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The Rational Mean

Excerpts from the book: ‘LA QUINTA OPERACIÓN ARITMÉTICA, Media Aritmónica’

(Translation: The Fifth Arithmetical Operation, Arithmonic Mean) ISBN: 980-12-1671-9.

Author: © Domingo Gomez Morin. Copyright. 1993-2009

© All rights reserved under international Copyright Conventions. 2009

arithmonic@hotmail.com

 

CONTENTS:  

A new general and unifying arithmetical concept, based on the operation called Rational Mean, which allows to generate not only Lucas’s, Bernoulli’s, Newton’s, Halley’s, Householder’s root-approximating algorithms but many other methods, also covering complex roots.

The Rational Mean embraces the well known Arithmetic, Harmonic, Geometric and Golden means. 

From the evidence at hand, this new arithmetical root-solving concept do not appear in neither any Chinese, nor Hindu, nor Arab, nor European text on numbers, since Babylonian times up to now. 

 

 LINKS AND REFERENCES  

Publications, books, papers, news, and web-pages referencing and/or containing

comments and analyses on these new arithmetical methods

 

  

Rational Mean definition

 

Given a set V of n rational numbers arranged according to their magnitudes:

 

 

are real values. On the other hand are real values and have the same sign.

 

The Rational Mean of all the elements of  V  is denoted as follows:

 

 

Cauchy proved that this operation always produces a mean value between  and  :

 

 

 

However Cauchy did not considered it as an arithmetical operation but just a curious property of those sequences of numbers, and consequently gave no name for it, so we will call it: Rational Mean. As can be observed the Rational Mean is not restricted to operate with irreducible fractions.

In Number Theory there is a particular case of the Rational Mean concept called: Mediant, which has been restricted to operate only between two irreducible fractions:

 

 

 

The Mediant is the fundamental principle which rules Farey fractions, Stern-Brocot sequences and simple continued fractions. There is more information on the precedents of the Mediant in the book: “La Quinta operación aritmética, media aritmónica”

 

By modifying the form of some fractions of the set V --but not its decimal value--, the Rational Mean yields another result, for example: multiplying  and    by the Form Factors  and  , that is:

 

 

 

This is a very important property of the Rational Mean, and consequently the term ‘Form Factor’  will play a very important role in all the rational processes which are based on the Rational Mean. There are some observations on the definition of this operation within the set of rational numbers which will be discussed later.

The Rational Mean is a new general and unifying concept which embraces as particular cases all the known means: the Arithmetic Mean, the Harmonic Mean, the Geometric Mean, the Golden Mean, etc.

The Arithmetic Mean

The Arithmetic Mean is the Rational Mean for a set of fractions with equal denominators.

The Harmonic Mean

The Harmonic Mean is the Rational Mean for a set of fractions with equal numerators.

 

 

 

 

Note: All the examples shown here can be easily extended for solving roots of any degree and equations

The term ‘Rational Process’ will be used to denote any numerical algorithm based on the Rational Mean

 

 

 

 

Arithmonic Mean Definition

The Arithmonic Mean is a very particular case of the

general and unifying Rational Mean concept

 

 

Given any set A of values, as for instance:

 

 

The Rational Mean of all the values of the set A is:

 

 

Let’s modify the forms of the fractions of A but not their decimal values, as follows:

 

 

 denotes the new transformed set of fractions.                                                     

Notice the sequence of interweaving blocks with equal denominators and equal numerators. Such sequence will be denoted as:

We have made the transformation  according to the sequence:  which represents interweaving-&-alternating blocks of equal denominators and equal numerators.

 

We define the Arithmonic Mean of the set of fractions A as the Rational Mean of the transformed set  and will be denoted as either   or :

 

 

Notice that the lower-hyphenated   is used to denote that the first block of the transformed set   have equal denominators.

On the other hand, the upper-hyphenated  is used to denote that the first block have equal numerators, as for example: Let’s transform the set A according to a new sequence  , as follows:

 

 

  The Arithmonic mean is the Rational Mean of the fractions of the new transformed set :

           

 

 

It is clear that the Arithmonic Mean embraces the Arithmetic mean and the Harmonic mean.

 

 

 

Preliminaries on the new arithmetical methods

(Nichomacus, Superparticular ratios)

 

All the ratios in the sequence:

 

 

were analyzed and classified as “Superparticular Ratios” by Nichomacus in his “Introducction to Arithmetic” (Ref.[iv]).

 

“in accordance with number by the forethought and the mind of Him that created all things,

 for the pattern was fixed, like a preliminary sketch,...”

(quoted text: Nicomachus, chap.VI, [1]).

 

Despite all the detailed analysis that he and other mathematicians did on such sequences, none of them seem to have noticed a very simple and important property which is directly related to the root-solving  issue, that in such sequence of ratios the product of each set of n=2, 3, 4, … fractions is always equal to 2 as shown in the following picture:

 

 

 

So each row of n fractions defines n approximations by defect-&-excess to the nth root of 2.

Notice that the value n is also given by the denominator of the first fraction in each set.

 (3/2, 4/3) defines two approximations by defect and excess to

(4/3, 5/4, 6/5) defines three approximations by defect and excess to

(5/4, 6/5, 7/6, 8/7) defines four approximations by defect and excess to 

And so on …

It is easy to notice that the product of all the fractions in each set is trivial and always equal to 2, I say “trivial product” because denominators and numerators trivially cancel each other out.

Thus, considering that  this pattern was certainly fixed like a preliminary sketch, from now on, will use the most basic principle of root solving: k values whose product is P represent  k  approximations, by defect and excess, to the k^th root of P

 

If we find a Rational Process --based on the Rational Mean-- which yields k values in each step closer to each other and  whose product is always  trivial and equal to P, then we can say we got a true natural and trivial arithmetical algorithm for approximating .

 

 

 

 

  

The Simplest Rational Processes

 

Let’s take a look now at some of the simplest Rational Processes which led the way to other high-order arithmonic processes.  As said, we will use sets of fractions whose product is trivial and equal to a number P, that is, sets of approximations by defect and excess to the desired root. Even though the generalization of this extremely simple method is trivial, sometimes numerical samples are the best way to grasp the very essence of the Rational Process.

 

Cube root of 2:

 

Given the initial set of three values whose product is trivial and equal to 2:

First step: let’s use the following notation for computing three rational means at once, that is, each one is computed for only the values within the curly brackets:

 

 

As you can see, the three rational means produced a new set of three fractions whose product is trivial and equal to 2, that is, we got three approximations closer to the cube root of 2.

The term  acts in the same way as the Form Factors we already mentioned at the beginning of this webpage.

If we repeat the same operations, this time by using the new set of values {9/7, 11/9, 14/11}, we will get as good approximations as we want to the cube root of 2.

 

Second step: Let’s evaluate another set of three rational means by using the new set {9/7, 11/9, 14/11}: 

 

The Form Factor remains always the same all through the rational process: (2/2) 

We got a new set of  three fractions whose product is trivial and equal to 2. 

 

We can continue with this process in order to produce new sets of three fractions, which represent approximations by defect and excess to the cube root of 2. The product of the fractions in each set will be always trivial and equal to 2.

Even considering its slow speed of convergence and that it sporadically produces some ‘best approximations’, it must be said that its simplicity and clarity would make bloom some stirring feeling in the soul of any mathematician. Indeed, it is amazing to see the Natural Order imprinted in such a simple rational process.  

 

 

 

 

 

Square Root: Trivial High-Order Rational Process

 

Square root of  P :

Given two fractions (positives values) whose product is P, that is, two approximations by defect and excess to :

 

 

According to the above Arithmonic Mean definition: Given the sequences:  and  let’s compute the following two arithmonic means  and .  In this very particular case both arithmonic means correspond to the arithmetic and harmonic mean. 

 

 

Two new expressions whose product is trivial and equal to P.

 It is easy to prove that you can use these new functions as independent iteration functions for computing the square root of P, there is an example on this below.

Notice that the first function yields the same results produced by  applying Newton’s method to x²-P. 

 

Actually, at this point, there is nothing new here, because from the historical evidences at hand it seems that ancient mathematicians certainly knew how to compute square roots by agency of the arithmetic and harmonic means. So in this particular case the Arithmonic Process yields, let’s say, the same ancient results for approximating the square root of P. 

But, wait a minute, let’s take a look at this particular arithmonic process from the perspective of the general and unifying Rational-Mean concept. Let’s see what we have done:

 

 

The forms of x/1 and P/x have been transformed using the Form Factors:  and , and the Rational Mean yields two expressions whose product is trivial and equal to P.

Thus, if you apply the same form-factors to this new pair of expressions and compute again the Rational Mean,  then you will get another new set of two functions whose product is always trivial and equal to P, as follows:

 

  

This way you get  two new iteration functions whose product is P, each one can be used independently for computing the square root of P, and you will find that both of them triples the number of exact digits in each iteration. The first one is the same than the one you get when applying the well-known Halley’s method. The second function also triples the number of digits.

By repeating the process, in the next step you get:

 

 

The first function brings convergence speed of the fourth order (Householder’s method), and the second one also multiplies by four the exact digits in each iteration towards the square root of P.

 If you continue this process, then you will get other high-order iteration functions which correspond to Householder’s method for computing the square root of P.

The reader should notice that all these trivial high-order functions have been developed here just by agency of the most simple arithmetic, while Newton’s, Halley’s and Householder’s methods required the construction of a huge structure composed of the whole Cartesian system, the decimal system, and Infinitesimal Calculus, and this certainly imprints an extra connotation to all these new trivial methods.

Let’s see another step of the process:

 

 

 

These new two functions multiply by five the number of exact digits in each iteration, let’s see how it works:

The subscript ‘i’  denotes the step number of the iteration process. Each new approximation x_i+1 to the square root of P is calculated by using a previous value x_i.

 

We can choose P=2 and the starting value x₀ = 1

 

 

At the first step of the iteration process we get:

 

 

Now we use: x₁ = 41/29 and we get:

 

 

 

By continuing this procedure, you will get the following errors for the first four approximations to the square root of two:

 

 

And so on…

Thus, in the fourth step we got 479 exact digits for the square root of 2.

It was not required neither any derivatives, nor the Cartesian system, nor any other elaborated concept coming out from Infinitesimal Calculus, but just the simplest arithmetic.

 

Of course, many questions arise from all this.

 

 

 

 

The Freshman-Problem and the Egostism-Problem

 

Why this trivial stuff have been never published in neither any math journal, nor any book on numbers,

nor any papyrus, nor any clay tablet since Babylonian times up to now?

 

It is important to cogitate on what is going on here, because it is weird to realize that such trivial high-order arithmetical stuff do not appear in any math text since ancient times.

 

What could be the reason?

 

 

Actually, there are two thoughts that come to my mind:

 

 The first thought is about the well known “Freshman Problem” which I prefer to call “Freshman Trauma”, I mean, when we were kids and our teacher was trying to teach us how to sum fractions for the first time, we always got reprehended every time we added numerators and denominators.

Thus, due to such a trauma it seems natural that many mathematicians could be severely born-apprehensive against what we call Rational Mean. Indeed, I have perceived this unconscious rejection in so many books devoted to the analysis of Farey fractions and the generation of convergents of the continued fractions, besides, the reader can easily confirm that --as a direct consequence-- the Mediant operation has been exclusively confined to very specialized books on Number Theory. Consequently, it is a truly unknown subject for students.

Actually, up to these days, only Cauchy and Charles de Comberousse were the only ones who skimmed over the general Rational Mean operation in order to demonstrate that it always produces a mean value, and that is the only precedent for the general Rational Mean operation.

By revisiting any book on the history of mathematics, the reader will realize that the Rational mean seems to have been a truly condemned operation since ancient times, and this is highly disturbing, mainly when considering that it embraces all the known means, and constitutes a powerful tool for generating root-solving algorithms and defining the arithmetical operations of irrational numbers.

I think that math teachers should take careful thought on the outstanding phrase that the American philosopher Charles Sanders Peirce used to remark the wonderful order imprinted on the rational sequences generated by agency of the Mediant which is a very particular case of the Rational Mean. Even though he never assigned a name to this operation and just called it: ‘form of relation’:

 

 

(Charles Sanders Peirce, Collected papers, Harvard University Press, 1933, Vol. IV, art. 681, pag. 580)

 

“…It is because [of] this form of relation of rational consequence

that numbers are of such stupendous importance in reasoning.

But the highest and last lesson which the numbers whisper in our ears

is that of the supremacy of the forms of relation

for which their tawdry outside is the mere shell of the casket…”

 

 

      The second thought is about the “Egotism Problem”. The trivial high-order methods produced by agency of the general Rational Mean concept imply the existence of a Natural Order imprinted in Number itself, I mean, the one foreseen by ancient Greeks. Any concrete evidence on the existence of a Natural Order implies the existence of a pre-determined sketch, and you are not neither its planer nor its creator. You are just a researcher who can only discover things that have been pre-established in Quantity. Consequently, you are not the leader and you must adopt a humble behavior, because people will not believe in any ideas from yours that fail to fit within the pre-determined sketch. Thus, you will never take full control of people, because at any time and without any guidance from you, they will be able to become active contemplators of the preexistent frame of Quantity and will achieve their own findings.

On the contrary, if many people were convinced that there is no Natural Order at all, and that they live in a realm of Chaos and Uncertainty, then someone could took advantage of all that by playing the role of Leader. He/She could convince people that he/she is the only one who can manage Chaos and Uncertainty by means of his/her very personal creations, artifices and tricks-&-patches, that is, things like the Cartesian System, the imaginary numbers, the general relativity theory, the fourth dimension, the evolution theory, string theory, etc.

In this way, being the only one who knows how to deal with Uncertainty he/she could be able to take full control of people. The whole idea becomes worst if we replace the word ‘he/she’ by the words ‘Consortium’, ‘Society’, ‘Company’, etc.

It is clear, that the option of a Universe of Chaos-&-Uncertainty is paradise for many groups of egotistic scientists, and they would assure themselves to hold the power by all means, as well as to smash any other solid theories opposed to Chaos-&-Uncertainty.  Of course, I do not mean that this behavior could be something planned by any group, on the contrary, I am sure that all that would be just the instinctive and spontaneous result of egotism and egoism, just pure individualism and longing for power.

 

 

 

Rational Process and Daniel Bernoulli’s method

 

There are many other simple rational processes for approximating roots, as for example:

 

Given the set of three values:

 

 

 

whose product is P, and computing three rational means, at once, in each step of the rational process as follows:

 

 

and so on..

 

notice the Form Factor: (P/P).

 

In each step of the process we got three expressions, whose product is P, and represent approximations closer to .

 

This rational process gives the same results of the well-known Daniel Bernoulli’s method, all this has been stated in the book: “La quinta operación aritmética”.  The process has slow convergence speed.

The Rational mean is a general and unifying concept, which embraces innumerable root-solving algorithms: Bernoulli’s, Newton’s, Halley’s, Householder’s as well as many other new ones, all this by agency of the simplest arithmetic.

 

 

The Golden Mean

The irrational number:  called the Golden Mean satisfies the golden proportion:  and is a solution to the equation x²+x-1=0, and is also related to the Fibonacci's sequence: 1, 1, 2, 3, 5, 8, 13,...

 

The Rational Process –based on the Rational Mean-- for approximating the irrational value of the Golden Mean can be developed as follows. Considering the Golden proportion:

 

 

We can choose the following set of initial values satisfying such proportion:

 

 

In each step, let’s compute two rational means by using the form factor: 2/2, as follows:

 

 

 We got two new ratios satisfying the Golden proportion.

 

The next step yields:

 

 

One more step yields:

 

 

and so on...

Each new pair of ratios will always satisfy the Golden proportion.

At each stage of the process we get two closer rational approximations to the Golden Mean. Notice that the numerical coefficients of all those expressions bring out Fibonnaci's sequence.

 

This rational process for computing the Golden Mean differs from the previous rational processes shown in this webpage, in the sense that it requires to satisfy a very specific proportion of the initial ratios, instead of their product.

 

 

 

 

  

New High-order arithmetical Root-Solving Algorithms

The Arithmonic Process

 

  

Cube root of any number  P  (including complex roots) :

 

Given any set of three numbers and the sequences ,

Calculate the following three arithmonic means:

 

 

 

 

Notice again the interweaving-&-alternating blocks of fractions with equal numerators and denominators according to the sequences ,  and the Arithmonic Mean definition.

Also Notice the trivial product:

 

 

 

Thus, we got three expressions that represents three approximations by defect and excess to .

 

From this point onwards, there are many ways we can choose in order to find iteration functions for computing , or even we could intend to continue by just computing another set of three arithmonic means.

 

Let’s see one way we might choose:

 

If we choose: , and , then the above three expressions take the form:

 

The second expression  is the same that results from applying Newton’s method to the equation x³-P = 0.

 

The third expression  is the same that results from applying Halley’s or Householder’s method to that equation

 

and it triples the number of exact digits in each iteration step.

 

Another way:

As said, instead of using these iteration functions one could continue the arithmonic process and produce another set of three expressions by making:

 

and repeating the computation of three arithmonic means as done in the first step.

As said, there are many ways of handling these methods.

 

 

Complex Cube Roots

 

Given the initial set of values: whose product is P.

 

Being  ,  and .

 

By computing the same three arithmonic means that were developed above for the cube root of P:

 

 

 

By repeating this step by using each new resulting set of approximations up to the fourth iteration, we will get:

 

 

and so on…

 

 

This way we approximated the first complex root of P : .

 

The second complex root   can be approximated by using

 

There are many other arrangements and ways of computing these complex roots by agency of the Rational Mean.

 

 

 

Additional comments on the Cube root of any number  P :

It is possible to get better functions than the well-known Householder’s method. For example, in the case of the cube root of any number P one can get --in the second step of the arithmonic process-- a function like this:

 

    Fifth order Arithmonic iteration function

 

Which multiplies by 5 the number of exact digits in each iteration, and clearly differs from those that come out from applying Householder’s method to the equation f(x)=x³ –P.

 

      Fourth order Householder’s iteration function

 

 Fifth order Householder’s iteration function

 

 

 

 

 Notice that in order to develop a Rational Process for computing, say, the cube root of P, you might use an initial set of values as for example:

 

 

Which are initial approximations by defect and excess to .

 

 

Or you might use:

 

Which are initial approximations by defect and excess to .

 

There are certainly uncountable arrangements to get different iteration functions and methods of approximating roots by agency of the Rational Mean.

 

 

Fourth root of any number  P :

 

Given the set  and the sequences  we can get iteration functions for approximating the fourth root of P, by computing the following four arithmonic means:

 

 

You may notice four different arrangements of A showing a fixed pattern, that is, the changes in the position of the elements of A is described by using arrows in the following figure:

 

 

 

So, for instance in the case of the fifth root the arrangement will be:

 

 

 

And so on…

 

In the case of the cube-root the Arithmonic Process also complies such pattern.

 

  

 

Seventh root of P :

 

By using the initial set:

 

and computing seven arithmonic means according to the sequences ,  for the following arrangements of A: 

 

 

 

 we get seven iteration functions for approximating  , whose product is trivial and equal to P :

 

 

 

 

As said, there are an uncountable number of ways to get other iteration functions. The rational mean is certainly a new general and unifying concept which gathers round issues that seemed unrelated up to this day.

 

 

 

 

 

High-order approximations by agency of square roots

 

There are uncountable ways of producing iteration functions by agency of the Rational mean, among all of them, another way is to choose the following set of three initial values for approximating the cube root of P:      

 

whose product is trivial and equal to P, so when applying the Arithmonic process for approximating the cube root of any positive  number P  then you can get  the following iteration functions at the first step:

 

 

 

The errors were computed for the cube root of 2 starting from the initial value x₀=1.

 

All these methods have been easily extended for roots of any degree. The second function in the table converges faster than the corresponding third-order Householder’s and Halley’s function. The other two functions converge faster than Newton’s, so in the fourth step these arithmonic functions double the number of exact digits produced by Newton’s method.

Amazing, indeed, just by means of simplest arithmetic.

 

 

 

 

The history of root-solving versus the new arithmetical algorithms

Mathematics of Trial-&-Error, Geometry, Chaos, and Tricks-&-Patches

versus

The Natural Mathematical Order

 

In order to grasp the importance of these new arithmetical methods, it is necessary to revisit the history of root solving.

In ancient Greek times, the study of the science of Quantity was mainly oriented to find the “Natural Order predetermined by the mind of the world-creating God” (Nichomacus, Ref. IV). So any trace of ‘Natural Order’ they could find in any numerical method was directly related to the existence of God, whereas all Trial-&-Error methods represented Chaos and inability to grasp the very essence of God.

One of the most important issues that ancient mathematicians had to deal with was to solve primitive roots. An ancient Babylonian approximation (1600 B.C.) to the square root of two with five exact digits was imprinted on clay:

 1+ 24/60 +51/60² + 10/60^{3 =} 1.4142129.

According to some authors, the ancient Babylonian method was the same of Heron of Alexandria. Starting from two initial approximations [x, P/x],  by defect and excess, and producing a new pair of values by agency of the Arithmetic Mean (x+P/x)/2 and its inverse multiplied by P (harmonic mean), and by repeating that at will, they got approximations to the square root of P.

The Indian mathematician and historian Radha Charan Gupta (Ref.[i]) has pointed out that such Babylonian approximation is exactly the same value found in India and dated about 500 B.C (Ref.[ii],[iii]).

Some ancient scholars from India, China and Greece developed geometrical and Trial-&-Error methods for finding some square-root approximations, however, it became a real challenge every time they tried to extend it for approximating Cube roots.   

Heron of Alexandria tried to find approximations to the cube root by agency of Geometry, as well a Menachmus who worked on the duplication of the cube, however, it was just a one-step process and it seems there is no evidence of any numerical result. Indeed, there were just some numerical examples on square-roots but no cube-roots approximations, at all, of course I am not talking about perfect cubes as those you can find in ancient Chinese and Indian texts.

Moreover, the ancient and re-known Chinese attempts on root-solving were just numerical examples on some particular square roots, mainly perfect squares, by using a Geometrical-Trial-&-Error method which turns into a true headache when extended for approximating cube roots. I would say that one cannot find any numerical Chinese or Hindu approximations of, say, the cube root of 2 (we are not talking about trivial perfect cubes, because there is plethora of them in ancient texts). The re-known ancient Chinese method, usually taught at school for approximating square and cube roots is based on the first ancient-Chinese geometrical attempts for approximating roots and is related to the following binomial expansions:

Square root:   P=(10a+b)² = 100a² + 2*10ab + b²,

Cube root: P=(10a+b)³ = 1000a³ + 3*100a² b +3*10ab²+ b³.

Nobody could deny that the very spine of this ancient procedure is Trial-&-Error checking, that is, you can only move forward in the process by previously making a Trial-&-Error checking. On the contrary, a true Natural Arithmetical Method should not use any checking, but just smooth, well-ordered and pre-determined arithmetical operations for approximating the root, that is, without the help of geometry nor any trial-&-error checking. Thus, we cannot consider those ancient methods as true Natural Arithmetical Methods.

 

The ancient faith on some “Natural Order” lasted for so long, but it was almost impossible to hide such a shocking arithmetical failure: No Natural Order ahoy¡, no natural methods exempt from Geometry and Trial-&-Error checking.

 

It follows some of those people who worked on root approximations, the list is not complete and might not be even exact, it only intends to serve as a general reference on the issue covering up to 1634, comments on more recent authors will be included hereinafter.

 

 

 

Long time before the rise of Cartesian System many people tried to find either numerical approximations or algebraic solutions. Mathematicians found algebraic solutions for equations up to fourth-degree, and stated the impossibility for higher-degree equations.

Diophante solved some very particular cases.

Some Chinese mathematicians solved some equation systems, and developed what many people call Horner’s method.

Arab mathematicians contributed with the false position rule, which actually came from India.

By the year 510 (A.C.) Hindu mathematicians solved some particular quadratic equations.

 

Arabs and Hindus also gave a geometrical treatment --by agency of conics— to quadratic and cubic equations, so they didn’t find any Natural Arithmetical root-solving method, being so difficult to get from any of their texts any simple numerical approximation to the most simple algebraic equation of third degree, as for example:  x³ =2. (See for example: Al-Khârizmi,  Bhâskara,  Âryabhata)

Fibonacci produced some tricks for the numerical solution of some very particular cases.

Finally , Viette, Newton y Daniel Bernoulli contributed so much to numerical solution of algebraic equations.

In reference to the algebraic solution of equations of the second, third and fourth degree the following authors contributed so much to the subject: Pacioli (1494),  Scipio del Ferro (1515), Cardano (1545), Ferrari (1545), Tartaglia(1545), Bombelli (1572), Vieta (1590), Descartes (1637), , Lagrange, Ruffini, Sturm, Galois, Abel, Wronski,  Edouard Lucas, and others.

 

It is very important to notice that in order to generate Newton’s method it was mandatory to create not only the decimal numbers, but also the Cartesian System and infinitesimal calculus. So considering the way it was established, Newton’s method could not be considered as a true Natural Arithmetical Method, and this also applies for all those old and well-known root-solving algorithms based on logarithmic computations. In the book there are included other important observations that arise when comparing Newton’s method and the Rational Process.

 

 

 

 

 

Rational Mean definition within the current stream of thought about rational numbers, decimal fractions and Cartesian System

 

Some of those who made some work on the Rational Mean (though gave no name to this operation) were: Nicolas Chuquet (1484), Haros(1802), Farey(1816), Cauchy, J. Wallis, C. S. Peirce, Stern-Brocot(1858-1860), Lester R. Ford, Pick and others.  

The number theorist Edouard Lucas pointed out that this operation was named: ‘Médiation’ by french people and it was shown in the works of Archimedes and ancient geometers from India.

 

Now, I recall again the outstanding phrase expressed by the American philosopher Charles Sanders Peirce when analyzing the wonderful order imprinted on the rational sequences generated by agency of the Rational Mean:

 

(Collected papers, Hardvard University Press, 1933, Vol. IV, art. 681, pag. 580)

 “…It is because [of] this form of relation of rational consequence that numbers are of such stupendous importance in reasoning. But the highest and last lesson which the numbers whisper in our ears is that of the supremacy of the forms of relation for which their tawdry outside is the mere shell of the casket…”

 

It is commonly argued that this operation (Rational Mean) is not well defined within the set of rational numbers, that is, the rational mean just works with ordered pairs of numbers. As an example, let's compute the following rational means:

 

Rm[3/2, 4/3]= 7/5        Rm[6/4, 4/3]= 10/7

 

We have used the same rational number 3/2=6/4 and got two different results. So, according to the fundamentals of the Cartesian-decimal System this operation is “not well defined” within the “set of rational numbers.

 

The terms “well defined” and “bad defined” do not mean this operation is classified as either “well” or “bad”, it is just that it works with ordered pair of numbers (fractions), not rational numbers, but it is important to notice that this is a definition imposed by the Cartesian-decimal current stream of thought.

The Cartesian system imposed the replacement of any rational number 3/2, 6/4, 9/6,… by its corresponding decimal value, i.e.:  1.5, that is, restricting 3/2, 6/4, 9/6,…  to a unique absolute decimal fraction 1.5.

 

Ancient mathematicians did not consider 3/2 being equal to 6/4, however, such concept did not help them to find natural arithmetical root-solving methods. They never brought to light any general and natural high-order arithmetical method for solving roots of any degree. All that baffled ancient Greeks because of their persistent geometrical point of view of all those issues relating Number.

The conclusion was that Arithmetic was an obstacle they should overcome in order to find general and natural root-solving methods.

This new geometrical-decimal-infinitesimal system led the way to the well-known Newton, Halley, and Householder root-solving methods and consequently this new system was consecrated as the most outstanding achievement of human race.

 

Not to my surprise, the issue on “bad” or “well” definition within the modern stream of thought of rational numbers, is intentionally misused by some egotistic and biased people in a futile attempt to avoid any discussion on the lack of precedents of the new Natural Arithmetical methods shown here, something that they find so hard to admit.

 

It is important to notice that the generation of convergents of the Lord Brouncker’s continued fraction, for approximating 4/Pi or the generalized continued fraction of the number e,  is just Rational Process ruled by the Rational Mean without any conditions whatsoever. Just Rational Means of the two preceding fractions which are not necessarily in their reduced forms. Another crude fact, indeed.

Notice that only the convergents of the Simple Continued Fractions are reduced fractions, but that is not the case with the General Continued Fractions. 

In this way, given the well-known and widely accepted continued fraction expression for the number ‘e’:

 

The convergents are: 2/1,  3/1, 8/3, 30/11, 144/53, 840/309, 5760/2119, 45360/16687 ….

The generation of the convergents starts with two initial values 2/1 and 3/1. Evaluating the Rational Mean of the two preceding values in the sequence, always modifying their forms by the Form factors which are actually the coefficients of the well-known continued fraction expression of the number e as follows:

 

And so on…

 

Notice that the form factors (1/1), (2/2), (3/3), (4/4), (5/5), (6/6),… are the coefficients of the well-known continued fraction expression of the number e.

 

Thus, the well-known and widely accepted continued fraction for the number e is just a rational process ruled by the Rational Mean. Notice that there are not just reduced fractions here. These continued fractions are just Rational Means. Thus, the continued fractions also seem to be “bad defined” within the set of rational numbers, indeed.

Some mathematicians assert that the arithmetic and harmonic means are “well defined” even though they are just Rational Means, because they are given the condition of previously making their denominators and numerators the same, but that is not the case of the example shown above.

As said, some egotistic people intentionally misuse the issue on bad-&-well definition in a futile attempt to avoid any discussion on the lack of precedents of these new Natural Arithmetical methods, and this could be an understandable reaction because all this is just striking, and also because it is strongly related to the issue on the aforementioned ‘Freshman Problem’

 

Let`s see another example on “bad definition” within the set of rational numbers:

 

The Mass Center X_CM of a set of masses: {M₁, M₂, ..., Mn} is basically the Rational Mean of those masses as follows:

 

 

 

The factors ,,,…, acts as the aforementioned Form Factors.

 

Thus, when we modify the form of a rational number representing any of those mass values, then we get a very different result for the Mass-Center function. From this mathematical point of view, the Mass-Center function brings different values when using the same rational number for any mass.

We can see now that the terms “well defined” or “not well defined” are just referring to the very particular and arbitrary current stream of thought about rational numbers.   

The true is that the Rational Mean rules all the following mathematical operations:

 

•The Harmonic mean: Rational mean between fractions having equal numerators.

•The Arithmetic mean: Rational mean between fractions having equal denominators.

•The Arithmonic mean: Rational mean between fractions having some of their denominators and numerators the same, according to a specific rule.

•The Geometric mean.

•Generation of convergents of generalized continued fractions.

•Algebraic and transcendental numbers.

•Bernoulli's, Newton's, Halley's, Householder’s methods for solving algebraic equations.

•Power series expansions (Maclaurin-Taylor series).

•Statistics.

•Gravity center

•Vectors sum

•Ford's circles

•Farey's fractions

 

 It is clear that all those statements on "bad-or-well defined" operations are just arbitrary impositions which should be ignored by all those who feel that the Cartesian-decimal system and its related methods are certainly not any sacred products from the inspiration of God, but just artifices created by human beings.

All those arbitrary Cartesian-decimal definitions imposed on the rational numbers should be revised and corrected.

The rational Mean is not only a “very well defined” operation within the set of rational numbers (those which have not been defined according to any Cartesian-Decimal dogma) but rather a general and unifying nexus which allow us to compute any mean value and achieve a true definition of the arithmetical operations of irrational numbers.  The Rational Mean embraces issues that up to these days seemed to be unrelated.

 

 

 

Conclusions

The Cartesian-decimal-system fundamentals have imposed the replacement of Number by its corresponding decimal value, i.e.:  3/2=6/4=1.5, that is, it has restricted 3/2 to the decimal result of dividing 3 by 2, and this constitutes a crude  transfiguration and degradation of Number.

 In the same way flowers bring us their multiple natural properties: beauty, color and scent, also Number brings out much more than just the decimal result of a division, it brings us a relative value (The form of the ratio) and a very specific location within a set of ratios, and also its absolute value.

Cartesian system has depersonalized Number confining it to just an absolute decimal value.

Indeed, it is really striking to realize that ancient mathematicians (Babylonians, Greeks, etc.) certainly had at hand the most elemental arithmetic tool (The rational mean) for achieving all those "advanced" algorithms that were consecrated as the most outstanding successes brought to light by the Cartesian system and decimal fractions. Believe It or Not!, based on all the evidence at hand, it seems that the extremely simple arithmetical methods shown in the book “The Fifth Arithmetic Operation” have no precedents at all, all through the very long history of roots solving.

Summarizing:

• The Cartesian system is just an artificial creation, which apart from being extrinsic to the natural properties of Number could be contributing to distort and vitiate the genuine image of Quantity. On the other hand, the new general Rational Mean concept is clearly pointing out a new scheme for representing Number.

• The arithmetical operations of irrational numbers can be easily established by agency of the Rational Process (based on the Rational Mean in accordance with Number itself), rather than by using Dedekind's and Cantor's judgments.

• The traditional continued fractions expressions are just a particular case (Second Order Continued Fractions) of a more general conception called: Generalized Continued Fractions (Fractal Fractions), which yield periodic representations for algebraic numbers of higher degrees. You will realize that any representation of irrational numbers of higher degree get distorted when using the traditional continued fractions, that is, the “Second Order Continued Fractions” . It remains so much to be investigated on this matter mainly the issue on generating best approximants for roots of higher degrees, but these new generalized continued fractions might be of some help in the quest for periodicity and best convergents.

• The rational mean is the most intelligible arithmetical operation to understand irrational numbers, and the arithmonic mean is a particular case and an essential operation for root-solving.

Based on the simple arithmetical methods brought to light by agency of the Rational Mean, whose properties have been passed over since ancient times, one can say that it is certainly a regrettable arrogance to think that any result (i.e.: imaginary numbers, cartesian system, general-relativity theory, etc.) coming out from a bunch of artifices extrinsic to Number could ever supersede the Natural Order determined

“in accordance with number by the forethought and the mind

of Him that created all things, for the pattern was fixed,

like a preliminary sketch,...” (Nicomachus, chap.VI, [1]).

 

 

 

 

 

Bibliography