Re: About algebraic and trancendental numbers

In article <eg5tsb$3i2$1@xxxxxxxxxxxxxxxxxxxxxx>,
israel@xxxxxxxxxxx (Robert Israel) wrote:

In article <1160144423.794039.220900@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
Hero <Hero.van.Jindelt@xxxxxx> wrote:
G. A. Edgar schrieb:

How to test whether a decimal
number is algrebraic or trancendental? Are some decimal numbers easier
to test than other?


Is there a single known example where, starting with a decimal
expansion, we can show the number is irrational and algebraic?

We can recognize rationals from the decimal. And we can recognize
in a few cases that a number is transcendental. But mostly
we cannot tell from the decimal expansion whether a number is
algebraic or transcendental.

What about the continued fractions? Can we tell from them whether a
number is algebraic or transcendental?

If the continued fraction terminates, the number is rational. Otherwise,
if the elements of the continued fraction are eventually periodic,
the number is a quadratic irrational. If the elements grow too rapidly,
then for some p > 2 there are infinitely many rationals a/b with
|x - a/b| < 1/b^p, and by Roth's theorem x is transcendental. But
I am not aware of any known results that distinguish between the
continued fraction of an algebraic number of degree > 2 and the
continued fraction of a typical transcendental number.

It is widely believed, but not proved, that if x is algebraic
of degree exceeding 2 then its partial quotients are unbounded.
Of course, this is also true of the partial quotients of a typical
transcendental number (provided you have a good definition of
"typical"), but it does give you some hope of recognizing some
atypical transcendentals.

--
Gerry Myerson (gerry@xxxxxxxxxxxxxxx) (i -> u for email)
.

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