Wondering about Domingo's Rational Mean book"

[dgomezm at etheron.net]

Very sorry, for this message appearing twice, my fault.


Subject:      Re: Wondering about Domingo's Rational Mean book"
Author:       Iain Davidson <Sttscitrans at tesco.net>
Organization: Tesco ISP
Date:         Sat, 10 Jun 2000 23:27:24 +0100


Domingo Gomez Morin wrote:
>> Given a set of two initial fractions:
>> [3/2, 4/3]
>> According to my definition and notation (see home page),
>> Rm[{3/2, 4/3},{2*3/2*2, 4/3}]=[7/5,10/7]
>> Rm[{7/5, 10/7},{2*7/5*2, 10/7}]=[17/12,24/17]
>> Rm[{17/12, 24/17},{2*17/12*2, 24/17}]=[41/29,58/41]
>> Rm[{41/29, 58/41},{2*41/29*2, 58/41}]=[99/70,140/99]
>> and so  on....
>> is the most simple example of a Rational Process for aproximating
>> the square root of 2 (yielding sets of two fractions whose product
>> is always 2).
>> Thus, as everyone can see, this very simple rational process
>> yields two column of values, the first one
>> (3/2, 7/5, 17/12, 41/29, 99/70,...) and
>> the second one
>> (4/3, 10/7, 24/17, 58/41, 140/99,...)


"Iain Davidson" <Sttscitrans at tesco.net> replied:

>I don't see why you choose [3/2, 4/3]
>as the two initial fractions and why you choose
>to multiply top and bottom by 2.
>What you seem to be doing is choosing two fractions
>a/b and c/d such that ac= 2l and bd = 1.

I always do prefer to bring numerical examples, this brings
more fun to many people.
The set of two initial fractions used in a rational process
for approximating the square root of any number P  is:
[a/b, P*b/a]

It is clear that  [3/2, 4/3] and consequently all the other new sets
satisfy the well known condition for generating reduced fractions  |ad-
bc| = 1.
This particular rational process is not only related to CFs
but to Daniel Bernoulli´s method for solving algebraic equations.
As a matter of fact Bernoulli´s method for  solving
second order algebraic equations is exactly the same procedure that
the CFs expressions of algebraic numbers of second order (Traditional
CFs).

I didn´t feel any need for explaining the above facts, specially when
considering that my favorite point is about the significance of the
lacking on any precedents on such trivial rational processes in the
whole history of mathematics,
worst when considering that by agency of the rational mean
(simple arithmetic) you can easily find Bernoulli´s,
Newton´s and Halley´s methods, among other infinite number
of new methods.
That is what I call "general and fundamental".
Unfortunately, it seems that you are not willing
to discuss such a very important point, very important indeed,
specially for all those readers-purchasers of math books  who
have never read anything about such trivial way (simple arithmetic)
for finding Bernoulli´s, Newton´s and Halley´s methods.

What do you think a humble reader-purchaser of math books
should say about that?

Just say: Oooops.....just two or three thousands years to reach that?

I think one should say:
Are all those math-books on warranty?
What about "rigorousness" in mathematics?

This a very important point for me and some others,
and hope no to be plunged into a one-way discussion.


>While the fractions get closer and closer to sqrt(2), there
>is no guarantee that best approximations will be found.


Well, let me try to explain all this in a more general way,
even though it is very difficult by using this media:

****(Notice that I won´t use the symbol _ as subscripts, I don´t
find it neccesary for the following explanation, so the
numerical value of the variables C1, C2, a1, a2, b1 means subscripts)

It is well known that for any  CF whose coefficients
are:
Sucesive numerators: b1, b2, b3, b4,...
Sucesive denominators: a1, a2, a3, a4,...

that is:  CF= b1 + b2/(a1+ b3/(a2+ b4/...

Being C1, C2, C3, ... the successive convergents of the CF
then each convergent can be computed in terms of the two
preceding convergents by using a lineal homogeneous
recurrence relation (LHRR) of the second order:

Thus being the first two convergents of the CF:
C1= b1/1,  		let´s denote this fraction: N1/D1
C2= (b1*a1 + b2)/ a1, 	let´s denote this fraction: N2/D2

then the third convergent is:

C3= (a2*(b1*a1 + b2)+ b3*b1)/ (a2*a1 + b3*1) =
N3/D3= (a2*N2+ b3* N1)/(a2*D2+b3*D1)

As you can see according to the general definition of
the rational mean stated in my page:
www.etheron.net/usuarios/dgomez/RmDef.htm
the convergent C3 is just the rational mean between the
two preceding convergents C1 and C2.
Of course, all this also applies for all convergents Ci.

Therefore,  all CFs are just a rational process ruled by the rational
mean, including of course the particular case of the
simple CFs (mediant, reduced fractions, b1=1, b2=1, b3=1,...).
All this is fully explained in my book and even though
I´m pleased to answer any question, it is hard to put all this
by using Newsgroup format. An english version of my book
is coming out soon and will be available via internet.



Any convergent, any reduced fraction,
any best approximation condition (|ad-bc| = 1), any new idea,
that is, anything  coming out from Cfs concept is fully embraced
by the Rational mean concept and the Rational Process.
The rational mean rules traditional CFs.
So if you have some doubts, something to say,  about the rational mean,
then your arguments are straightaway directed to the CFs, Bernoulli´s,
Newton
and Halley´s method, because all of them are ruled by the rational
mean as stated in my book and summarized in my home page:
www.etheron.net/usuarios/dgomez/
All of them are just rational processes ruled by the rational mean.

I find of so much importance to consider the striking fact that western
culture needed more than two or three thousands years to find
Bernoulli´s,
Newton´s and Halley´s methods by means of the infinitesimal calculus
and the cartesian system, also that all  their most outstanding
mathematicians never realized that such methods could have been
easily developed by using simple arithmetic.
That´s just striking, specially when considering that any
algebraic equation can be formulated in terms of a "parametric"
equation (may be the term "multiple proportion" would be more
adequate),
a very important point that allows us to find the roots
of any algebraic equation by agency of the rational mean, that is,
by means of simple arithmetic.
See for example, the most simple (trivial) example on this matter
at the end of my page:
www.etheron.net/usuarios/dgomez/arithmonic.htm
which deals with the golden mean (assuming x=p/q you can formulate
the equation x^2+x=1 in terms of a "parametric" equation (proportion),
then applying the rational process as shown in the web page, so any new
set of two approximations of x is always satisfying the golden
proportion.


Just striking, indeed, specially for a humble purchaser
of "rigurous" math books containing so much rigorous talk about
best approximations and roots solving.



>No, you are making the claim "Again: the rational process embraces
_all_
>CFs."
>There are CFs based on generalizations of Euclid"s algorithm that find
>simultaneous approximations to three quantities, say, Cubrt(4):Cubrt
(2): 1
>So how does the rational process embrace these ?



I think that by means of this very long message
(very sorry for its length) I have clearly stated
that the rational mean rules the CFs, so they are just part of a
rational process of second order.
So, again,  if you have some doubts, something to say,
about the rational mean, then your arguments are immediately
directed to the traditional CFs, Bernoulli´s, Newton and
Halley´s method, (among others) because all of them are
ruled by the rational. That´s it. I think we could save a lot of time
if you were could  read the contents of :
www.etheron.net/usuarios/dgomez/RmDef.htm
and
www.etheron.net/usuarios/dgomez/Roots.htm
which explain sufficiently the rational mean and rational process
definition

Now, considering that this not a one-way discussion I hope
you take some time to formulate here those CFs method you mentioned
for finding a sequence of best approximations for the cube root of
any number, I´m sure it will be amusing and very useful not only
for me but for the whole audience of this newsgroup.
Also I would like to get any response about the other topics
I have mentioned on the precedents and history of developing
Bernoulli´s, Newton´s, and specially Halley´s method
by means of simple arithmetic, if I´m wrong on this then
there must be, at least, two or three thousands years, sorry,
I meant to say "thousands of references" (books, articles, etc)
on such a very simple arithmetical matter.
:-)


Domingo




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