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From: Richard Damon <richard@damon-family.org>
Newsgroups: comp.theory
Subject: Re: Repeating decimals are irrational
Date: Tue, 26 Mar 2024 22:17:56 -0400
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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.