Re: Transcendental Dimensions



Timothy Murphy wrote:
> Timothy Little wrote:
>
> > Dimensions of fractals often have the form (log p / log q) for some
> > integers p and q. If the ratio isn't "obviously" a rational then it's
> > irrational and hence transcendental. If log p / log q is a rational
> > then p^n = q^m with m > 1, which tends to be pretty obvious. That's a
> > fairly uncommon relationship, requiring that p and q be different
> > powers of a common base.
>
> Irrational is not the same as transcendental.
> (A number can be irrational without being transcendental.)

Yes -- but this is not a general case. Look up Gelfond-Schneider
Theorem.
If p and q are integers (and greater than 1), then log(p)/log(q) is
either
an integer or it is transcendental. Alternatively, if log(p)/log(q) is
algebraic, then it must be an integer. Since all integers are
rational,
if log(p)/log(q) is not rational (i.e. if it is irrational), then it
must
indeed be transcendental.

Michel.

.

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