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From: wij <wyniijj5@gmail.com>
Newsgroups: comp.theory
Subject: Re: Repeating decimals are irrational
Date: Wed, 27 Mar 2024 19:14:05 +0800
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On Tue, 2024-03-26 at 22:17 -0400, Richard Damon wrote:
> On 3/26/24 10:45 AM, wij wrote:
> > 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=C2=A0 digits may be infinitely long }
> >=20
> > =C2=A0=C2=A0=C2=A0 Note: This definition implies that repeating decimal=
s are irrational number.
> > =C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0 Let's list a com=
mon magic 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=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=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=C2=A0 (3) =
9x=3D9
> > =C2=A0=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=C2=A0 Ans: There is no=
 axiom or theorem to prove (1) =3D> (2).
> >=20
> > =C2=A0=C2=A0=C2=A0 Note: If the steps of converting a number x to <fixe=
d_point_number> is not
> > =C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0 finite, x is not=
 a ratio of two integers, because the following
> > =C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0 statement is alw=
ays true: =E2=88=80x,a=E2=88=88=E2=84=9A, x-a=E2=88=88=E2=84=9A
> >=20
> > ---End of quote
> >=20
> >=20
>=20
> So, if 10 * 0.999... isn't 9.999... what is it?
> and if 9 + 0.999... isnt 9.999... what is it?
>=20
> And why aren't the same numbers the same numbers.
>=20
> So, either your "wij-Reals" just fail to have the normal mathematical=20
> operations defined or you have a problem with the proof.
>=20
> Numbers defined with no rules on how to manipulate them are fairly=20
> worthless.

Right, but you are actually refering to yourself.=20
Your math. has big holes you don't realize (and refuse to). You are fooled =
by
your math.