# Re: PDQ Unleashed--update

On Thu, 25 Jun 2009 12:00:02 -0500, Rodger Rosenbaum wrote:

Back in October 2004,
Joe Horn posted the ultimate program for finding "best fractions".
His program found fractions that previous programs had missed.

I see on Wikipedia:
http://en.wikipedia.org/wiki/Continued_fraction#Best_rational_approximations

a nice exposition that discusses the all-important "semiconvergents"
for finding those "intermediate" fractions.

Did I ever mention this brief but profoundly elegant text
on Continued Fractions by A. Ya. Khinchin, written in 1935:

http://www.amazon.com/Continued-Fractions-Ya-Khinchin/dp/0486696308
http://www.amazon.com/Continued-Fractions-Ya-Khinchin/dp/0226447499

In that text, Khinchin describes two kinds of "best" approximations,
and derives deep conclusions with the most elegant simplicity,
long before any other reference I can find on this topic.

Here's another discovery and insight by Khinchin,
about continued fractions, which is also elegantly simple,
yet at the same time, other simple questions it raises
remain in the realm of the unsolved:

Khinchin's constant
http://en.wikipedia.org/wiki/Khinchin's_constant

Khinchin
http://en.wikipedia.org/wiki/Aleksandr_Yakovlevich_Khinchin

A contributed review of Khinchin's "Continued Fractions":

A Y Khinchin was one of the greatest mathematicians
of the first half of the twentieth century.
His name is is already well-known to students of probability theory
along with A N Kolmogorov and others from the host of important theorems,
inequalites, constants named after them.
He was also famous as a teacher and communicator.
Several of the books he wrote are still in print in English translations,
he combines a complete mastery of his subject
with an ability to explain clearly
without sacrificing mathematical rigour.

In this short book the first two chapters contain a very clear development
of the theory of simple continued fractions, culminating in a proof of
Lagrange's theorem on the periodicity of the continued fraction representation
of quadratic surds. Chapter three presents Khinchins beautiful and original work
on the measure theory of continued fractions.
The proofs of the theorems in this chapter are also entirely elementary.

I recommend this book to anyone who loves mathematics.

---

Like H. S. M. Coxeter's "Introduction to Geometry," I also found the above text
to be a work of beauty in its own field, yet another direction of approach,
in addition to art, music, and anything else which engages the individual awareness,
to comprehend the awesome creation in which we exist.

"A mathematician, like a painter or a poet, is a maker of patterns.
If his patterns are more permanent than theirs,
it is because they are made with ideas."

"The mathematician's patterns, like the painter's or the poet's,
must be beautiful; the ideas, like the colours or the words,
must fit together in a harmonious way. Beauty is the first test;
there is no permanent place in the world for ugly mathematics."

- G. H. Hardy, "A Mathematician's Apology"

I would say that all of the world's most profoundly perceptive scientists
have had this same deep awareness, including Newton and Einstein
[numerous quotes previously posted, and here omitted]

..
.

## Relevant Pages

• Re: PDQ Unleashed--update
... Joe Horn posted the ultimate program for finding "best fractions". ... In that text, Khinchin describes two kinds of "best" approximations, ... about continued fractions, which is also elegantly simple, ... I suspect she was unfamiliar with the work of various later mathematicians who ...
(comp.sys.hp48)
• Re: Continued Fractions
... continued fractions usually refer to fractions that have ... in 4 cycles total, vs. the 8 cycles required by your expression ... addends of the second expression be p,q,r,s,t. ... Can continued fractions be simplified? ...
(sci.math)
• Re: query on number theory
... with continued fractions. ... the denominator of these fractions. ... >clear that binary representation cannot represent all the fractions ... fraction in finitely many digits that can be so represented in base ten. ...
(sci.math)
• Re: Bit fiddling calculating fraction
... The classic book on continued fractions ... bracketed on the left and right by the convergent and a specific one of the ... intermediate fractions. ... strongest statement possible because he wasn't that interested in the ...
(comp.lang.c)
• Re: WMs Matheology S 030
... In modern set theory, there is no difference. ... A contradiction between mathematics and set theory ... Bijections between different infinite sets can lead to paradoxes like ... fractions between 1 and 2 and, if not yet inserted, the fraction q2 ...
(sci.logic)