Nuevos métodos aritméticos para raíces (español)

New Generalized

Continued Fractions

New Geometrical constructions (SIP)

Curiosity on two Mersenne numbers

Links and References

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. 1993-2008

____djesusg@gmail.com___

CONTENTS:

A brief summary on new extremely simple high-order root-solving algorithms developed by agency of the 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 iteration algorithms, as well as the Arithmetic, Harmonic, Geometric, Arithmonic, and Golden mean, among others.

From the evidence at hand, all these simple arithmetical high-order methods have appeared 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 get 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 though this rational process has slow speed of convergence and sporadically gives some ‘best approximations’, its simplicity and clarity would make bloom some stirring feeling in the soul of all those who truly feel something about Number and its harmonies, 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.

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…

So, in the fourth step we got 479 exact digits for the square root of 2, just by means of the most simple arithmetic. We did not required neither any derivatives, nor the Cartesian system, nor any cumbersome-&-mysterious concept coming out from Infinitesimal Calculus.

Of course a lot of questions arise from all this new trivial arithmetical methods, the first one:

¿Why nobody ever taught me about such trivial stuff at school?

¿Why there is no trace of such trivial high-order methods in any “rigurous” math journal since antiquity up to now, but at the same time many “rigorous” journals have published a plethora of cumbersome methods for appoximating square and cube roots which are by far slower and less general than these new trivial arithmetical methods?

¿Why most of them use to boast of their math-rigurosity but they have never published such trivial methods since babylonian times up to now?

¿Since Babylonian times up to now?

At this point, I recall 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 Mediant which is a very particular case of the Rational Mean. Even though I must add that he didn’t assign any name to this operation but just used to call it as: ‘form of relation’:

(Collected papers, Harvard University Press, 1933, Vol. IV, art. 681, p. 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…”

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 always satisfying 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 of the simplest ways 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 the same

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

As said, instead of using these iteration functions one could continue the artihmonic 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.

At this point I must say that I do not consider “imaginary numbers” as true representatives of Quantity but just kind of glued stamps used to mark some real numbers so they can be individually handled. I mean, they should not be considered as another different realm of numbers, because the stamp i is just a kind of glued “post it” and consequently such complex numbers always become inevitably handled by the same basic rules of Quantity.

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: