Allow me to introduce a very simple arithmetical operation: The Arithmonic Mean, which surprisingly and from the evidences at hand, has not been used in mathematics since ancient times up to now. I use the word 'Surprisingly', because it is a generalization of the 'Mediant' operation which is well known by number theorists and it leads the way to

the generation of high-order root approximating methods:

The Arithmonic Mean is 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 A₁, the first two numerators and the last two denominators for the set A₂, and the first two denominators and the last two numerators for the set A₃, always keeping the values of the fractions the same:

The Rational Mean (Generalized Mediant: The sum of all the numerators and denominators) of the three fractions for each set A₁, A₂, A₃, 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.

Because, it has no precedents in the whole story of root solving methods I think it could be of some interest to this audience.

There are two videos and a webpage on this subject in the net

Thanks and best regards,

Domingo Gomez Morin

The Rational Mean