From: janis_papanagnou+ng@hotmail.com   
      
   (I probably regret answering to your post.)   
      
   On 2026-01-25 18:20, wij wrote:   
   >   
   > You need to prove 4/33 exactly equal to 0.1212..., not approximation.   
      
   Is that all you want proven; a specific example?   
      
   This appears to be as trivial as the more general approach that James   
   gave and that you (for reasons beyond me) don't accept (or don't see).   
      
   First   
        __   
      0.12   or   0.1212...   
      
   are just finite representations of real numbers; conventions. And 4/33   
   is an expression representing an operation, the division. You can just   
   do that computation (as you've certainly learned at school decades ago)   
   in individual steps, continuing each step with the remainder   
      
      4/33 = 0     => 0   
      40/33 = 1    => 0.1   
      remainder 7   
      70/33 = 2    => 0.12   
      remainder 4   
      40/33 = 1    and at this point you see that the _operations_ *repeat*   
      
   so the calculated decimals (1 and 2) will also repeat. And sensibly we   
   need a finite representation (see above) to express that.   
      
      Albert Einstein (for example) said: „Die Definition von Wahnsinn ist,   
      immer wieder das Gleiche zu tun und andere Ergebnisse zu erwarten“.   
      
   Are you expecting the sequence of decimals differing at some point?   
      
   If not you see that the number represented by the convention "0.1212..."   
   equals to the number calculated or expressed by "4/33".   
      
   Janis   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)