Re: Repeating decimals are irrational

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
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Real Nunmber(ℝ)::= {x| x is represented by n-ary <fixed_point_number>, the
       digits may be infinitely long }
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       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).
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       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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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,

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I did not say that. ℝ just numbers expressible by <fixed_point_number>.

Near the top of the paper is:
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+-------------+

Real Number |

+-------------+
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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...)
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What are you referring to?
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      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.
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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...)

 

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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.

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