Arithmonic Mean as an essential arithmetical operation for roots solving. New Properties and observations on Number.

 

���������� Roots Solving: Bernoulli�s, Newton�s, Halley's, Householder's and many other new algorithms trivially found just by agency of arithmetic (The Rational Mean),  no Cartesian system, no decimals, no derivatives. No precedents, at all.

 

���������� Generalized Continued Fractions: Traditional continued fractions as particular cases of a new general concept "Generalized Continued Fractions". (Fractal Fractions)

 

These pages are just a brief introduction to the book:

�LA� QUINTA� OPERACION� ARITM�TICA�

(Translation: The Fifth Arithmetical Operation)

ISBN:980-07-6632-4. 200 pages, spanish language.

Copyright �. All rights reserved under international Copyright Conventions.

�Author: D. G�mez.

 

 

 

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Comments

 

Some authors have pointed out that "Arithmetic" was the main obstacle ancients should overcome in order to solve problems involving what we call nowadays "roots-solving methods of higher degree", and that such analytic algorithms could only be found, formulated and explained by agency of the modern Cartesian system and infinitesimal calculus. We can see now that ancients certainly had at hand the most simple arithmetical tool (The Rational Mean, The Fifth Arithmetical Operation) for solving all those problems involving higher degree equations. It is really striking to realize that since ancient times mathematicians could have easily carried out such an elemental operation and roots solving methods but --from all the evidences-- they didn't!.

Based on the extremely simple arithmetical processes and wonderful properties of Number shown in the book and its introductory web pages: Rational Mean Definition-&-Evaluation, Roots Solving and Continued fractions), �it is so hard to realize these so simple arithmetical methods do not appear in any book on numbers since ancient times up to now.

�Indeed, there are very good news here, specially, for young people because from now on, by means of simple arithmetic they will be able to learn at primary or secondary school the "most advanced" analytical methods (Halley�s, Newton�s, Bernoulli�s, Power series expansions and many other new algorithms.

 

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Copyright � 1993-2002

All rights reserved under international Copyright Conventions.

No part of this page may be reproduced, stored or transmitted in any form or by any means without the prior permission of the author: D. G�mez.

Last revision: 2002