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From: Richard Damon <richard@damon-family.org>
Newsgroups: comp.theory
Subject: Re: Repeating decimals are irrational
Date: Wed, 27 Mar 2024 22:38:32 -0400
Organization: i2pn2 (i2pn.org)
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On 3/27/24 10:18 PM, wij wrote:
> On Wed, 2024-03-27 at 22:09 -0400, Richard Damon wrote:
>> On 3/27/24 10:01 PM, wij wrote:
>>> On Wed, 2024-03-27 at 21:05 -0400, Richard Damon wrote:
>>>> On 3/27/24 8:56 PM, wij wrote:
>>>>> 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/RealNumber-en.txt/download
>>>>>>>
>>>>>>> ...
>>>>>>> Real Nunmber(ℝ)::= {x| x is represented by n-ary <fixed_point_number>, the
>>>>>>>        digits may be infinitely long }
>>>>>>>
>>>>>>>        Note: This definition implies that repeating decimals are irrational number.
>>>>>>>              Let's list a common magic proof in the way as a brief explanation:
>>>>>>>                (1) x= 0.999...
>>>>>>>                (2) 10x= 9+x  // 10x= 9.999...
>>>>>>>                (3) 9x=9
>>>>>>>                (4) x=1
>>>>>>>              Ans: There is no axiom or theorem to prove (1) => (2).
>>>>>>>
>>>>>>>        Note: If the steps of converting a number x to <fixed_point_number> is not
>>>>>>>              finite, x is not a ratio of two integers, because the following
>>>>>>>              statement is always true: ∀x,a∈ℚ, x-a∈ℚ
>>>>>>>
>>>>>>> ---End of quote
>>>>>>>
>>>>>>>
>>>>>>
>>>>>> So, if 10 * 0.999... isn't 9.999... what is it?
>>>>>> and if 9 + 0.999... isnt 9.999... what is it?
>>>>>>
>>>>>> And why aren't the same numbers the same numbers.
>>>>>>
>>>>>> So, either your "wij-Reals" just fail to have the normal mathematical
>>>>>> operations defined or you have a problem with the proof.
>>>>>>
>>>>>> Numbers defined with no rules on how to manipulate them are fairly
>>>>>> worthless.
>>>>>
>>>>> The update was available:
>>>>> https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber-en.txt/download
>>>>>
>>>>> Hope, it can solve your doubt.
>>>>>
>>>>
>>>> But the name "Real" is still very bad.
>>>>
>>>> Particularly since you seem to say that any number that can't be
>>>> expressed in a finite number of digits in SOME base, is not a number in
>>>> your system,
>>>
>>> I did not say that. ℝ just numbers expressible by <fixed_point_number>.
>> Near the top of the paper is:
>>
>>
>> +-------------+
>>> Real Number |
>> +-------------+
>>
>>
>>
>>
>>>
>>>> since they can not be explicitly defined, OR HAVE MATH DONE
>>>> ON THEM, since
>>>>
>>>> 0.9999.... * 10 = 9. and somethnig not defined after it. (it isn't even
>>>> .999...)
>>>>
>>>
>>> What are you referring to?
>>>
>>
>>
>>
>>       IOW, by repeatedly multiplying 0.999... with 10, you can only see 9,
>>       the structure of the rear end of 0.999... is never seen.
>>
>>
> Will you explain more specific? I did not mention anything "0.9999.... * 10 = 9. and somethnig not
> defined after it. (it isn't even
> .999...)"

The line above was taken directly from the paper that I downloaded by 
clicking on the link.

You say, and I quote:

0.999.... * 10 = 9. and somthing not defined after it. (it isn't even 
..999...)




> 
>>
>>>> So, your system seems more to be just the rationals. and you don't seem
>>>> to provide a clear set of axioms of what you allow to be done with these
>>>> numbers.
>>>
>>>
>>>
>>
> 
>