wij <wyniijj2@gmail.com> writes:

> On Friday, 26 August 2022 at 18:33:52 UTC+8, Ben Bacarisse wrote:
>> wij <wyni...@gmail.com> writes:
>>
>> > On Friday, 26 August 2022 at 08:16:18 UTC+8, Ben Bacarisse wrote:
>> >> Skep Dick <skepd...@gmail.com> writes:
>> >>
>> >> > At this point I am not really sure formalization will help us out of
>> >> > this mess...
>> >> It's your mess. I am quite content with the conventional interpretation
>> >> of 0.999... You, I think, don't want to agree that 0.999... = 1 but you
>> >> can't say anything useful about what else it might mean. All you've
>> >> done in fire questions at me as if I should be able to tell you what you
>> >> mean.
>> >>
>> >
>> > As said. 0.999... or "Repeating decimal" suffer from pattern
>> > interpretation problems:
>> Only here. The meaning of ... after digits is almost universally
>> agreed. So much so that other notations should be used for any other
>> meaning.
>
> Do you agree that derivation like this is not a proof?
> (high schools use such deductions to 'prove' repeating decimals are
> rational).

What a shame. But it's irrelevant to the discussion here.

> Proof: 1=0.999...
> x=0.999...
> 10x= 9.999...
> 10x= 9+x
> 9x=9
> x=1

Eh? The fact that 0.999... = 1 comes from (a) defining field in
question (the reals), (b) defining what the ... means (an infinite sum),
(c) defining what an infinite sum is (the limit of the sequence on
partial sums), and finally, showing that the limit of the sum in
question is 1.

>> > (1) 0.999...= 0.(9) = 0.(99)= 0.(999)
>> Yes.
>
> You should say no. They are not what "∑(n=1,∞) 9/10^n)" mean.

They all mean the same. The ()s denote repetition of digits. They have
no effect on how the sum of digits is determined.

>> > (2) 0.999...= 0.((9)(99)) // Andy Walker provided such
>> > interpretation
>> No. Just write 0.((9)(99)) if that's what you mean (and explain the
>> concept either in terms of games or order relations).
>
> The example is given by Andy. It could be valid to mean something.

Not my concern. Cranks often misrepresent what others have said and I
am not going to defend something I never said.

>> > (3) 1-1/∞= 0.999...
>>
>> Introducing a new number called ∞ by which one can divide other numbers
>> does not produce any clarity. It produces even more ambiguity. What I
>> would like to see from a "alterntivist" is clarity. If that requires a
>> new number, so be it, but the set of numbers needs to be defined along
>> with the operations on those.
>
> The symbol ∞ I use is perfectly defined, not like yours.

Yet I have not see a definition of it from you. Why is that?

>> > 1-2/∞= 0.999...
>> > 1-3/2^∞= 0.999...
>> > 1-4/10^∞= 0.999...
>> Without further explanation I will conclude that n/∞ = 0 and that n/k^∞
>> = 0. That does not sound very useful, but what else can I conclude from
>> your lack of explanation?
>
> You are using your ambiguous ∞ again.

Don't be silly. It's not "my" ∞. You used it in a context in which I
can determine that the sub-expression (e.g. 3/∞) is zero. You give no
meaning for 3/∞ other than it must be zero, since you now know that
0.999... = 1. I will make any deductions I like based on the fact that
0.999... = 1 so you really should start using some new notation what
whatever it is you want to be "not quite 1".

>> | (4) 0.999...= 0.9 + 0.09+ 0.009 + 0.0009+ ...
>> | = 0.09 + 0.9 + 0.0009 + 0.009+ ...
>> | = 0.04+0.05 + 0.4+0.5+ 0.0004+0.0005+ 0.004+0.005+....
>> | = 0.44+0.55 + 0.0044+0.0055+...
>> = 1.
>> | = Σ(n=1,∞) f(n) // this f(n) can be nearly anything and yields
>> | different result.
>> |
>> | See the snippet [Infinite series] in previous post about
>> | re-grouping/re-arrange issues of infinite series.
>> No. See any good textbook on how to calculate these limits.
>
> The limit theory perfectly says THE LIMIT of lim(x->c) f(x) is L, why you
> keep pretending I don't know limit better than you do?

You don't know how to define the other sort of limit. It's staggering
that you will accuse several people who appear to have degrees in
mathematics of not knowing how limits are defined simply because you
think there is only one.

> I am questioning f(c)=L (EQUAL),

No one is talking about such a limit. Pick up a book. Learn about the
asymptotic limit and then start being more respectful of the years some
people have put into learning about this material.

> like 1/∞= 2/∞= 3/∞. Do all graph of
> n/x join at any remote point at infinite remote point? There is no
> value x in ℝ such that 1/x=2/x=3/x, not even ∞ (not even valid in your
> math.).

Indeed. So whatever is you meant by ∞ when you wrote that 1-2/∞=
0.999... was not in ℝ. Will you ever say what it is? I doubt it. That
would involve doing some real work.

> I don't think you can find any valid logic except brainlessly
> reciting the limit theory, as so far exhibited.

Yes, I am stuck having to report on the result of four centuries of
mathematical investigation into the reals, limits and calculus.

>> > (5) More interpretations are possible
>> *Sigh* I'd like to see just /one/ alternative interpretation. All
>> anyone posts is bad algebra and one-line hints. Do the work. Define
>> your extension to the reals and explain how algebra is to be done in
>> this new set.
>
> I don't see you really understand algebra.

Of course you don't.

>> > Which one does "0.999..." really mean?
>> 0.999... means lim(k->oo) Sum(k=1,n) 9/10^k. That limit is 1. It will
>> mean that even if you stop being lazy and define what you want it to
>> mean. All you will have done is add confusion. You should define (for
>> example) 0.999___ and then you can prove that 0.999... =/= 0.999___.
>
> It is you who are lazy, just reciting textbook to pass exam. for ???.

But I am not claiming anything new or interesting. Should I join the
massed ranks of cranks that post here and make something up, just so you
can't call me lazy? No, that would be daft. So, yes, I will be lazy.
I will continue to recite the accumulated understanding I got from my
many years of education.

>> > Note that these expressions eventually will be translated to
>> > procedure/operation of natural numbers and then, physical entities.
>> >
>> > As to possible point of current discussion, a consensus that
>> > "Repeating decimal does not specify a unique entity" should be
>> > established.
>> You are not a dictator. You can't take away the meaning I (and vast
>> numbers of boring old mathematicians) give to the symbols 0.999... Even
>> when your great work is published it will (for a century or so at least)
>> still mean lim(k->oo) Sum(k=1,n) 9/10^k.
>>
>> You can introduce ambiguity by not using the symbols like the rest of
>> us, or you can be a proper mathematician and introduce an unambiguous
>> way to say what you want. Of course, not only is that real work, you
>> also have the up-hill struggle to persuade anyone to be interested.
>
> Let's see. I estimate it won't take long. Because I try to be
> reasonable.

No, refusing to say what you mean is not reasonable.

>> > However, such expressions "1-1/∞" or "1-1/10^∞" or
>> > "∑(n=1,∞) 9/10^n" can specify a unique entity, because elements in
>> > these expressions can be defined.
>> If only! Yes they /can/ be defined. I just wish someone would get off
>> their... er, I mean, stop prevaricating and do the work to define them
>> and their operations.

Still no definition of what you mean of course.

> A= Σ(n=1,∞) 1/2^n = 0.999... Agree?

Yes, because 0.999... = Σ(n=1,∞) 9/10^n = 1.

> If you really understand algebra, A!=B is logically unavoidable.

The B you defined before: B= Σ(n=1,∞) 9/10^n, has the same sum.

Ultimately, there is no getting around this basic disagreement. And if
you knew how to find the sum (i.e. the limit of the sequence of partial
sums) you would know that the sums are equal. This is an argument on
the level of 2+2 != 1+3. The truth of the matter follows from the
conventional meaning of the symbols.

>> > As to whether "1-1/2^∞" and "1-1/10^∞" are equal or not, from the snippet:
>> > Ex3: "0.999..." usual 'repeating decimal' cannot denote a unique number.
>> > Let A= Σ(n=1,∞) 1/2^n = 0.999...
>> > B= Σ(n=1,∞) 9/10^n = 0.999...
>> >
>> > Assume A=B
>> Or, more simply, prove that A = B = 1.
>> > <=> 1-1/2^∞= 1-1/10^∞ // converted from the formula of geometric series
>> > <=> 1/2^∞= 1/10^∞
>> > <=> 5^∞=1
>> > <=> false
>> Not with the conventional meanings of any of these symbols.
>
> I just use a perfectly defined symbol '∞', nothing more.

Ah. You think that's a definition. I see the problem. It isn't.

> If you really understand algebra, the definition of '∞' is sufficient:
> 1. ∀n∈ℕ, n<∞
> 2. The multiplicative inverse of ∞ is 1/∞, the additive inverse is
> -∞

Even this is not a definition. Have you read anything about extensions
to the reals? That would show you how this sort of thing needs to be
defined. (And why are you talking about ℕ?)

>> > Conclusion: A and B denote different numbers. (a physical device
>> > computing the truth value of A=B is a deterministic process, yields
>> > false, no way 'equal').
>> Conclusion: you don't know what the symbols mean and you don't dare
>> define new ones with the meaning you;d like them to have.
>
> The meaning the symbol ∞ used by you (and in textbook) is vague, all you
> can say is "it is a concept", "it does not exist"..., nothing very
> significant.

Unfortunately some notations use ∞ metaphorically (the case that's
confusing you is lim{x->∞}), but the proper definition of the limit is
precise and does not refer to ∞ at all. But it seems you've never seen
such a definition because you still think there is only one kind of
limit.

--
Ben.