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Generalized continued fractions

A new general concept on continued fractions extracted from the book: “LA QUINTA OPERACI�N ARITM�TICA, Revoluci�n del N�mero” (Title translation: The Fifth Arithmetical Operation, Number Revolution) ISBN: 980-07-6632-4. Copyright �. All rights reserved under international Copyright Conventions. Author: D. G�mez.

 

 

CONTENTS:

• Generalized Continued Fractions (Fractal Fractions)
• Conclusions

 

 

Generalized Continued Fractions

Fractal Fractions

Given the algebraic equation f(x) = 0 of n-th degree:

Its maximum modulus root can be expressed in terms of a generalized continued fraction (Fractal Fraction), as follows:

By replacing the fractional part at each stage by the expression:

all this yields the following general expression for the maximum modulus root of f(x):

The minimum modulus root of the equation:

is given by the following generalized continued fraction:

Example:

Being a₀ = -1, a₁ = -2, the representation of the maximum modulus root of the equation:

as a generalized continued fraction is:

We can see that the traditional continued fraction expression of the irrational:

is just a second order expression of the new generalized continued fraction concept.

It is necessary to redefine the representation of irrational numbers by means of traditional continued fractions.

Another example:

Given the equation (x+1)³ =2, or the same -x³-3x²-3x+1=0, whose minimum modulus root is

. Thus, being a₁ = -3, a₂ = -3, a₃ = -1, the generalized continued fraction expression for this root is:

A very interesting expression bringing a periodical representation of a cubic irrational.

The convergents of this fractal fraction are:

yielding successive approximations to

.

This sequence of convergents is ruled by the following lineal homogeneous recurrence relation:

y_n=3y_n-1 + 3y_n-2 + y_n-3

It is important to notice that even the generalized continued fractions are just a special case of the Rational Process (Based on the rational mean).

If one try to represent the cube root of 2 by means of the traditional continued fractions (Second order continued fractions as we should call them) then we'll get a distorted representation (non-periodic coefficients) of this irrational number, as follows:

whose convergents are:

 

Conclusions

It is clear that traditional continued fractions should be better called: “Second Order Continued Fractions”. As we have seen in the above numerical examples, when trying to represent a cubic irrational by means of a second order continued fraction (traditional concept) then one get a disfigured image of the irrational. It is necessary to redefine the traditional representation of irrational numbers by means of continued fractions.

Generalized Continued Fractions (Fractal Fractions) are directly related to the Rational Process (Rational Mean)

 

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Last revision: 2002