Re: 0.9999... = 1?
From: Tim Mellor (timm_at_amsta.leeds.ac.uk)
Date: 10/29/04
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Date: 29 Oct 2004 06:43:23 -0700
ulfwikstrom@netscape.net (Ulf Wikstr?m) wrote in message news:<4a10db93.0410281726.37f992c1@posting.google.com>...
> Hi!
> I just wrote a loong article here, that just disappeared. Great work
> Ulf!
>
> Anyhow, to me, this whole discussion about whether 0.999... =/= 1
> seems just like a matter of notation.
>
True.
> I think a/b, a and b being integers is the "real" notation for
> rationals.
Well, there is a similar problem here too:
2/3 and 4/6 are different strings of symbols, but we want them to
correspond to the same rational.
The rationals are actually (field isomorphic to) a quotient of Z times
Z* (i.e. Z\{0}), where we have a/b is equivalent to c/d iff ad = bc
Similarly the reals can be represented by (and are field isomorphic
to) a quotient of sequences (legnth omega) of integers between 0 and
9. The equivalence relating to the quotient is defined via limits (of
the finite truncations). It seems (if they are not just trolls) that
this is something many people are not willing to try and grasp.
Mathematicians don't use this notation of course (except parhaps very
informally). It confuses people, (and besides and off hand I can't
think of a nice rule for generating the decimal expansion of any non
rational number).
> We can add numbers together using decimal expansions:
> 1/3 + 1/3 = 2/3
> 0.333... + 0.3333... = 0.6666...
> Since 2/3 really has the decimal expansion 0.666..., it works in this
> case.
>
> Some people says that it should not work for some cases, such as 2/3 +
> 1/3.
>
> Those people are WEIRD! Do not talk to them! ;)
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