From: jameskuyper@alumni.caltech.edu   
      
   On 26/01/2026 16:51, wij wrote:   
   ...   
   > https://sourceforge.net/projects/cscall/files/MisFiles/RealNum   
   er2-en.txt/download   
   >           3. 1/3 = 0.333... + non-zero-remainder (True identity. How to   
   deny?)   
   > How would you deny it, and call the cut-off 'equation' identity?   
      
   I deny it easily - the remainder is exactly 0.   
      
   > You cut off non-zero-remainder to stop repeating, so yes, you see the part   
   you   
   > want to see, i.e. the front part without "...", and forgot the definition   
   > "infinitely repeat" is invalidated.   
      
   There's no non-zero-remainder to cut off, nor is there any need to stop   
   repeating. It repeats endlessly, and it is only because of the endless   
   repetition that the remainder is 0. If it ever ended, the remainder   
   would be non-zero, as you claim.   
      
   The flaw is in your property 2, which claims that an infinite sum of   
   rational numbers is not a rational number. That's unambiguously not the   
   case in the standard real number system. Your proof of that claim is   
   based upon asserting that the numerator and denominator of each step in   
   the infinite series has a larger number of digits (which isn't   
   necessarily true - but that's unimportant), but ignores the fact that   
   the limit of an infinite sequence can have smaller numerators and   
   denominators than the terms that make up the sequence.   
      
   ...   
   >         Prop 2= Repeating Q+Q infinitely does not yield rational number.   
   >                 (precisely, positive rational number)   
      
   I dispute the validity of the proof of Prop 2.   
      
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