From: wyniijj5@gmail.com   
      
   On Mon, 2026-01-26 at 21:07 +0100, David Brown wrote:   
   > On 26/01/2026 16:51, wij wrote:   
   > > On Mon, 2026-01-26 at 01:25 +0100, Janis Papanagnou wrote:   
   > > > (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   
   > > Not quite sure what you mean.   
   > >    
   > > https://sourceforge.net/projects/cscall/files/MisFiles/RealN   
   mber2-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?   
   >    
   > Have you ever heard of the concept of "limits" ?  You might want to    
   > learn something about them before embarrassing yourself.   
      
   What do you know about the concept of "limits"? (You invented? Don't try to be   
   the next one, again. I remember the other expert in this forum has humiliated   
   himself   
   once, not sure which one, if I can safely predict. And I ignored the other   
   reply,   
   because it is too obvious, I leave as record)   
      
   > > 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.   
   > > Let n(i) be the repeating number 0.999... The range [n(i),1] remains 1-1   
   > > correspondence to [0,1] in each step, nothing changed except scale. Or you   
   > > suggests every zooming of the small area of Mandelbrot set will be 'empty'   
   or   
   > > uniform or 'stop' for some mysterious reason.   
   > >    
   > > I assume you disagee my point in the previous post that every denial must   
   > > refute Prop 1= Repeating N+N infinitely does not yield natural number.   
   > >         Prop 2= Repeating Q+Q infinitely does not yield rational   
   number.   
   > >                 (precisely, positive rational number)   
   > >    
      
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