From: david.brown@hesbynett.no   
      
   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/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?   
      
   Have you ever heard of the concept of "limits" ?  You might want to   
   learn something about them before embarrassing yourself.   
      
   > 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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