Re: Repeating decimals are irrational

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
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---End of quote
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>

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So, if 10 * 0.999... isn't 9.999... what is it?
and if 9 + 0.999... isnt 9.999... what is it?
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And why aren't the same numbers the same numbers.
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So, either your "wij-Reals" just fail to have the normal mathematical
operations defined or you have a problem with the proof.
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Numbers defined with no rules on how to manipulate them are fairly
worthless.

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The update was available:
https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber-en.txt/download
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Hope, it can solve your doubt.
>

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But the name "Real" is still very bad.
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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
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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.

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.