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Topic: Generalized Continued Fractions.
Replies: 0

 Domingo Gomez Morin Posts: 215 Registered: 12/3/04
Generalized Continued Fractions.
Posted: Oct 14, 1999 9:44 PM

PERIODICAL REPRESENTATION OF ALGEBRAIC IRRATIONALS BY MEANS
OF GENERALIZED CONTINUED FRACTIONS.

IÃÂ´m including below two generalized continued fractions (Fractal
Fractions), I have found that the first one corresponds to the
minimum-modulus root of any algebraic equation, the second one to the
maximum-modulus root of any algebraic equation.
In case these fractions get distorted please take a look at:

http://www.mathsoft.com/asolve/constant/pythag/dgm.html
http://www.mathsoft.com/asolve/constant/pythag/prtlcnvg.html
http://www.mathsoft.com/asolve/constant/pythag/cf.html

MINIMUM MODULUS ROOT:
In the first fractal fraction (minimum-modulus) you have :
x = 1 / FractalFraction .
for a an algebraic equation of the form:
a_1*x + a_2*x^2 + a_3*x^3 + ... + a_n* x^n = 1
being "FractalFraction" the result of iterating (n=2,3...) the
following expression:

1
---------------
a_n
a_1 + -------
a_1

applying the following expression at any stage j :

. a_j+1
. a_j + --------
a_j a_1
----- = --------------------
a_1 a_2
. a_1 + --------
. a_1

which yields:
.
. 1
x = ----------------------------------------------------
. a_5
. a_4 + ------ ...
. a_1
. a_3 + -----------------
. a_2
. a_1 + ------ ...
. a_1
. a_2 + ------------------------------
. a_3
. a_2 + ------ ...
. a_1
. a_1 + ----------------
. a_2
. a_1 + ------ ...
. a_1
. a_1 + -------------------------------------------
. a_4
. a_3 + ------ ...
. a_1
. a_2 + ----------------
. a_2
. a_1 + ------ ...
. a_1
. a_1 + -----------------------------
. a_3
. a_2 + ------ ...
. a_1
. a_1 + ----------------
. a_2
. a_1 + ------ ...
. a_1
.
.

As a numerical example, by using (x+1)^3 =2 we get a
periodical continued fraction representation for
the cube root of 2 (third order irrational):

___
3 / 1
\/ 2 = 1 + ---------------------------------------
. 1
. 3 + ------------------------
. 1
. 3 + -----------
. 3 ...
. 3 + ------------------
. 3 + 1 ...
. 3 + --------------------------------
. 1
. 3 + ------------------
. 3 + 1 ...
. 3 + -----------------------
. 1
. 3 + ------
. 3 ...
. 3 + ------------------
. 3 + 1 ...

**********************O-O**********************

Now, by using equation (x+1)^2=2,
with, a_1 = 2 , a_2 = 1, ... , a_n = 0
this yields the conventional continued fraction expression
for the square root of 2.

. 1
1 + --------------
. 1
. 2+ ---------
. 2...
(Actually, from now on, conventional continued fractions
should be properly called: "Second Order Continued Fractions").
**********************O-O**********************
Conventional continued fractions (Second order continued fractions)
are just a special case of the nth-order continued fraction shown
above.

Algebraic irrationals DO HAVE their proper periodical
continued fraction representation.
**********************O-O**********************

MAXIMUM MODULUS ROOT:
In the second fractal fraction (maximum-modulus) you have:
x= FractalFraction.
for the equation:
a_0 + a_1*x + a_2*x^2 + ... + a_(n-1)*x^(n-1) + x^n = 0

This time, being "FractalFraction" the result of iterating (j=
1,2,3,....)
the following:

a_n-j-1
a_n-j + ---------------
a_n-1

by aplying at any stage j:

. a_n-j-1
. a_n-j + -------------
a_n-j a_n-1
----- = ---------------------------
a_n-1 a_n-2
. a_n-1+ --------------
. a_n-1

which yields:
.
. a_n-5
. a_n-4 + ------- ...
. a_n-1
. a_n-3 + -----------------------
. a_n-2
. a_n-1 + ------- ...
. a_n-1
. a_n-2 + ----------------------------------
. a_n-3
. a_n-2 + ------- ...
. a_n-1
. a_n-1 + -----------------------
. a_n-2
. a_n-1 + ------- ...
. a_n-1
x = a_n-1 + -----------------------------------------------
. a_n-4
. a_n-3 + ------- ...
. a_n-1
. a_n-2 + -----------------------
. a_n-2
. a_n-1 + ------- ...
. a_n-1
. a_n-1 + ---------------------------------
. a_n-3
. a_n-2 + ------- ...
. a_n-1
. a_n-1 + -----------------------
. a_n-2
. a_n-1 + ------- ...
. a_n-1
.

I hope you donÃÂ´t get distorted this message, anyway you can take a
look
at the web pages I mentioned at the begining of this message.
Kindest regards.
Any references and comments are welcome.
Domingo Gomez Morin