Re: Repeating decimals are irrational

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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>

>
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.
>
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, 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...)
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.