Re: A simple question about integers

From: Gottfried Helms (helms_at_uni-kassel.de)
Date: 09/15/04 

Date: Wed, 15 Sep 2004 11:53:00 +0200

Am 15.09.04 09:33 schrieb Angus Rodgers:

>
>
> That's not necessarily a problem, because there is no problem
> with the concept of a formal power series - considered simply
> as the sequence of ``detached coefficients'' of the powers of
> a notional ``indeterminate'' (say \theta).
Yes; I tried to express this fact with the notion of "fixed index",
which can be translated into the assigning of a theta^n, where
n is the index-number.

>
> (This may be nonsense, of course - I stopped trying to
> follow it through, when I realised you couldn't always
> carry 1 when all but finitely many digits are = r - 1,
> and you couldn't restrict yourself to sequences without
> this property, as is shown by the example above.)
again yes. That alone is likely to spoil the whole idea.

>
> I don't think this is obvious. One could at least *try*
> to define a structure of ``infinite nonnegative integers'',
> considered as infinite decimal expansions (or expansions
> to another base); define a < b if and only if there exists
> c such that a + c = b; and then try to prove things about
> this relation (such as totality, i.e. either a < b, a = b,
> or a > b) - with the hope of then being able to embed this
> ``big N'' structure into a ``big Z'' structure, in much the
> same way as Z can be constructed from N.
This construct behaves exactly like the rational numbers between
0 and -1, at least the periodic constructs, and periodic
constructs with a finite additive part behave like such
rationals with the same additive part.

I don't know what that means in respect of irrational numbers
of that range, where -at best- a more or less complex rule to
compute digits can be given.
Consider such a string with reversed sequence of digits from
that of frac(sqrt(2)) (base decimal). Can it be called "greater"
or "smaller" than such derived from frac(sqrt(7))?
With each finite "approximation" ;-) this relation changes arbi-
trarily.(one should better say "diversion" instead "approximation")

I note, there is a "valuation" concept referring to such
strings, but I never read deeper into it.

>
> But none of this is any good if you have a ``number'', n,
> for which you cannot even define n + 1 - so it all looks
> pretty abortive, to me at least.
Yes, I agree.
>
> (Pity - it almost looked like fun for a while.)
>
Yes; when in (i think) 1996 I stumbled into this NG first time,
there was such a discussion going on, and this incidently met
some fiddlings that I had undergone just that time. I didn't
find any use for it after a certain time and left it. It was a
nice game for that period and at least it was useful to get in
touch with some properties and conditions of basic number theory.

Gottfried Helms