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.

Right, but you are actually refering to yourself.
Your math. has big holes you don't realize (and refuse to). You are fooled by
your math.