Re: cube root of a given number

On 16 Jul, 06:13, Gottfried Helms <he...@xxxxxxxxxxxxx> wrote:
Am 15.07.2007 05:30 schrieb arithmeticae:

If you really like to analyze the most simple high-order root-solving
algorithms then you should take a look at:

http://mipagina.cantv.net/arithmetic/rmdef.htm

It is striking to realize that these new extremely simple artihmetical
algorithms do not appear in any text on numbers since Babylonian times
up to now.

Yes, I'd second that. It surprises me, that this method is
not more widely discussed. It is -at least- an amazing
approach in his simpliness and in its line of proceeding,
even if it should not be efficient.

Hope, it will make its way into some books, at least as
an annotation, or in journals/books which are dedicated
to recreational and surprising mathematics.

I don't think the claim that these methods are in any way
new stands up to scrutiny.
The idea of Farey dissections is clearly not new.
It is mentioned in Hardy and Wright for example.
Hurwitz wrote a paper "Ueber die Irrationalzahlen"
in the 1890s which describes a "mediant" method based on Farey
fractions that produces best rational approximations.
Monkmeyer and Mahler have examined generalizations
of Farey fractions, essentially a higher order
"mediant" method, intended to produce best rational
simultaneous approximations to a set of irrationals.
Can Morin find best rational approximations
to cubrt(2), cubrt(4) with his methods ?
He has not been able to do so in the past.




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