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

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, these new arithmetical methods have not been found in the mathematics literature since ancient times up to now.

Note: Articles, comments and references to these methods can be found at  Other Articles&Links webpage

The well known mathematician Cauchy proved that this operation always produces a mean value between a₁/b₁ and a_n/b_n :

When constructing the ordered sequence of all rational numbers by agency of this operation (Rational Mean, Mediation, Mediant) the famous American philosopher Charles Sanders Peirce  wrote the following outstanding remark about this very special mean operation, in his  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…”

Cauchy considered this mean operation (which hereafter we will call Rational Mean) as a very curious property of numbers.  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 Rational Mean is not restricted to operate with 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”. Farey and Stern-Brocot Tree deal with the generation of rational numbers by agency of the Mediant. 

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 maintaining their decimal values, as follows:

The term  denotes the new transformed set of fractions. Notice the sequence of interweaving blocks of equal denominators or numerators, such sequence will be denoted as follows:         

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”:

“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]

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

          (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

Basic 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.  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.

Cube root of 2:

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

As you can see, the initial 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.

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: High-Order Rational Process

Square root of  P :

Given two positive fractions whose product is P, that is, two approximations by defect and excess to :

 According to the 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 from India and Greece certainly knew how to compute square roots by agency of the arithmetic, harmonic, and rational mean.

So in this particular case the Arithmonic Process yields, let’s say, the same ancient results for approximating the square root of P.  This method for computing square roots have been reinvented by many authors all through the history of mathematics, and sadly, some of them seemed to be totally convinced that they were on something really new.

However, 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: (x/x) and (P/P) 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.

As an example, 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, the absolute 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 didn’t required neither any derivatives, nor the Cartesian system, nor any other elaborated concept coming out from Infinitesimal Calculus, but just a truly simple arithmetical concept.

It is striking to realize that these simple and eficient methods do not appear in the literature since ancient times up to now, also from the evidences at hand, these algorithms have never been taught at any academy. 

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 the cube root of P.

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 are Fibonnaci numbers. 

This rational process for computing the Golden Mean differs from the previous methods shown in this webpage, mainly because 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 just compute 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 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 x³-P = 0, 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 using each new resulting set of approximations as the initial set of values and repeating the previous step, 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.

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:

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 :

Approximating ,   x0=1
Iteration function	Error
	2.72*10-2 1.97*10-3 1.05*10-5         3.01*10-10
	3.88*10-2 3.73*10-3 3.75*10-5       3.81*10-9
	2.09*10-2 7.77*10-4 1.09*10-6       2.16*10-12
	7.02*10-3 4.52*10-5 1.85*10-9       3.10*10-18
	4.09*10-3 2.26*10-7 3.77*10-20      1.76*10-58
	1.32*10-2 1.65*10-4 2.46*10-8          5.49*10-16
	2.08*10-2 7.94*10-4 1.14*10-6        2.37*10-12

There are an uncountable number of ways to get other iteration functions. The rational mean is certainly a new general and unifying concept that 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

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]) 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 them 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 and the current definition of rational numbers

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.

The Rational Mean is considered as not well defined within the set of rational numbers, that is, it  works with ordered pairs of numbers. In order to illustrate this, let's compute the following rational means:

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

We used the same rational number 3/2=6/4 and got two different results. So, according to the modern definition of rational numbers this operation is “not well defined” within the “set of rational numbers, that is true, but we must be acquainted with the fact that such statement is based on an arbitrary  definition which serve as the base for the cartesian system.

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, however 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, all this corresponds to the modern definition of rational number.

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 probably because of their persistent geometrical point of view of any issue relating Number, and consequently they concluded  that Arithmetic was an obstacle that should be overcome in order to find general and natural root-solving methods.

This new geometrical-decimal-infinitesimal point of view 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.

It is important to notice that the generation of convergents of the Lord Brouncker’s continued fraction, for approximating 4/Pi and the generalized continued fraction of the number e  are just Rational Process ruled by the Rational Mean. They ar just Rational Means of the two preceding fractions not necessarily expressed in their reduced forms. 

Also notice that only the convergents of the Simple Continued Fractions are reduced fractions, however that is not the case for 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:

Each factor   acts just as Form Factors.

when modifying 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.

Thus the terms “well defined” or “not well defined” are just referring to the current definition of the rational numbers.   

The Rational Mean is a natural principle that can be found in the following mathematical topics:

•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.