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From: wij <wyniijj5@gmail.com>
Newsgroups: comp.theory
Subject: Re: Repeating decimals are irrational
Date: Wed, 27 Mar 2024 08:40:04 +0800
Organization: A noiseless patient Spider
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On Tue, 2024-03-26 at 17:28 -0700, Keith Thompson wrote:
> wij <wyniijj5@gmail.com> writes:
> > On Tue, 2024-03-26 at 17:01 -0700, Keith Thompson wrote:
> > > wij <wyniijj5@gmail.com> writes:
> > > > On Tue, 2024-03-26 at 16:22 -0700, Keith Thompson wrote:
> > > > > wij <wyniijj5@gmail.com> writes:
> > > > > > On Tue, 2024-03-26 at 15:55 -0700, Keith Thompson wrote:
> > > > > [...]
> > > > > > > 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 th=
at
> > > > > > > 0.333..., where the "..." denotes the limit as the number of =
3s exceeds
> > > > > > > without bound, is exactly equal to 1/3.
> > > > > > >=20
> > > > > > > [...]
> > > > > > >=20
> > > > > >=20
> > > > > > I already stated it the file:
> > > > > > https://sourceforge.net/projects/cscall/files/MisFiles/RealNumb=
er-en.txt/download
> > > > >=20
> > > > > OK, in that file you wrote:
> > > > >=20
> > > > > =C2=A0=C2=A0=C2=A0 Ex1: A=3D lim(n->=E2=88=9E) 1-1/n=3D lim(n->0=
=E2=81=BA) 1-n=3D lim 0.999...=3D1
> > > > > =C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0 B=3D lim(n->=E2=
=88=9E) 1+1/n=3D lim(n->0=E2=81=BA) 1+n=3D lim 1.000..?=3D1
> > > > >=20
> > > > > What do you mean by that question mark?=C2=A0 Do you agree that b=
oth limits
> > > > > in your example (A and B) are equal to 1?
> > > > >=20
> > > >=20
> > > > Agree.
> > >=20
> > > OK.=C2=A0 Do you agree that 0.333..., where ... denotes the limit as =
the
> > > number of 3s increases without bound, is exactly equal to 1/3?
> > >=20
> >=20
> > Nop.
>=20
> I'll assume that's a typo for "No" or "Nope".
>=20
> So you're saying that 0.333... is not exactly equal to 1/3.
>=20
> It seems odd that you agree that 0.999... is exactly equal to 1, but
> 0.333... is not exactly equal to 1/3.
>=20

I say the limit of 0.999... is 1, not 0.999... is 1. (this is also what you=
 asked)
Read the definition carefully from trustworthy website.

Be noted that this is still nothing to do with what we are discussing.
If you understand "circular reasoning", limit cannot define the number it i=
s using.

> Is 0.333... a real number?=C2=A0 Is 1/3 - 0.333... a real number?=C2=A0 W=
hat is
> the value of 1/3 - 0.333...?
>=20
> > > And that that number multipled by 3 is exactly equal to 1?
> > >=20
> > > And that 0.999... is exactly equal to 1?
> > >=20
> > > And that all the numbers we're discussing are both real and rational?
> >=20
> > Can you make statement really like math?
>=20
> Please rephrase that.=C2=A0 I don't understand what it means.
>=20
> > You still keep asserting your belief, not a proof but strangely formed =
statement !!!
> > You cannot read proof !!! Go home and learn more.
>=20
> I haven't offered a proof because (a) there's no point in doing so if we
> don't have enough mutual understanding, and (b) I'm not a mathematician,
> so I'm not confident of my ability to construct a rigorous proof.=C2=A0 I
> might give it a try later, but I see no point in doing so now.