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New Geometrical constructions (SIP)
Publications, links and references to these new arithmetical methods
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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 Gómez Morín. Copyright. 1993-2006
All rights reserved under international Copyright Conventions
CONTENTS: A brief summary on new
extremely simple high-order root-solving algorithms established by agency of the
simple arithmetical operation: ‘Rational Mean’ which is a general and unifying
principle of Quantity that rules not only Bernoulli’s, Newton’s, Halley’s,
Householder’s methods but many other new high-order iterating functions, as well
as all known means (Arithmetic, Harmonic, Geometric, Golden, Mediant, etc.). From the evidence at hand, none of these simple
arithmetical methods appear in neither any Chinese, nor Hindu, nor Arabic,
nor European, nor Inca, nor Mayan text on numbers, since Babylonian times up to
now.
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Suggested first reading:
and the
with publications and comments on some of these new arithmetical methods.
All the examples shown in this page can be easily extended for solving roots of any degree
The Rational Process is a numerical algorithm based on the Rational Mean
Given a set V of n rational numbers arranged according to their magnitudes:
, and
holding equal signs.
Then, the expression:
is the Rational Mean of all the elements of V.
The Rational Mean represents a mean value between (a1/b1) and (an/bn):
That this operation always yield a mean value was proved by Cauchy, however he didn’t give it any name, even more, he didn’t assign it the category of an arithmetical operation but just a property of the sum of those sequences.
In Number Theory there is a particular case of the rational mean called ‘The mediant’ which has been restricted to operate only between two irreducible fractions:
and it 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
changing the form (not its value) of some fractions in the set the Rational
Mean yields another different result, for example, by changing the form of
and 
using the ‘Form Factors’:
and 
:
This is a very important property of the Rational Mean and consequently the term ‘Form Factor’ will play a very important roll for all the rational processes which are based on this arithmetical operation.
Note: The evaluation of several Rational Means may be denoted as follows:
Notice that in the last group of fractions the term 
was modified by the ‘Form Factor’: 
, this was done in order to bring an example on the Rational
Mean notation.
The Arithmetic Mean
The Arithmetic Mean can be obtained by evaluating the Rational Mean for a set of fractions with equal denominators.
The Harmonic Mean
The Harmonic Mean can be obtained by evaluating the Rational Mean for a set of fractions with equal numerators.
The New Arithmonic Mean
The ‘Arithmonic Mean’ is a very particular case of the
general and unifying concept called ‘Rational Mean’
Given any set A of values:
The form of all those values can be modified in order to make a first group of them having equal denominators, a subsequent group having equal numerators, another subsequent group having equal denominators, and so on, that means that equal numerators and equal denominators will take it in turns.
For example, in the following group:
We can modify the forms (not their values) of the fractions in this way:
Notice that the name of the new set is:
.
As you can see, this new set has:
A first set of two fractions with equal denominators: 5/3, 6/3
A second set of three fractions with equal numerators: 6/3, 6/12, 6/8
A third set of two fractions with equal denominators: 6/8, 12/8
And two fractions with equal numerators: 12/8, 12/4
We say that the set A was
transformed according to the arithmonic sequence: 
and this is the reason for the
new set to be named (The arithmonic transformation of A)
The lower-hyphenated 
means that the first group must
have equal denominators.
An upper-hyphenated 
means that the first group must
have equal numerators
For example, the set A can be modified
according to the sequence 
as follows:
The sequences and can be chosen at will.
Arithmonic mean definition:
The Arithmonic mean
of any set A according to either the sequence or
the sequence is
the Rational Mean of the corresponding modified set or 
.
For example, given a set: 
and the
sequence 
Then the arithmonic mean of the set 
according to that sequence is:
Here we can see a first group of two fractions with equal numerators. A second group of two fractions with equal denominators. A third group of two fractions with equal numerators, and finally two fractions with equal denominators.
The arithmonic Mean will allow us to generate high-order root-solving algorithms.
There are many cases where the Arithmonic Mean is equivalent to
either the Arithmetic or the Harmonic mean, as for example in the case of
the basic sequences: , of course there are many other cases.
s
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]).
Despite all the detailed analysis he and other mathematicians did on such sequences, nobody seemed to have noticed a very simple and important property which is directly related to the root-solving issue. In this way, notice that the product of each set of n=2, 3, 4, … fractions is always equal to 2 as shown in the following picture:
n: number of fractions in the set: Root index.
So each set 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 the set.
It is for sure that if Nichomacus or any other ancient wise mathematician had ever noticed such an important property of Number then nowadays we were facing a very different root-solving story, and consequently enjoying another Philosophy of Mathematics founded on Number itself instead of the Cartesian-decimal-Infinitesimal system. Actually, Nichomacus and his predecessors certainly had at their reach the simple arithmetical operation: ‘Rational Mean’ which was the only arithmetical tool they needed to find all kind of Natural Processes for generating irrational numbers, however, they didn’t, and that was a striking arithmetical flop.
Indeed, it is astonishing and so worrying to realize that neither ancients (who used to talk so much about God, some of them even suggesting to have talked to Him) nor any other mathematicians (Including Greeks, Arabs, Hindu, Chinese, Incas, and Mayas) ever noticed such forceful evidence of a pre-established Natural Order on Number, mainly when considering they always had at their reach --since Babylonian times up to now-- the most elementary arithmetical tool for handling such outstanding property of Number, I’m talking about the: ‘Rational Mean’, whose general and unifying properties are briefly summarized in these web pages.
Nowadays, the phrases ‘Natural Order’ and ‘Natural Method’ have been banned by modern scientific consortia in the same way as the Holly Office did in past times, fortunately, due to international human rights conventions scientific consortia cannot practice corporal punishments on persistent offenders, but have arranged to turn their back on them by other means. Thus, such forbidden phrases like ‘Natural Order’, ‘Natural versus Trial-&-Error Method’ cannot be neither handled nor understood by many modern scholars (including some self-called “believers”), some of them even feeling ashamed of themselves when being appealed --from time to time-- by such illegitimate words. Nobody should be surprised for all this, because this is the direct consequence of an ancient Greek arithmetical flop. Indeed, from all the evidences at hand and based on the new Natural Arithmetical methods summarized here, all those ancient Greek efforts on root solving were just Geometrical-Trial-&-Error methods and a wordy-promising philosophy of Arithmetic, even though they certainly brought to light wonderful arguments plenty of humanism and logic, they found no Natural Arithmetical root-solving methods, at all, they found no Natural Order for generating irrational numbers, and as a direct consequence they led the way to the creation of: the Decimal System, the Cartesian System and our modern Mechanicist Philosophy, as well as the establishment of a new mainstream of thought where Natural Order doesn’t exist but just those personal creations and opinions of egotistic pseudo-scientists (Cartesian-Technicians) devoted to produce mod cons and lots of money.
Moreover, nowadays so many authors have proclaimed in so many papers, books and other publications that the Cartesian System, decimal numbers and infinitesimals were the only superior way to finally achieve general high-order algorithms, whereas Arithmetic and ancient beliefs on a “Natural Order” were just an obstacle for mathematical progress. So it is not a surprise, at all, to find some people who prefer to take no notice that these new extremely simple High-order Arithmetical Methods do not appear in neither any European, nor Chinese, nor Hindu, Nor Arab, nor Inca, nor Mayan, nor any other text on numbers, since ancient times up to now, a truly shocking issue that really rock the boat. Some others prefer not to hear that the Rational Mean was used and analyzed by: Nicolas Chuquet (1484), Cauchy, Charles de Comberousse, Haros(1802), Farey(1816), John Wallis, Charles Sanders Peirce, Stern-Brocot(1858-1860), Lester R. Ford, Pick, etc., and that this so important arithmetical operation was always treated basically as a curious property.
The new extremely simple Natural Arithmetical Methods based on the ‘Rational Mean’, which are briefly summarized in this web-page constitute a strong evidence of Natural Order in the Science of Quantity: The Simplest Arithmetic. Actually, these methods are not “new” but have always been and shall always be within the realm of the immutable Science of Quantity, never changing, no matter any “Superior” opinions lying over the shoulders of some “giant” mathematicians.
Let’s see now how to apply the observations of superparticular ratios to the evaluation of primitive roots.
The Simplest Rational Processes
Let’s take a look now at some of the simplest and fundamental Rational Processes which led the way to other high-order arithmonic processes. In this particular case, we will use the most important property of superparticular ratios: A set of fractions whose product is 2, that is, a set of approximations by defect and excess to the desired root. Even though the generalization of this simple method is also trivial I think that numerical samples are the best scenario to get a basic grasp 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 evaluate the following three rational means:
As you can see, we have produced another set of three fractions whose product is trivial and equal to 2, that is, we obtained three approximations more closer to the cube root of 2 (Notice that the form factor we used in the last group of fractions is: 2/2)
If we continue this rational process by evaluating in each step other three rational means by using the values produced in the previous step, then we will get the desired approximation to the cube root of 2.
Second step: Evaluating another set of three rational means:
The Form Factor remains the same: (2/2)
We got a new set of three fractions whose product is trivial and equal to 2: (20/16)*(25/20)*(32/25)=2
We can continue with this process in order to get new sets of three fractions:
Third step:
Note: (45/36)*(57/45)*(72/57)=2
Fourth step:
Note: (102/81)*(129/102)*(162/129)=2
And so on.
Each set of three fractions represents approximations by defect and excess to the cube root of 2, the product of the fractions is always trivial and equal to 2.
Even though this rational process has low convergence speed and sporadically gives some ‘best approximations’, its simplicity and clarity would have to make bloom some stirring feeling in the soul of all those who truly feel something by the study of Number, mainly when considering that from all the evidences at hand this extremely simple procedure does not appear in any text on Numbers since ancient times up to now.
The Natural Order imprinted in such a simple rational process could not be ignored by someone who truly feels some respect for the science of Quantity, that is, for mathematics.
At this point, I would suggest the reader to ask his favorite math-teacher or expert on numerical methods about any precedents on these extremely simple methods. The reader will realize they are absolutely unaware of these rational processes, so the main question is: Why these trivial methods have not appeared in any text on numbers since ancient times up to now?
As an example of the incredible simplicity and supremacy of this form of relation: The Rational Mean, let’s take a look at the simplest high-order Rational Process for computing the square root of a number P.
Given a set of two fractions whose product is P:
Based
on the above definition of the Arithmonic Mean and given the sequences: .gif){kind=small}
compute
the following two arithmonic means .gif){kind=small}
which in this very particular case will be respectively
the arithmetic and the harmonic mean. At this point there is nothing new,
because it seems that historians have found that ancients certainly knew how to
compute square roots by agency of the arithmetic and harmonic
means. So in this particular case the Arithmonic Process yields the same well
known iterating functions for approximating the square root of P:
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It
is trivial to prove that you can use these new functions as independent
iterating functions for computing the square root of P, also all we know that
the first one is the same than the one produced by applying Newton’s
method to the equation .gif){kind=small}
.
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:
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In order to get those two functions whose product is trivial and equal to P, it is just necessary to modify the forms of x/1 and P/x by agency of the Form Factors: x/x and P/P , and then compute the rational mean, pretty simple.
Now, it is trivial to see that if you continue by applying such form-factors then you will get another new set of two functions whose product is always trivial and equal to P.
Just try another step:
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This way you get two iterating functions whose product is also 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.
In the next step of this Rational Process you will get a function which multiplies by four the number of exact digits in each iteration, the first one will be the same than the one you get when applying the re-known HOUSEHOLDER’S METHOD (fourth order) for the square root, and the second one will be another function which also multiplies by four its exact digits.
If you continue this process, then you will get similar high-order functions to those you get from Householder’s method for computing the square root of P, along with another reciprocal iterating function.
Let’s the reader realize how disturbing it could be to confirm that all these naive rational processes have no precedents in the whole history of root-solving, that is, since Sumerians up to now. All this is truly inexplicable because it was very difficult to even deliberately hide the wonderful Natural Order imprinted into these rational processes.
So the question is:
¿Why all these trivial Natural Arithmetical Methods do not appear in any text on numbers since ancient times up to now?
¿Why we didn’t learn such trivial Natural-Arithmetical stuff at school?
Indeed, the main question here is: Why?
Based on evidence at hand, I could not say that those godlike coteries deliberately ignored all these simple arithmetical methods, but one can definitely say that they never made any serious quest for the Natural Order underlying the Science of Quantity.
Actually, since many centuries ago some selected coteries mainly made up of influential egotistic people -- in most cases dressed up as representatives of some religions-- have successfully blocked and crushed any incoming ideas relating the existence of ‘Natural Order’. By doing so, they have persuaded so many people to believe that because everything in the world have been created by Chance then you have no choice but just follow their superior ideas, i.e.: The Cartesian System and the Infinitesimals.
This way, all the members of those selected coteries became the owners of the mainstream of thought and their superior ideas constitute fashionable laws of the universe. They pretend to be our guide, that is, to be our spiritual leaders before such chaotic universe.
Of course, it is clear that those coteries deny the existence of any 'Natural Order' for the deliberately purpose of seizing power, they just want people praising and thinking about all their very personal ideas instead of making a serious quest for Natural Order.
It is so sad to realize that mathematics have been caught into such an egotistic-Cartesian trap since long time ago. Now, just imagine how many other NATURAL and TRIVIAL methods remain ignored all around the Science of Quantity, actually it could be just the tip of the iceberg.
The re-known American philosopher Charles Sanders Peirce was quite right when stating the following striking phrase about the Rational Mean (though he didn’t assign any name to this operation):
(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…”
A true outstanding statement brought by him while observing the wonderful order imprinted on the rational sequences generated by agency of the Rational Mean.
Notice that problems related to the root-solving issue have been the very spine of mathematics since ancient times up to now, so it really makes you wonder about the huge amount of other TRIVIAL arithmetical methods which remain still ignored all around the Science of Quantity. It makes you wonder if Newton really knew what he were talking about when he entitled his book as: “The Mathematical Principles of Natural Philosophy”. ¿Natural?. Indeed, his use of the transcendental word ‘NATURAL’ is a real joke, mainly when considering that from all evidence he was unaware of all these truly NATURAL AND TRIVIAL ARITHMETICAL METHODS.
Rational Process and Daniel Bernoulli’s method.
The rational mean as a general and unifying concept:
There are many other simple rational processes for approximating roots, as for example:
Starting from the initial set of values whose product is P, and computing
three rational means (as shown) in each step we can get three other
approximations closer to 
.
This rational process gives the same results that come from applying the well known Daniel Bernoulli’s method, all this has been analyzed and stated in my book: “La quinta operación aritmética”. So we can see that the Rational mean is a general and unifying concept which embraces innumerable root-solving algorithms.
Golden Mean (Divine Proportion)
The irrational number:
known as the Golden Mean (a number satisfying the golden proportion: p/q : q/(p+q) is a solution of the equation x2+x-1=0 and is related to the Fibonacci's sequence: 1, 1, 2, 3, 5, 8, 13,...
The Rational Process for approximating the irrational value of the Golden Mean is as follows:
Considering the Golden
proportion 
We can develop the following very simple rational process, which by the way, leads the way to the generalization of approximating the solution of any algebraic equation by agency of the rational mean:
In each step, let’s compute two rational means and notice that the second one always comes from modifying the form of the second fraction using the form factor: 2/2:
and so on...
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.
New High-order arithmetical Root-Solving Algorithms
The most basic principle of root-solving algorithms which we will be considered here is:
‘k ‘ values whose product is P represent ‘k’ approximations, by defect and excess, to the kth root of P.
If we find a rational process (based on the rational mean) which yields in each step k values –closer to each other-- whose product is always trivial and equal to P, then we can say we got a true natural and trivial arithmetical algorithm for roots solving.
In this way, let’s see some examples on those rational processes.
First, we will see an example on the fourth root, the reader will find so easy to extend such method to the cube root as well as any other high-degree roots. Indeed, the case of the cube root is so trivial that I found more interesting to show another example on the seventh root. Of course these methods can also be used for solving algebraic equations of any degree.
Example on the fourth root of any number P :
Basic Arithmonic Process:
Given
the set 
and two sequences 
By computing four arithmonic means for each of the following arrangements of A4:
The lines indicate the pattern of changes in the position of the elements of A4
The pattern is similar for roots of any degree. (See below: Seventh root of P)
It yields the following four ‘Reciprocal Arithmonic Means’ :
Look at the numerators and denominators, notice how they are trivially canceling each other, so evaluating the product of all those expressions requires no arithmetical operations, at all, this is the reason they are called ‘Reciprocal Arithmonic Means’.
Therefore,
we can see four expressions whose product is trivial and equal to: 
If we assign numerical values to the elements of the initial ser A4 then those values become four approximations by defect and excess to the fourth root of P.
By computing other four arithmonic means using those four expressions produced in the previous stage it will yield closer approximations to the fourth root of P, and by repeating this procedure at will, it will yield the desired approximations to the root.
This simple procedure will be called: ‘Basic Arithmonic Process’
Secondary Arithmonic Process:
It can be proved there is no need for computing four arithmonic means in each step of the process when using a set of the form:
(note: there are many other different choices for assigning elements to A4 )
With this very special set of initial values, those four reciprocal arithmonic means take the following forms:
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Note: The errors were computed for the fourth root of 2 starting from the initial value x0=1.
We could continue this process substituting the variable x of the set A4 with any chosen function of the table, then computing other four arithmonic means, and by repeating the same procedure, we could get the desired approximations, however, if you operate any of the functions of the table as an independent iterating function it will yield exactly the same result. That’s why in the table are shown the errors (true value - approximation) produced when iterating each function up to the fourth step. In this way, the total work is drastically reduced. This procedure will be called ‘Secondary Arithmonic Process’.
In the case of the ‘Basic Arithmonic Process’ for approximating the fourth root, the convergence is guaranteed by four defect-&- excess approximations getting closer in each iteration. It is clear that computing four arithmonic means in each step would seem to be a lot of calculations, however, there are so many ways to reduce them to just one column of values as has been already stated in the book “La quinta operación aritmética”. From this point of view, the ‘Basic Arithmonic Process’ can be classed as a truly Natural Method.
In the case of the ‘Secondary Arithmonic Process’ (which is somehow related to Cartesian-infinitesimal methods), in each step of the process and before computing those required four arithmonic means -- we introduced a very special artifice which consists in the implementation of the set (x, x, x, P/x3 ), so for this case the convergence can not be guaranteed. We must have in mind that in each step of the process we are arbitrarily choosing any of the resulting arithmonic means and substituting it as the variable x in the arbitrarily imposed set (x, x, x, P/x3). This is the reason Newton’s method can be classed as an artificial algorithm (not a natural method) whose convergence cannot be guaranteed. Newton’s method is just another artificial-geometric algorithm coming out from the artificial-geometric system called: Cartesian System.
Notice that the implementation of the set (x, x, x, P/x3) plays the same role than that of the tangent to a curve in Newton’s method, and as previously said it leads to an uncertainty on the convergence. From this point of view, the ‘Secondary Arithmonic Process’ can not be classed as a natural method and consequently this also applies for Halley’s and Householder methods.
Observations on the functions in the table (Secondary Arithmonic process):
The
first function is that which results from applying Newton’s method to the
equation 
The last one has slightly faster convergence than Newton’s and corresponds to the harmonic mean of A.
The other two fifth degree functions have faster convergence speed
Note: Depending on the set and the sequence used to compute the Arithmonic Mean, it is evident that sometimes the result may coincide with the well-known Arithmetic and Harmonic Means.
When this rational process is used in a similar way to compute any odd-indexed root, as for example the cube root, it yields -- in the first step -- functions that triple the number of exact digits in each iteration, that is, the equivalent to Halley’s method. Quite interesting are all those innumerable ways for developing rational processes based on the Arithmonic Mean.
By using the Rational Mean one can obtain the same functions that result from applying Householder’s method, that is: high-order algorithms, as well as many other new iterating functions
Cube root of any number P :
In other cases it is possible to get better functions than Householder. For example, the reader might verify that in the case of the cube root of any number P one can obtain --in the second step of the arithmonic process-- a function like this:
Fifth order Arithmonic iterating function
Which
multiplies by 5 the number of exact digits in each iteration, and clearly
differs from those that come out when applying Householder’s method to the
equation 
.
Fourth order Householder’s iterating
function
Fifth order Householder’s iterating function
Odd-indexed roots: Approximations by agency of square roots
One can try other set of initial values whose product will be always trivial and equal to P, so when approximating the cube root of a number P you can get the following iterating functions in the first step of the arithmonic process:
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(Note:
Arithmonic iterating functions for the cube root of P |
Error |
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The errors were computed for the cube root of 2 starting from the initial value x0=1.
As you can see in the table, we can compute an odd-indexed root by agency of square roots. All this can be applied to 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.
Seventh root of P:
By
using the initial set 
and computing seven arithmonic means according to the
sequences .gif){kind=small}
which correspond to the following seven arrangements of A7
, which actually are very simple because only the term P/x6 changes
its position in the set:
.gif)
(The lines indicate the pattern of changes in the position of the elements of A7)
(Notice the similarity with the corresponding arrangements of the above example on the fourth root)
we get seven reciprocal arithmonic means whose product is trivial and equal to P :
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Simplest Arithmetic |
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