Path: ...!news.nobody.at!eternal-september.org!feeder3.eternal-september.org!news.eternal-september.org!.POSTED!not-for-mail
From: wij <wyniijj5@gmail.com>
Newsgroups: comp.theory
Subject: Re: Repeating decimals are irrational
Date: Wed, 27 Mar 2024 07:05:42 +0800
Organization: A noiseless patient Spider
Lines: 57
Message-ID: <02aa964280a00d97c3d3d23af84f5582d5497ff7.camel@gmail.com>
References: <e9009d933dc0c3008201ba6cfced892d235192c8.camel@gmail.com>
	 <87wmppuhn8.fsf@nosuchdomain.example.com>
	 <fa1bc63730570ff93fb7b43f81e735453829bab3.camel@gmail.com>
	 <87sf0dueli.fsf@nosuchdomain.example.com>
	 <27544510d7559860979611a504b6ad0d9aebfd1c.camel@gmail.com>
	 <87o7b0vja4.fsf@nosuchdomain.example.com>
	 <1b980d564d961c4c290b5bf722835f2f2f284969.camel@gmail.com>
	 <87jzloveiw.fsf@nosuchdomain.example.com>
	 <ca4c6e9d56ec2929d0c4b1319627f0ae17c7cfc1.camel@gmail.com>
	 <87frwcvc19.fsf@nosuchdomain.example.com>
MIME-Version: 1.0
Content-Type: text/plain; charset="UTF-8"
Content-Transfer-Encoding: quoted-printable
Injection-Date: Tue, 26 Mar 2024 23:05:43 +0100 (CET)
Injection-Info: dont-email.me; posting-host="e58f2b2de37d2d0fd85c976e2be071c6";
	logging-data="2518815"; mail-complaints-to="abuse@eternal-september.org";	posting-account="U2FsdGVkX1/R9xOwZfhIe8xxCNiMRmFK"
User-Agent: Evolution 3.50.2 (3.50.2-1.fc39)
Cancel-Lock: sha1:YOVmWF5afoPvn1CRFs5haxNklIQ=
In-Reply-To: <87frwcvc19.fsf@nosuchdomain.example.com>
Bytes: 3693

On Tue, 2024-03-26 at 15:55 -0700, Keith Thompson wrote:
> wij <wyniijj5@gmail.com> writes:
> > On Tue, 2024-03-26 at 15:01 -0700, Keith Thompson wrote:
> > > wij <wyniijj5@gmail.com> writes:
> > > > On Tue, 2024-03-26 at 13:18 -0700, Keith Thompson wrote:
> [SNIP]
> > > > > Still, rational numbers can be represented in decimal, and you al=
ready
> > > > > acknowledged that "the digits may be infinitely long" for real nu=
mbers.
> > > > > So 0.333..., where the sequence of 3s is unending, is a valid
> > > > > representation of the rational number 1/3, yes?=C2=A0 It's the nu=
mber itself
> > > > > that's rational, regardless of which of several valid representat=
ions
> > > > > you choose to express it.
> > > > >=20
> > > > > Do you agree so far?
> > > >=20
> > > > I just understand the front part of your question.
> > > > No, 1/3 and 0.333(3) are not exactly equal. See the note "=E2=88=80=
x,a=E2=88=88=E2=84=9A, x-a=E2=88=88=E2=84=9A"
> > > > The semantic of "repeating" already says the number 1/3 cannot be c=
ompletely
> > > > exhausted by subtracting 3/10^n. The proposition "repeating decimal=
 is rational"
> > > > is simple false by semantics.
> > >=20
> > > And that's where you're quite simply wrong.
> > >=20
> > > 1/3 and 0.333(3) are exactly equal, and rational.=C2=A0 "=E2=88=80x,a=
=E2=88=88=E2=84=9A, x-a=E2=88=88=E2=84=9A" is
> > > not violated; the difference is 0, which is also rational.=C2=A0 You =
do not
> > > have an internally consistent model of real or rational numbers.
> > >=20
> >=20
> > Then, I would ask you to provide proof, not just assertion that you are=
 right.
> > My model is not perfect, but it does not matter, I am sure now you cann=
ot really
> > distinguish good or bad.
>=20
> Do you understand what a mathematical limit is?=C2=A0 Can you explain it?
>=20
> If you understand limits, you should be able to understand that
> 0.333..., where the "..." denotes the limit as the number of 3s exceeds
> without bound, is exactly equal to 1/3.
>=20
> [...]
>=20

I already stated it the file:
https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber-en.txt/do=
wnload