It
took thousands of years for mathematicians to finally achieve modern
high-order root-approximating methods for algebraic equations, i.e.:
Newton's, Householder's methods. In the mean time, many attempts were
made always by the agency of trial-&-error methods, geometry, and
finally the cartesian system and infinitesimal calculus.

All through the
math history, there are no traces of any general natural arithmetical
root-approximating method exempted from either any trial-&-error
checks or geometry. Notice that even Newton's method is based entirely
on geometry. This work shows that by means of the Simplest Arithmetic
one can develop all those renown methods (among many others) in a very
simple way, without any need of trial-&-error checks, nor geometry,
nor derivatives, nor infinitesimal calculus:

The
Arithmonic Mean is an arithmetical operation, a particular case of the
Rational Mean (Generalized Mediant), and it can be used for generating
high-order root-approximating functions (Iteration functions), as an
example:

Given an initial set A of three fractions whose product
is trivial and equal to P (three approximations by defect and excess to
the cube root of the number P), all of them arranged in three sets as
follows:

The
product of the fractions in the set is trivial because the numerators
and denominators cancel to each other so the resulting product is
obvious.

By computing the Arithmonic Mean for each set, it yields
another set of three fractions whose product is also trivial and equal
to the number P.

The arithmonic mean requires to previously modify
the form of the fractions, by equating pairs of denominators and
numerators according to a very simple rule, keeping their values the
same, and subsequently applying the Rational Mean (Generalized Mediant:
The sum of the numerators and denominators) :

Equating the first two denominators and the last two numerators for the set *A*₁, the first two numerators and the last two denominators for the set *A*₂, and the first two denominators and the last two numerators for the set *A*₃, always keeping the values of the fractions the same:

The Rational Mean (Generalized Mediant: Sum of all the numerators and denominators) of the three fractions for each set *A*₁, * A*₂, *A*₃, are:

Summarizing, the corresponding arithmonic means of A₁, A₂, A₃ are:

A
new set of three fractions whose product is also trivial and equal to
P. Notice that numerators and denominators are trivially canceling each
other, producing the obvious product: *P*. Three new approximations by defect and excess to the number *P*.

By
repeatedly applying this procedure, it yields high-order approximations
to the cube root.

Moreover, it can be extended to yield high-order
iteration functions similar to those of Newton's, Halley's,
Householder's methods.

**High-order Root-Approximating Methods**

In
order generalize the above procedure let's arrange the initial set of
three approximations to the cube root of P, by using the following set
of three expressions:

*x*, *x*, * **P*/*x*²

Then,
by repeatedly computing the three arithmonic means as stated above,
then in each iteration step it will yield a new set of three
expressions whose product is always trivial and equal to the number P,
that is, sets of rational functions for approximating the cube root of P
at any convergence rate, as can be seen in the following table:

The product of all the rational functions in any row is P, and the convergence rate of each function in the table corresponds to its row number: quadratic (2nd row-Iteration 1), cubic (3rd. row-Iteration 2), quartic(4th. row-Iteration3, quintic, ...

Each of them can be used as an independent iteration function for approximating the cube root of P, as for instance the rational function located in row: 5, column: 3:

**Considering the very long story of root-solving methods, this is really striking ¡¡¡**

** **A
very rich but unexplored field. Actually, there are no precedents on
this matter since ancient mathematics up to now, so from the evidences
at hand, what you will read in the book does not appear in any other
math-book abroad.

**The new methods are accompanied with convergence
proofs. There is an uncountable number of variants for these methods,
the book shows some of them also embracing the well-known Bernoulli's
and Lucas' root-approximating methods, among some new others.**

**The methods are also arranged in matricial form and extended to complex numbers and the general algebraic equation.**

**Generalized Continued Fractions**

**Apart
from other stuff that might be of interest to the reader, the book also
includes a new type of Generalized Continued Fraction for the number
Pi, e, Golden Mean, among others, i.e.:**

**This special continued fraction serves as the book's cover.**

The
book also includes a chapter on some new geometrical constructions
(partitions), as well as a chapter on mathematical foundations of music,
specifically on consonance.

The chapter on Music includes some observations on the mathematical fundaments of music and some observations on consonance.

At
the end of the sidebar, you can find a link for a Synthesizer
programming code elaborated using wolfram language (Mathematica) that
can be freely downloaded and improved. This code is not included in the
book. It allows to play chords of up to six sounds at any frequency,
including piecewise functions for the attack, decay, sustain and release
parameters for each sound. It also incorporates interpolate functions
with up to 12 harmonics which can be edited and saved at any time for
modeling musical instruments timbre at any frequency. The reason for
including this Synthesizer is due to the fact that almost all current
synthesizers are based exclusively on the customary frequencies of
musical notes, so it is so hard for any researcher to make any analysis
on consonance with any frequencies chosen at will.