Rational Mean: (Generalized Mediant)

Brief definition, some examples and observations extracted from the book: “LA QUINTA OPERACIÓN ARITMÉTICA, Revolución del Número” (Title translation: The Fifth Arithmetical Operation, Number Revolution) ISBN: 980-07-6632-4. Copyright ©. All rights reserved under international Copyright Conventions. Author: D. Gómez.

CONTENTS:

 

 

 

 

The Rational Mean

(Generalized Mediant)

Rational Mean Definition

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

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The expression:

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is a mean value (Rational Mean, Rm) between the extreme values (a1/b1), (an/bn):

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A more general notation for multiple rational means can be expressed this way:

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Example:

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The following case:

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is also the rational mean between the same n rational numbers, this time by modifying the form of each fraction using the factors: F1, F2,..., Fn.

We can say that given any set of n values taken at will (integers, rational or irrational) arranged in an increasing order:

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and given another set [F1, F2, F3, . . ., Fn] of any positive values (integers, rational or irrational), then the expression:

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is the rational mean between the values v1,v2,v3, ... ,vn


 

General Rational Operation (Gro) Definition:

We can go further by defining the (Gro) General Rational Operation (not necessarily a mean value) between n arbitrary values v1,v2,v3, ... ,vn as stated in the following example: Given n values v1=a1, v2=(-a2/b2), v3=(a3/b3), v4=-a4, . . . , v5=-an , the Gro between those values is:

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Moreover, Given n rational functions: f1(x)/g1(x), f2(x)/g2(x), f3(x)/g3(x), ..., fn(x)/gn(x), the General Rational Operation (not necessarily a mean value) is:

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as said, not necessarily a mean value. In some cases the General Rational Operation becomes a Rational Mean between those n rational functions.

Any algorithm based on the Rational Mean will be called: Rational Process.

The analysis of the rational mean (Rm) has been restricted –all through the history of mathematics– to the specific case called Mediant (n=2) and some curious properties of Farey series, Ford's circles, Stern-Brocot tree and the generation of the convergents of the simple continued fractions:

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Comments

Some experts usually state that this operation is not well defined within the set of rational numbers, that is, the rational mean works with ordered pairs of numbers. As an example, let's compute the rational mean (mediant):

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

and the rational mean between the same values this time using 6/4 instead of 3/2:

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

Same values, same operation, however, different results.

Now, if we argue that this operation is not “well defined“ within the set of rational numbers, then we must also say that the following mathematical operations are not well defined, mainly when considering that the rational mean is the fundamental principle which rules them:

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 an specific rule.

Geometric mean.

•Generation of convergents of generalized continued fractions.

•Algebraic and transcendental numbers.

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

•Power series expansions (Maclaurin-Taylor series).

•Definition of the arithmetical operations of irrational numbers.

•Statistics.

•Gravity center.

•Ford's circles

•Farey's fractions

Contrary to the mainstream of thought of many modern mathematicians, it is clear that the issue on the "definition" of this operation within the set of rational numbers should be considered from a very different point of view, I mean, a new vision which clearly differs from the cartesian system principles.

As said, contrary to the Cartesian system fundamentals, Number should not be considered just as an "absolute" value (the term "absolute" means "decimal value"). In the same way flowers bring us their multiple natural properties: beauty, color and scent, also Number bring out much more than just an absolute value: It bears a relative value (The form of the ratio) and a very specific location within a set of ratios which plays an important role in the development of roots solving algorithms. The relevance of all these properties of Number will become evident to the reader mainly when using the new Arithmonic Mean. Cartesian system have depersonalized Number confining it to just an absolute value (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 has been consecrated as the most outstanding successes brought to light by the Cartesian system and decimal fractions. Believe It or Not!, based on all the evidences, it seems that the extremely simple arithmetic methods shown in the book “The Fifth Arithmetic Operation” have no precedents at all, all through the very long story on roots solving.

Based on the new elemental rational processes and many other considerations you will realize that:

•The Cartesian system cannot be considered as a fundamental system of Natural Philosophy but just as an artificial creation which apart from being extrinsic to the natural properties of Number also contributes to distort and vitiate the genuine image of Quantity. Contrary to the mainstream of the Cartesian system, Number should not be considered just as an "absolute" value (the term "absolute" means "decimal value" commonly used in Cartesian system). In the same way a flower bring us its multiple natural properties: beauty, color and scent, also Number bring outs much more than just an absolute value: It bears a relative value (The form of the ratio) and a very specific location within a set of ratios. The importance of all these properties of Number will become evident to the reader mainly when using the new Arithmonic Mean.

•The arithmetical operations of irrational numbers can be easily defined by agency of the Rational Process (based on the Rational Mean in accordance with Number, not by any other but Number itself), rather than by using Dedekind's and Cantor's opinions and 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 degrees get distorted when using the traditional continued fractions, that is, the “Second Order Continued Fractions”

•Surprisingly, there are few precedents on the analysis of the rational mean, most of them on the special case: Mediant. Some people who worked with this operation: Nicolas Chuquet (1484), Haros(1802), Farey(1816), Cauchy, J. Wallis, C. S. Peirce, Stern-Brocot(1858-1860), D. Hidalgo (1963). Lester R. Ford's circles and Pick's theorem are also related to this operation.

•The rational mean is the most elemental arithmetical operation for roots solving, it also rules the power series expansions and the generation of transcendental numbers.

•The Arithmonic Mean is an essential and very simple operation which --from all the evidences--have been passed over all through the whole history of mathematics.

•Based on all those important arithmetical methods, number properties and observations that have been passed over during so long time, one can say now that it is certainly a pathetic arrogance to think that any result (i.e.: imaginary numbers, higher dimensions, cartesian system, relativity theory, etc.) coming out of the opinions and judgments of any mathematician, could ever supersede the natural order and wonderful properties 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,...” (quoted text: Nicomachus, chap.VI, [1]).

 

 

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Copyright © 1993-2002

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Last revision: 2002