The mediant of any two fractions a/b, c/d is usually defined to be (a+c)/(b+d)
> The MEDIANT works only with reduced
It depends what you mean by "works"
3/5 = 6/10 2/3 = 10/15 5/8 <> 16/25 This just means that the interval between the two fractions is being divided in different ratios.
If a/b = pa/pb c/d = qc/qd
(a+c)/(b+d) (pa + qc)/(pb +qd) divide the interval between the fractions in different ratios.
This would not matter in your case as you are neglecting the key property of Farey intervals All you require is that the distance between the fractions enclosing the irrational you want to approximate gets smaller, i.e. the fractions get nearer the approximand.
The Farey mediant is defined on two consecutive elements of a Farey sequence s/t, u/v with tu-sv =1. This is essential to find best rational approximations. as the Farey mediant is then the fraction with the smallest denominator between s/t and u/v.
A Farey mediant is a mediant, and also a " rational mean" in your terminology, but the converse isn't necessarily true a mediant or " rational mean" does not have to be a Farey mediant
Changing the name of (a+c)/(b+d) does not change its properties. M(a/b,c/d)= (a+c)/(b+d) is not a well-defined binary operation. even if you call it "moolyming" This is simply a matter of common sense. If M(2,5) = 6, then it would be reasonable to expect that M(1+1,2+3) =M(4/2,15/3) =6.
It has nothing to do with the "Cartesian/decimal system" whatever you mean by that term.
>According to modern mathematicians the MEDIANT is a "WELL DEFINED" >operation within the set of rational numbers because it works with >reduced fractions, while the "RATIONAL MEAN" (which does not work >exclusively with reduced fractions) IS NOT WELL DEFINED within the >set of rational numbers.
The Farey mediant is well-defined as is the reduced fraction mediant but everyone expects OPERATIONS to be well-defined. the "square operation" SQR(n) should return the same result on identical numbers. SQR(4) = SQR(2+2) = SQR(12/3).
I don't see that this matters fundamentally for your method. The problem is purely linguistic. By using the words "arithmetical operation" you are making a claim about the "rational mean" which is clearly not true, because "operation" means something quite specific in a mathematical context.
The same problem may be occurring when you claim that your method produces "best approximations"
p/q is a best approximation to sqrt(2), if there is no other rational p'/q', q' <q closer to sqrt(2).
I know that your methods can't produce best approximations because they do not use Farey fractions. It is just chance if a best approximation is found.
It doesn't really matter whether your mediants are in this sense well-defined or not. The consequence of not using reduced fractions is that the "rational mean" will not necessarily equal the Farey mediant and some best approximations will be lost (best case) or no best approximations will be found at all.
>I have never said that my using the Mediant or the Rational Mean is > new. This was not clear to me, but is now. The mediant is not exactly a new concept.
>I have never said that, you are only generating confusion when >stating that, on the contrary, my webpages and book contains full >information on the precedents on the use of the Mediant. In my >webpages I show that there have been some attempts to compute roots >by agency of the MEDIANT, moreover, it is well known that the MEDIANT is the fundamental rule for the generation of convergents in the continued fractions of second order (as I use to call them)
Yes, the Farey mediant eventually gets you to continued fractions which give "ultragood" rational approximations. But your method does not do this. It is not based on Farey mediants.
Continued fractions have very special properties. That is why they have been studied so intensely. Every convergent is an "ultragood" rational approximation and they distinguish between rationals, quadratic irrationals and higher irrationals. Any "higher dimensional" generalization should produce "ultragood" rational approximations and, say, distinguish between rationals, quadratic irrationals, cubic irrationals and higher irrationals. I mentioned x^3 +Ay^3 +AAz^3 -3Axyz = 1 because it is a higher analogue of Pell's equation. If your "higher order" methods prodiuce best approximations they should be able to find solutions to the cubic Pell. I have not seen you demonstrate this.
>What I have said is that the EXTREMELY SIMPLE HIGH-ORDER >ROOT-SOLVING >METHODS shown in my web pages are brand new and have no precedents >in the whole history of mathematics, and I think that math historians >should cogitate on such a crude fact.
This is a historical claim and may well be true. Although one poster has claimed to have found this in a book published in 1945.
I don't know how you are going to search through every book published since Babylonian times to confirm your claim.
If mediant-based methods have been used since the Middle Ages, it does not seem implausible that your method is a rediscovery. It is a common phenomenon.
>You assert that I have said: "I can solve the cube version of Pell's >equation", and you are forcing me to ask you to show such a link to >a posting from mine contining such phrase. All that i have said is >that the methods shown in my webpages certainly produce best > approximations
No, that they cannot do.
>and can yield high order convergence speed as shown in the very >simple example on the square root I posted to you.
Yes, that may well be true.
You make at least two mathematical claims about your method that are false. (arithmetical operation, best approximations). Your historical claim may well be true, but at least one poster states he has a reference predating your claim. It would be interesting if he could post the method he found in the Handbook