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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 23:02:49 -0400
Organization: i2pn2 (i2pn.org)
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On 3/27/24 10:45 PM, wij wrote:
> On Wed, 2024-03-27 at 22:38 -0400, Richard Damon wrote:
>> 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...)
>>
>>
> I uploaded again: What/where were you referring to?
> --------------------------
> 
> +-------------+
> | Real Number |
> +-------------+
> 
> n-ary Fixed-Point Number::= Number represented by a string of digits, the
>     string may contain a minus sign or a point:
> 
>       <fixed_point_number>::= [-] <dstr1> [ . <dstr2> ]
>       <dstr1>::= 0 | <nzd> { 0, <nzd> }
>       <dstr2>::= { 0, <nzd> } <nzd>
>       <nzd> ::= (1, 2, 3, 4, 5, 6, 7, 8, 9)  // 'digit' varys depending on n-ary
> 
>     Two n-ary fixed-point number (same n-ary) x,y are equal iff their
>     <fixed_point_number> representation are identical.
> 
> Real Nunmber(ℝ)::= {x| x is finitely represented by n-ary <fixed_point_number>
>     and those that cannot be finitely represented }
> 
>     Note: Numbers that is not finitely representable cannot all be explicitly
>           defined, this is the property of real number based on discrete symbols
>           (like quantum?). E.g.
> 
>           A= lim(n->∞) 1-3/10^n = 0.999...
>           B= lim(n->∞) 1-2/2^n  = 0.999...
>           C= lim(n->∞) 1-1/n    = 0.999...
>           ...
> 
>           IOW, by repeatedly multiplying 0.999... with 10, you can only see 9,
>           the structure of the rear end of 0.999... is not seen.

             ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

> 
>     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: To determine whether a repeating decimal x is rational or not, we can
>           repeatedly subtract the repeating pattern p(i) from x.
>           If x-p(1)-p(2)-...=0 can be verified in finite steps, then x is
>           rational. Otherwise, x is irrational, because, if x is rational, the
>           last remaining piece r(i)= x-p(1)-p(2)-... must exactly be the
>           repeating pattern p(i). But, by definition of 'repeating', r(i) cannot
>           be pattern p(i). Therefore, repeating decimal is irrational.
> 
> 
>>
>>>
>>>>
>>>>>> 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.
>>>>>
>>>>>
>>>>>
>>>>
>>>
>>>
>>
> 
>