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From: Keith Thompson <Keith.S.Thompson+u@gmail.com>
Newsgroups: comp.theory
Subject: Re: Definition of real number =?utf-8?Q?=E2=84=9D?=
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Date: Tue, 02 Apr 2024 16:33:43 -0700
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Mike Terry <news.dead.person.stones@darjeeling.plus.com> writes:
> On 02/04/2024 19:29, Keith Thompson wrote:
>> Mike Terry <news.dead.person.stones@darjeeling.plus.com> writes:
>>> On 02/04/2024 02:27, Keith Thompson wrote:
>>>> olcott <polcott333@gmail.com> writes:
>>>>> On 4/1/2024 6:11 PM, Keith Thompson wrote:
>>>>>> olcott <polcott333@gmail.com> writes:
>>>>>> [...]
>>>>>>> Since PI is represented by a single geometric point on the number line
>>>>>>> then 0.999... would be correctly represented by the geometric point
>>>>>>> immediately to the left of 1.0 on the number line or the RHS of this
>>>>>>> interval [0,0, 1.0). If there is no Real number at that point then
>>>>>>> there is no Real number that exactly represents 0.999...
>>>>>> [...]
>>>>>> In the following I'm talking about real numbers, and only real
>>>>>> numbers -- not hyperreals, or surreals, or any other extension to the
>>>>>> real numbers.
>>>>>> You assert that there is a geometric point immediately to the left
>>>>>> of
>>>>>> 1.0 on the number line.  (I disagree, but let's go with it for now.)
>>>>>> Am I correct in assuming that this means that that point corresponds
>>>>>> to
>>>>>> a real number that is distinct from, and less than, 1.0?
>>>>>
>>>>> IDK, probably not. I am saying that 0.999... exactly equals this number.
>>>> "IDK, probably not."
>>>> Did you even consider taking some time to *think* about this?
>>>
>>> PO just says things he thinks are true based on his first intuitions
>>> when he encountered a topic. He does not "reason" his way to a new
>>> carefully thought out theory or even to a single coherent idea. Don't
>>> imagine he is thinking of hyperreals or anything - he just "knows"
>>> that obviously any number which starts 0.??? is less than one starting
>>> 1.??? - because 0 is less than 1 !! Or whatever, it really doesn't
>>> matter.
>> I don't think he's explicitly said that any real number whose
>> decimal
>> representation starts with "0." is less than one starting with "1." --
>> but if said that, he'd be right.
>
>   0.999...  = 1.000...  (so he'd be wrong)
>
>> What he refuses to understand is that the notation "0.999..." is not
>> a
>> decimal representation.  The "..."  notation refers to the limit of a
>> sequence, and of course the limit of a sequence does not have to be a
>> member of the sequence.  Every member of the sequence (0.9, 0.99, 0.999,
>> 0.9999, continuing in the obvious manner) is a real (and rational)
>> number that is strictly less than 1.0.  But the limit of the sequence is
>> 1.0.  Sequences and their limits can be and are defined rigorously
>> without reference to infinitesimals or infinities,
>
> Ah, I see - you're trying to say that 1.000... is a decimal
> representation, but not 0.999...?, which would make sense of why you
> think PO would be right above.  That's a new one on me, but I don't go
> for that argument at all.

No, I was trying to say that "0.999" is a decimal representation
(representing exactly 999/1000, or 9/10 + 9/100 + 9/1000), but
"0.999..." is something fundamentally different, where the "..." denotes
a limit.

But now that I think about it, "most" real numbers don't have a finite
decimal representation.  Rational numbers that are an integer multiple
of an integer power of 10 have finite representations, other rational
numbers have an indefinitely repeating representation that denotes a
limit (e.g., "0.333..."), and irrational numbers can only be
approximated arbitrarily closely in decimal.

> 0.999... is a decimal representation for the number 1, shortened by
> ... which means "continuing in the obvious fashion" or equivalent
> wording.  I.e. 0.999... is the decimal where every digit after the
> decimal point is a 9.  It represents the number 1, as does 1.000....
> Yes, there are two ways to represent the number 1 as an infinite
> decimal.  Not a problem.

Good point.  And I've probably muddied the waters enough.  (If olcott
claims that this discussion means he's right, I'll just ignore him.)

> Anyhow, I have a BA in mathematics, so I understand limits etc..  :)
> I was posting to explain why you're wasting your time trying to
> explain abstract ideas to PO, but it's fine with me if people want to
> do that for whatever reason.
> 
> Mike.
> ps. of course, someone could make a rule that infinitely repeating 9s
> in a decimal expansion is outlawed, but that's not normal practice
> AFAIK.  People just accept there are two representations of certain
> numbers.
>
>> It can be genuinely difficult to wrap your head around this.  It
>> *is*
>> counterintuitive.  And thoughtful challenges to the mathematical
>> orthodoxy, like the paper recently discussed in this thread, can be
>> useful.  But olcott doesn't offer a coherent alternative.
>> [...]

-- 
Keith Thompson (The_Other_Keith) Keith.S.Thompson+u@gmail.com
Working, but not speaking, for Medtronic
void Void(void) { Void(); } /* The recursive call of the void */