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From: wij <wyniijj5@gmail.com>
Newsgroups: comp.theory
Subject: Re: Repeating decimals are irrational
Date: Wed, 27 Mar 2024 00:15:53 +0800
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On Tue, 2024-03-26 at 08:39 -0700, Keith Thompson wrote:
> wij <wyniijj5@gmail.com> writes:
> > Snipet from https://sourceforge.net/projects/cscall/files/MisFiles/Real=
Number-en.txt/download
> >=20
> > ...
> > Real Nunmber(=E2=84=9D)::=3D {x| x is represented by n-ary <fixed_point=
_number>, the
> > =C2=A0=C2=A0 digits may be infinitely long }
> >=20
> > =C2=A0=C2=A0 Note: This definition implies that repeating decimals are =
irrational number.
>=20
> How does a definition that doesn't mention rational numbers imply
> anything about which numbers are rational?
>=20
First of all, it is not really my definition (strict meaning of the word=
=20
definition). What I showed is a reasonable proof of what the real number re=
ally=20
'practically' used world-wide (not the ones in academic theory).
<fixed_point_number> is just a representation of real number specified for
convenience for math. proofs and discussion of numbers.=20
Your 'rational number' might mean a sub-class defined latter.

> > =C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0 Let's list a common ma=
gic proof in the way as a brief explanation:
> > =C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0 (1) x=3D 0=
..999...
> > =C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0 (2) 10x=3D=
 9+x=C2=A0 // 10x=3D 9.999...
> > =C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0 (3) 9x=3D9=
=C2=A0=C2=A0=C2=A0=C2=A0=20
> > =C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0 (4) x=3D1
> > =C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0 Ans: There is no axiom=
 or theorem to prove (1) =3D> (2).
> >=20
> > =C2=A0=C2=A0 Note: If the steps of converting a number x to <fixed_poin=
t_number> is not
> > =C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0 finite, x is not a rat=
io of two integers, because the following
> > =C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0 statement is always tr=
ue: =E2=88=80x,a=E2=88=88=E2=84=9A, x-a=E2=88=88=E2=84=9A
> >=20
> > ---End of quote
>=20
> Is 1/3 a rational number?
>=20

Yes, by definition?

> Is 1/3 a real number?
>=20
> If 1/3 is a real number, what is its representation according to your def=
inition?
>=20

Yes, 1/3 is a real number, it's n-nary <fixed_point_number> representaion i=
s infinitely long.
Infinitely long number is harder to explain by now. I think this part can b=
e=20
skip for the moment (no present theory can make this very clear and satisfa=
ctory).