From: wyniijj5@gmail.com   
      
   On Mon, 2026-01-26 at 01:20 +0800, wij wrote:   
   > On Sun, 2026-01-25 at 11:25 -0500, James Kuyper wrote:   
   > > On Sun, 25 Jan 2026 13:28:31 +0800, wij wrote:   
   > >    
   > > > On Sat, 2026-01-24 at 23:06 -0500, James Kuyper wrote:   
   > > > >    
   > > > > This generalizes to work with any recurring decimal. When the   
   > > > > recurrence is n digits long (in the above example, n=2), just   
   > > > > multiply by 10^n.   
   > > >    
   > > > Such arithmetic is called approximation ...   
   > >    
   > > So what is, in your opinion, the exact representation of 4/33 as a   
   > > decimal fraction? Marking it as infinitely repeating means that it is   
   > > not in any sense an approximation - it is exact.   
   >    
   > You need to prove 4/33 exactly equal to 0.1212..., not approximation.   
   > But,...   
   >    
   > >  If it were an   
   > > approximation, there would be a finite difference between 4/33 and   
   > > 0.1212.... How big is that difference? No matter how small of a finite   
   > > difference you might think it has, going to infinitely many decimal   
   > > digits makes the actual difference smaller than that.   
   >    
   > Yes, the same as infinity/infinitesimal.   
   >    
   > > Keep in mind that, in the absence of specification to the contrary,   
   > > we're talking about the standard real number system, not one of those   
   > > alternatives that extends the real number system by adding   
   > > infinitesimals, such as the surreals or the hyperreals.   
   >    
   > True if you are not referring to the current 'standard real number system'.   
   > What else? (doesn't sound to contain much information)   
   >    
   > The standard real number system is not a constant thing, it will correct   
   > itself. Not even religion is constant.   
   >    
   > To be explicitly 'C' related, I remember people talking about what INF    
   > should mean (an option may be merely for signifying overflow), I would   
   > suggest reserve the bit, not mixing with other meaning.   
      
   Prop2 had rewrote (easier for older high schools to understand).   
      
   https://sourceforge.net/projects/cscall/files/MisFiles/RealNumbe   
   2-en.txt/download   
   ....[cut]   
   Prop 2: ℚ+ℚ=ℚ (the sum of a rational number and a rational number is   
   still a   
           rational number). The statement holds only for a finite number of   
           addition steps.   
     Proof: The addition of positive rational number q is strictly increasing. The   
            result is either 'divergent' or 'convergent'. Both contain infinitely   
            long 'natural number' (otherwise they cannot be called 'convergent' or   
            'divergent').   
   ----------   
      
   Therefore, real number contains infinity (so infinitesimal).   
   So, INFINITY is another real number, don't mix it with something else.   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)