From: chris.m.thomasson.1@gmail.com   
      
   On 1/28/2026 1:27 AM, Janis Papanagnou wrote:   
   > 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).   
   >   
      
   A real with infinite precision can hold an irrational?   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)