From: janis_papanagnou+ng@hotmail.com   
      
   On 2026-01-28 08:29, David Brown wrote:   
   > On 27/01/2026 23:52, Lawrence D’Oliveiro wrote:   
   >> On Tue, 27 Jan 2026 16:31:47 -0000 (UTC), Waldek Hebisch wrote:   
   >>   
   >>> Actually, part that is needed here is ancient, due to Eudoksos.   
   >>> Namely, real numebers a and b are equal if and only if comparing   
   >>> them with rational numbers gives the same result.   
   >>   
   >> They didn’t know about “real” numbers back then. He was talking about   
   >> “irrational” numbers.   
   >   
   > Indeed.  They certainly knew about irrational numbers - [...]   
      
   Erm.., no; as hinted upthread, not about "irrational *numbers*".   
   (They worked with geometric entities, relations between these.)   
   Modern texts sadly give a wrong impression, because they're not   
   using the ancient methods to explain the Greek's mathematics but   
   try to explain it with modern formulas, number systems, algebra.   
      
   The sqrt(2) sample, as before for squares, pentagons, etc., they   
   were looking for "gemeinsame Maß-Teilstrecke", as we call it here.   
   If you see (modern) expressions like "AC^2 : AB^2 = a^2 : b^2"   
   be aware that these terms and "operators" are just the algebraic   
   counterparts of the respective geometric entities the Greeks had   
   worked with. The graphical square ('^2') or relations (':'). And   
   terms like even/odd in commensurability expressions were defined   
   by "common unit subsections of lines" (not sure about the correct   
   English term).   
      
   (Have a look into Euklid's "Elements" to understand how they did   
   their ancient mathematics.[*] Also a few modern books try to use   
   more authentic representations; e.g. Nikiforowski/Freiman in the   
   introductory "Predecessors" chapter. But the examples there are   
   taken from Euklid's "Elements", so no wonder that it's authentic.)   
      
   Janis   
      
   [*] You'll find historic translations scanned and made available   
   online (e.g. at archive.org).   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)