Re: Repeating decimals are irrational

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.