Re: A simple question about integers

From: Denis Feldmann (denis.feldmann_at_wanadoo.fr)
Date: 09/15/04 

Date: Wed, 15 Sep 2004 12:42:17 +0200

Gottfried Helms wrote:
> 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.

Still, it is a pity you are fixed on decimal representation. In base -11,
you get something perfectly reasonable and very useful : the ring Z_11 of
11-adics integers, with the important property ...AAAAA (the analog of your
...9999) +1=0, i.e ....AAAAA= -1 , and (even more interesting , maybe)
...6666 =1/2, for instance. (it is still not a field, as 10 has no inverse,
though; the completion of Z_11 is Q_11 (the 11-adics numbers), with a formal
writing *completely* symetrical to that of IR, i.e. a typical member of Q_11
is ...12A457899A1 . 124A4 )

>
> Gottfried Helms

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