Path: ...!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: Real Number --- Merely numbers whose digits can be infinitely
 long
Date: Thu, 02 May 2024 12:46:20 +0800
Organization: A noiseless patient Spider
Lines: 91
Message-ID: <a2cf90b6ca7e1dd7dd2ed0da5e6710ea5f7adc20.camel@gmail.com>
References: <c10c644441b2307e828f8392fb6993a78c580ee4.camel@gmail.com>
	 <87edaobfm4.fsf@bsb.me.uk>
	 <0ad60eee1517af22b54bcdac3f4947895c9fa559.camel@gmail.com>
	 <87o79p5h45.fsf@bsb.me.uk>
	 <0e898ea58ba39da3c6d3a2a4cbd9b198d4b5c37a.camel@gmail.com>
	 <87ttjhkn6q.fsf@nosuchdomain.example.com>
	 <b903715ba20b5f40fb4bbcd1640e8ade97a233ac.camel@gmail.com>
	 <87plu4lw6o.fsf@nosuchdomain.example.com>
	 <87le4slvuv.fsf@nosuchdomain.example.com>
MIME-Version: 1.0
Content-Type: text/plain; charset="UTF-8"
Content-Transfer-Encoding: quoted-printable
Injection-Date: Thu, 02 May 2024 06:46:21 +0200 (CEST)
Injection-Info: dont-email.me; posting-host="8f3dd3269221b3f901ad2ab676f3c5fb";
	logging-data="3829713"; mail-complaints-to="abuse@eternal-september.org";	posting-account="U2FsdGVkX1+XvY/9Xxxej7RF/q+OAc21"
User-Agent: Evolution 3.50.2 (3.50.2-1.fc39)
Cancel-Lock: sha1:BDE2nCdku/iqOuyV07WW1jA3C4Q=
In-Reply-To: <87le4slvuv.fsf@nosuchdomain.example.com>
Bytes: 5153

On Wed, 2024-05-01 at 20:46 -0700, Keith Thompson wrote:
> Keith Thompson <Keith.S.Thompson+u@gmail.com> writes:
> > wij <wyniijj5@gmail.com> writes:
> > > On Wed, 2024-05-01 at 18:38 -0700, Keith Thompson wrote:
> > > > wij <wyniijj5@gmail.com> writes:
> > > > > On Wed, 2024-05-01 at 22:58 +0100, Ben Bacarisse wrote:
> > > > > > wij <wyniijj5@gmail.com> writes:
> > > > [...]
> > > > > > > =C2=A0=C2=A0=C2=A0=C2=A0 <fixed_point_number>::=3D [-] <wnum>=
 [ . <frac> ]=C2=A0 // excluding "-0" case
> > > > > > > =C2=A0=C2=A0=C2=A0=C2=A0 <wnum>::=3D 0
> > > > > > > =C2=A0=C2=A0=C2=A0=C2=A0 <wnum>::=3D <nzd> { 0 | <nzd> }
> > > > > > > =C2=A0=C2=A0=C2=A0=C2=A0 <frac>::=3D { 0 | <nzd> } <nzd>
> > > > > > > =C2=A0=C2=A0=C2=A0=C2=A0 <nzd> ::=3D 1 | 2 | 3 | 4 | 5 | 6 | =
7 | 8 | 9 // 'digit' varys depending on n-ary
> > > > > > >=20
> > > > > > > =C2=A0=C2=A0=C2=A0 Ex: 78, -12.345, 3.1414159
> > > > > >=20
> > > > > > So what's the point of defining these strings that represent a =
subset of
> > > > > > the rationals?
> > > > >=20
> > > > > <fixed_point_number> is a super set of rationals.
> > > > [...]
> > > >=20
> > > > An extraordinary claim.
> > > >=20
> > > > Do you agree that 1/3 is a rational number?=C2=A0 How is 1/3 repres=
ented in
> > > > your <fixed_point_number> notation?
> > > >=20
> > >=20
> > > I already told you: 1/3=3D 0.1 (3-ary <fixed_point_number>)
> > > Substitute the n in n-ary with the q in p/q, every p/q is representab=
le=20
> > > by <fixed_point_number>.
> > > And, the rule of <frac> can generate infinitely long fractions, read =
it carefully!
> >=20
> > That kind of notation almost universally refers to *finite* sequences o=
f
> > symbols.
> >=20
> > If you intend it to be able to specify infinite sequences, that's fine,
> > but it's not inherent in the notation you've presented.=C2=A0 I also wo=
nder
> > how an infinitely long <frac> can have <nzx> as its last element.
> >=20
> > So <frac> can be infinitely long.=C2=A0 Can <wnum> be infinitely long?
> >=20
> > I presume that the "n-ary" base can be any integer greater than or equa=
l
> > to 2, and that the digits can range from 0 to n-1.=C2=A0 That means you=
'll
> > need arbitrarily many distinct symbols for the digits in large bases.
> > That's all fine, but it would be good to state all this explicitly.
> >=20
> > There are already perfectly good mathematical methods for constructing
> > the integers, the rationals, and the reals.=C2=A0 Your method of using =
base-n
> > notation to *define* the reals and/or rationals seems superfluous.=C2=
=A0 It
> > can probably be done consistently, but I fail to see how it's useful.
>=20
> And something I thought of immediately after I posted the above:
>=20
> You need to use different bases to represent all rational numbers, but
> the base isn't part of your notation.=C2=A0 Your grammar matches "0.1", b=
ut
> how do I know whether than's 1/10, 1/3, or 1/1729?
>=20
Do you use different bases to represent all rational numbers?

> 0.2 (base 10) and 0.1 (base 5) represent the same number.=C2=A0 0.2 (base=
 10)
> and 0.1 (base 4) do not.=C2=A0 Your notation doesn't seem to have any way=
 to
> indicate this.=C2=A0 How can we know that 0.2 (base 10) and 0.1 (base 5) =
are
> equal without using the real numbers that you're trying to *define*?
>=20
How should I know your numbers (1/10, 1/3, or 1/1729) are in base-12 or bas=
e-16
if you also did not say it explicitly?

> Or are you assuming that real numbers already exist, and you're defining
> this notation on top of that?=C2=A0 If so, what's the point?
>=20

Your request is valid but not practically reasonable.