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Re: Rational Mean (Mediants) and the Irrational Numbers.
Posted:
Jul 7, 1995 2:26 PM


Invisible friends, Cordiales saludos Milo,
(From an .math thread)
I said :  "1. The Harmonic Mean is just the 'Rational Mean' between fractions having equal numerators. (This EXTREMELY SIMPLE observation seems to have no historical precedents at all." 
You said :  "False. The product of the arithmetic and harmonic means were discussed by Plato..[cut] 
False ???!!!
Milo, the Harmonic Mean of a set of 'n' numbers X1, X2, X3, ..., Xn has been defined  for the whole history of mathematics as :
Harmonic Mean = n/ (1/X1 + 1/X2 + 1/X3 + ... +1/Xn)
However, the Harmonic Mean can be defined in a rather simple way than the above. Let's see a very simple example :
Given two common fractions, for example, 5/4 and 10/7, you can compute their Harmonic Mean by just modifying their forms so their numerators become equal, then, by just applying the 'Rational Mean' you will get the harmonic mean between 5/4 and 10/7.
Thus, by computing the Rational Mean between 10/8 and 10/7 (Equal numerators, 5/4=10/8). We get : Harmonic Mean = Rm(10/8 , 10/7) = 20/15 = 4/3
Now, considering that the harmonic mean has been "wellknown" since ancient times, it is really surprising that this so simple definition (BASIC PRINCIPLE) has not been considered in any math book for the whole history of mathematics. (I would be happy to see any math book containing any analysis on this subject, so please, correct me if I'm wrong). Worst, I cannot find any excuses for all this, mainly when considering the following two remarks : 1. This so simple definition leads the way to many useful properties and methods based on the Rational Mean. 2. Suppose you are taking a mathexam and your teacher realize you are not really acquainted with some BASIC PRINCIPLES related to the subject he is asking for, then, it is for sure you will not be able to make your apologies and certainly won't pass such exam.
Of course, the above two remarks also apply to the posting I sent to the mathhistorylist on June 28 (Entitled: 'Mediants and continued fractions) , regarding the arithmetical operations of the irrational number, i.e, SQRT(2) * SQRT(3) = SQRT(6). I can't hardly believe such a simple definition (Based on the Rational Mean) on the arithmetical operations of the irrational numbers hasn't been considered (By Dedekind, Cantor, Kronecker, etc.) in any real analysis. (I think someone should correct me if I'm wrong, Indeed, I would really appreciate that!)
But, don't worry Milo, my point should be an extraordinary incentive for you, mainly because if modern mathematicians don't know anything about this so simple observations on the Rational Mean and the irrational numbers, then you can be sure there are certainly many other simple facts they (We all) are not acquainted with simplest mathematics, specially, with ancient mathematics (Egyptian fractions).
I hope you have understood my point on the Harmonic Mean.
Domingo Gomez
****************************************************************** "All the 'Means' are 'Mediants', say, 'Rational Means'" Domingo GÃÂ³mez MorÃÂn http://www.etheron.net/usuarios/dgomez/default.htm Caracas, Venezuela. ******************************************************************
 From: Milo Gardner <gardnerm@ecs.csus.edu> To: Domingo Gomez Morin <dgomezm@etheron.net> Cc: Mathematics History <mathhistorylist@maa.org> Subject: Re: Rational Mean (Mediants) and the Irrational Numbers. Date: SÃÂ¡bado 5 de Julio de 1997 12:14 PM
Hola Domingo:
Thank you for continuing this thread on the sci.math USENET. There are several points of your position that MAA mathhistorylist subscribers may wish to consider.
Regards,
Milo Gardner Sacramento, Calif.
On 2 Jul 1997, Domingo Gomez Morin wrote:
> > > Given 'n' common fractions a1/b1 , a2/b2 , ...., an/bn in an > increasing order (a1/b1 < an/bn), > then : > a1 + a2 + a3 + ...+ an > a1/b1 <  < an/bn > (*) > b1 + b2 + b3 + ...+ bn > > that is, (a1 + a2 + a3 + ...+ an) / ( b1 + b2 + b3 + ...+ bn) is a > mean value > between a1/b1 & an/bn. > ( a1, a2,.... ; b1, b2,...., have same signs) > > This very simple operation was analyzed by Cauchy, though he gave no > name for this operation. > The very special case : > (a1+a2)/(b1+b2) (**) > which is a mean value between a1/b1 & a2/b2 has been always > restricted to the generation of reduced fractions (Farey/Haros > fractions) and is commonly called 'The Mediant'. > It is well known that the 'Mediant' rules the generation of the > convergents of the continued fractions. > > I have found that such a simple operation (*) is CERTAINLY a > fundamental principle of Quantity. > It seems to rule everything relating Quantity!!!.
I see the development of Greek geometry as an extension of Egyptian fraction rules for rational numbers. That is, when Pythagoras and/or Pythagoreans developed the need to accept irrational numbers computational methods were required.
Seen on this level, the simple substitution of prime numbers p, q from pre600 BC rules like:
1. 1/p = 1/p*1 = 1/p(1/2 + 1/3 + 1/6)
(lines 12, 14, 15, 16, and 18 of the EMLR; and 2/101 from the RMP)
2. 1/p = 1/n(n/p), such as 1/8 = 1/5(5/8) = 1/5(1/2 + 1/8) = 1/10 + 1/40
(lines 1, 2, 3, 11, 19, 20, 21, 22, 23, 24, 25 and 26 from the EMLR)
1. 2/p  1/a = (2a p)/ap
(all RMP 2/nth table 2/p series, for p < 101, plus 2/95 = 2/19*1/5)
2. 2/pq = (1/q + 1/pq)2/(p + 1) (all RMP 2/nth table 2/pq series except for 2/35, 2/91 and 2/95)
3. 2/pq = (1/p + 1/q)2/(p + q)
(2/35 and 2/91 in the RMP and in Greek times by the generalized
n/pq = 1/pr + 1/qr, where r = (p + q)/n, such as the Akhmim P.
as we have discussed several times A = arithmetic mean
and H = harmonic mean is contained in this algorithm, as follows:
2/AH = 2/pq in the p = 7 and q = 13 special case.)
were replaced by distances (and other interesting mathematics as set down by Euclid.
> However, the importance of this so simple operation has been despised > by mathematicians since ancient times. The analysis of this operation
I would not say that the historical aspects of the above were despised. All that I would say is that we have no record/records about this type of partitioning method being used. At this point, with the highly developed series found in the Akhmim P., such as detailed by Kevin Brown, I would not say that ancient mathematicians did not think of the innovations listed above. We simply can not say, one way or the other.
> has been restricted in almost all math books to the VERY SPECIAL case > of 'Farey Fractions' and is commonly treated just as a curiosity!.
The whole field of Egyptian fractions tends to be treated as a modern curiosity  by modern math historians  a point that is very curious, to me.
> If one try to look for any sci.math thread on this operation, it will > be easy to realize mathematicians do not attach to much importance to > this operation.
That is true.
> Moreover, if one try to look for any historical precedents on the > analysis of this operation, it will be easy to realize mathematicians > have not made so much of this operation (Apart from the very special > case of Farey Fractions). >
You appear to be correct with respect to modern math historians. Again, since the field of ancient Egyptian fractions appears not to have been seriously attempted to be read, as clearly outlined above (at least since Hultsch in 1895 and his view of the RMP 2/p rule), we may only have to wait until fresh ideas from pre600 BC ancient number theory texts are published by journal like Historia Mathematica.
> > Now, being a civil engineer (specialty Structural Engineering) not a > mathematician but a humble purchaser of math books I find all this > really SURPRISING!!!, mainly because, it is hard to understand why > most of the following very SIMPLE and FUNDAMENTAL observations have > been neither considered in any mathbook nor taught in any math > school, for the whole history of mathematics :
Domingo, I agree with you. Keep up the posts.
> ________________________________________________ > PRELIMINARIES : > > i. Considering that neither Cauchy, nor Chuquet, nor Farey/Haros, > gave a name to the aforementioned general case (*) : > (a1 + a2 + a3 + ...+ an) / ( b1 + b2 + b3 + ...+ bn) > I decided to name it : > 'The Rational Mean.', or, 'Rm' > Hereafter I will use this name, so the name 'Mediant' will continue > being related to the generation of reduced fractions (Farey/Haros > fractions). > > ii. All procedure involving the 'Rational Mean' will be called : > "The Rational Process". > > ________________________________________________ > OBSERVATIONS ON THE RATIONAL MEAN : > All I have found relating the 'Rational Mean' can be summarized by > just saying : > "All the Means are Rational Means" > or, > "Numbers can be easily expressed in terms of a Rational Process" > > The following are the most important aspects of the Rational Mean I > have found : > > 1. The Harmonic Mean is the 'Rational Mean' between fractions having > equal numerators. (This EXTREMELY SIMPLE observation seems to have no > historical precedents at all, it is really striking!)
False. The product of the arithmetic and harmonic means were discussed by Plato. Note that I do not consider Plato as a serious mathematician. He jumbled the number theory of his day to such as point  that anyone that accepts Plato as a mathematician  can not read Greek number theory that was based on Egyptian traditions  that went back 1,500 years  and more.
> 2. The Arithmetic Mean is the 'Rational Mean' between fractions > having equal denominators.
This one is clearly historical. > 3. The Geometric Mean can be easily computed by agency of the > 'Rational Mean'.
The last problem in the RMP provides clear hints that this idea was well known to Egyptians.
> 4. The Rational Mean is the fundamental principle which rules the > continued fractions of second and higher orders. (See observation > 14.). There are precedents on this matter related to the Mediant and > continued fractions of second order.
By the time of Diophantus, the Chinese Remainder Theorem was well known in Hellenized Egypt  the GrecoRoman Egypt. Continued fractions were used for several purposes, as you outline.
> 5. Algebraic numbers and transcendental numbers can be easily > computed by agency of the 'Rational Mean'. This operation serves not > only to compute square, cube roots, ... but for solving algebraic > equations and computing transcendental numbers. Some proofs and > examples on this matter are shown in my paper: "New Elements for The > Irrational Numbers" published by JTFM, Berlin, september, 1996. > (Please, see my homepage)
I will do that.
> The 'Rational Process' also leads the way to Daniel Bernoulli's method > for solving algebraic equations and many others procedures. > There is much more to say on this matter. > Based on the above phrase "All the Means are Rational Means" > I could say that all modern and ancient methods for computing roots > are ruled by the Rational Mean
But the subject should be studied, right?
> 6. Of course, the 'Golden Mean' can be also easily computed by agency > of this operation. >
AH was called by ancient scribes as the Golden Proportion, right?
> 7. MORE IMPORTANT : Irrational numbers and their arithmetical > operations can be EASILY defined by agency of the Rational Mean, as > stated in my last posting to the MathHistoryList (maa.org) entitled > "mediants and continued fractions " (June, 28). > There you can see the most simple proof of what Dr. Fowler calls > "DedekindÃÂ´s Theorem SQRT(2) *SQRT(3) = SQRT(6)" (See American > Mathematical Monthly, Vol. 99, No. 8, Oct. 1992) > > 8. The Rational mean can be easily defined by agency of compass and > straightedge > > 9. The 'Center of Gravity' is just a Rational Mean!!!. > > 10. Given two equations : > ax + by = c > cx + dy = e > By dividing both equations, what do you get? > > The Rational Mean c/f between a/d & b/e !!! > > ax + by c >  =  > dx + ey f > > 11. Now, if you take a look at the linear transformations > y > (px +q) / (rx + s) , x > (Pz +Q) / (Rz + S) > of the theory of algebraic Invariants all what you see are just > 'Rational Means' !!!. > > 12. The power series expansion are also ruled by the same principle > of Quantity : The 'Rational Mean'. > As an example let's see the following rational Process which yields > rational approximations to the number 'e'. > > Gi+1 = Rm[{F1i , F2i} , {(Mi/Mi)F1i, F2i}] (***) > > (Sorry for this notation, I can't do it better with email) > Subscript 'i' : iteration number (1,2,3....). > > F1 , F2 : Rational approximations (Common fractions) by defect and > excess to the number 'e'. > > Mi = i +1 > The term (Mi/Mi)F1i means 'Mi' multiplying both the numerator and > denominator of the common fraction F1i. > > Gi+1 : Each new set of two rational approximations to the number 'e' > ('i+1' si supposed to be a subscript). > > The rational mean is computed for each set of two common fractions > enclosed within a pair of curly braces. > The initial values for F1 and F2 are : 1/1 and 1/0 > The output of the Rational Process (***), that is, the sets of two > Rational approximations Gi+1 to the number 'e' are : > > 1 1 >   > 1 0 > > 2 3 >   > 1 1 > > 5 11 >   > 2 4 > > 16 49 >   > 6 18 > > 65 261 >   > 24 96 > > 326 1631 >   > 120 600 > > 1957 11743 >   > 720 4320 > > 13700 95901 >   > 5040 35280 > > > One can easily prove that the left column of values are the same > output as that of the well known power series expansions of the > number 'e' : > e = 1 + 1/1! + 1/2! + 1/3! + ... + 1/n! = > = 2 + 1/2! + 1/3! + ... + 1/n! > > and the right column corresponds to the series : > > (Right column) > e = 3  0!/(2!)^2  1!/(3!)^2  2!/(4!)^2  .... > > > 13. Finally, based on the properties of the 'Rational Mean' and the > 'Rational Process' it is also surprising to realize that > mathematicians always refer to continued fractions just as if they > were just rational processes of second order, that is, just Mediant > procedures (**). However, there are certainly Rational Processes > between three, four, five,.... common fractions, and consequently > there are also continued fractions of third, fourth, .... order. > > In this way, let see an example on continued fractions of third order > for the cube root of 2 : > > ___ > 3 / 1 > \/ 2 = 1 +  > . 1 > 3 +  > 1 > 3 +  > 3 > 3 +  ... > 3 + 1 > 3 +  > 1 > 3 +  ... > 3 + 1 > 3 +  > 1 > 3 +  > 3 > 3 +  ... > 3 + 1 > > The rational mean has lots more wonderful properties!. > There is certainly much more to say on these and other directions but > I think it is enough for the moment. > As I have previously said it really surprises me that the > importance of this simple operation have been neither considered in > any mathbook nor taught in any math school, for the whole history of > mathematics!. > Remember I'm not a mathematician but just a civil engineer, a humble > purchaser of math books. So, IÃÂ´m eager to understand why those math > books do not contain any information on this very SIMPLE matter. > I hope to get some comments from sci.math fellows and experts on this > matter. > Many thanks for your help!. > > Domingo GÃÂ³mez MorÃÂn. > ________________________________________________ > > > **************************************************************** > "All the 'Means' are 'Mediants', say, 'Rational means'" > Domingo GÃÂ³mez MorÃÂn > http://www.etheron.net/usuarios/dgomez/default.htm > Caracas, Venezuela. > **************************************************************** > >




