\mnb150ÿ{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fswiss\fprq2 System;}{\f3\fswiss\fprq2 Helvetica;}{\f4\fswiss\fprq2 Arial;}{\f5\fmodern\fprq1 Courier New;}} {\colortbl\red0\green0\blue0;\red0\green0\blue255;\red0\green128\blue0;\red255\green0\blue0;\red0\green0\blue128;} \deflang1031\pard\ri4\plain\f5\fs20\cf0\b ________________________________________________________________________________ \par \par \plain\f5\fs20\cf0 Inhalt....: Das Heron-Verfahren \par Kategorie.: Arbeitsblatt \par Mathematik: Analysis, Geometrie R^2 \par MuPAD.....: 3.0.0 \par Datum.....: 2002-01-17 \par Autoren...: Vera Verspohl \par Funktionen: DIGITS, plot, plot::Polygon2d, plot::Scene2d \par \plain\f5\fs20\cf0\b ________________________________________________________________________________ \par \plain\f4\fs36\cf0\b \par \plain\f4\fs40\cf0\b Das Heron-Verfahren\plain\f4\fs36\cf0\b \par \plain\f4\fs22\cf0 \par \plain\f4\fs24\cf2 Wir wollen demonstrieren, wie man einige Schritte des \plain\f4\fs24\cf2\i Heron-Verfahrens\plain\f4\fs24\cf2 zur Berechnung \par der Wurzel einer Zahl mit MuPAD durchf\'fchren kann. Ferner wird auf die geometrische Idee, \par welche sich hinter diesem Verfahren verbirgt, eingegangen. \par \plain\f4\fs28\cf0 \par Dieses Verfahren ist uns von den Griechen \'fcberliefert. Es wird jedoch \par angenommen, dass es bereits den Sumerern bekannt war. \par \par Es handelt sich um ein geometrisches Verfahren zur Berechnung von \par Quadratwurzeln, denn die Griechen assoziierten das Produkt zweier Zahlen \par mit dem Fl\'e4cheninhalt eines Rechtecks oder eines Quadrats. \par \par Da wir Wurzeln berechnen m\'f6chten, ist es sinnvoll, die Rechengenauigkeit \par heraufzusetzen. Wir wollen zun\'e4chst mit 25 Gleitkommastellen arbeiten. \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f5\fs28\cf3 {\pntext\f1\'b7\tab}DIGITS:= 25: \par \pard\ri4\plain\f4\fs28\cf0 \par Beim Heron-Verfahren wird angenommen, dass der Fl\'e4cheninhalt eines \par Quadrats bekannt ist und dass nun die Seitenl\'e4nge ermittelt werden soll. Dies \par wird erreicht, indem ein fl\'e4chengleiches Rechteck mit bekannten Seitenl\'e4ngen \par schrittweise an ein Quadrat angen\'e4hert wird: \par \par Ein Rechteck \plain\f5\fs28\cf3 p\plain\f4\fs28\cf0 mit dem Fl\'e4cheninhalt 15FE und ein fl\'e4chengleiches \par Quadrat \plain\f5\fs28\cf3 q\plain\f4\fs28\cf0 : \par \plain\f4\fs22\cf0 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f5\fs28\cf3 {\pntext\f1\'b7\tab}p:= plot::Polygon2d([[0,0], [5,0], [5,3], [0,3]], \par \pard\li600\ri1\fi-300\plain\f5\fs28\cf3 Color= RGB::Blue): \par \plain\f4\fs28\cf3 \par \plain\f5\fs28\cf3 q:= plot::Polygon2d([[0,0], [sqrt(15),0], [sqrt(15),sqrt(15)], \par [0,sqrt(15)]], Color = RGB::Orange, \par LineStyle = Dashed): \par \par r:= plot::Scene2d(p,q): \par \par plot(r)\plain\f5\fs22\cf3 \par \par \par \pard\ri4\plain\f4\fs22\cf0 \par \plain\f4\fs28\cf0 F\'fcr das Rechteck wurden hier exemplarisch die Seitenl\'e4ngen 5FE und 3FE \par gew\'e4hlt. \par \plain\f4\fs22\cf0 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f5\fs28\cf3 {\pntext\f1\'b7\tab}p:= plot::Polygon2d([[0,0], [5,0], [5,3], [0,3]], \par \pard\li600\ri1\fi-300\plain\f5\fs28\cf3 Color= RGB::Blue): \par \par plot(p);\plain\f4\fs28\cf3 \par \pard\ri4\plain\f4\fs28\cf0 \par Um dieses Rechteck in ein Quadrat zu verwandeln, muss die L\'e4nge x gek\'fcrzt \par und die H\'f6he h verl\'e4ngert werden. Hierf\'fcr bildet man den Mittelwert: \par \tab \tab \tab \tab \tab \tab {\pict\wmetafile8\picw2758\pich1701\picscalex99\picscaley99\picwgoal1571\pichgoal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}\plain\f4\fs28\cf0 \par Um einen neuen y-Wert (\plain\f5\fs28\cf3 y1\plain\f4\fs28\cf0 ) zu erhalten, dividiert man nun den Fl\'e4cheninhalt \par durch den neuen x-Wert (\plain\f5\fs28\cf3 x1\plain\f4\fs28\cf0 ), damit das Produkt \par {\pict\wmetafile8\picw2154\pich1225\picscalex99\picscaley99\picwgoal1227\pichgoal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}\plain\f4\fs28\cf0 \par den Fl\'e4cheninhalt ergibt: \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f5\fs28\cf3 {\pntext\f1\'b7\tab}x1:= (3+5)/2; \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f5\fs28\cf3 {\pntext\f1\'b7\tab}y1:=float(15/x1); \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f5\fs28\cf3 {\pntext\f1\'b7\tab}p1:= plot::Polygon2d([[0,0], [x1, 0], [x1,y1], [0,y1]], \par \pard\li600\ri1\fi-300\plain\f5\fs28\cf3 Color = RGB::Blue): \par \par q:= plot::Polygon2d([[0,0], [sqrt(15),0], [sqrt(15),sqrt(15)], \par [0,sqrt(15)]], \par Color=RGB::Orange, LineStyle = Dashed):\plain\f4\fs28\cf3 \par \plain\f5\fs28\cf3 \par r1:= plot::Scene2d(p1,q): \par \par plot(r1) \par \plain\f5\fs22\cf3 \par \pard\ri4\plain\f4\fs28\cf0 Das sieht schon besser aus - berechnen wir nun den n\'e4chsten N\'e4herungswert. \par Daf\'fcr berechnen wir den Mittelwert von \plain\f5\fs28\cf3 x1\plain\f4\fs28\cf0 und \plain\f5\fs28\cf3 y1\plain\f4\fs28\cf0 : \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f5\fs28\cf3 {\pntext\f1\'b7\tab}x2:= (x1+y1)/2; \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f5\fs28\cf3 {\pntext\f1\'b7\tab}y2:=15/x2; \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f5\fs28\cf3 {\pntext\f1\'b7\tab}p2:= plot::Polygon2d([[0,0], [x2,0], [x2,y2], [0,y2]], \par \pard\li600\ri1\fi-300\plain\f5\fs28\cf3 Color= RGB::Blue): \par \par q:= plot::Polygon2d([[0,0], [sqrt(15),0], [sqrt(15),sqrt(15)], \par [0,sqrt(15)]], \par Color = RGB::Orange, LineStyle = Dashed): \par \par r2:= plot::Scene2d(p2,q): \par \par plot(r2);\plain\f5\fs22\cf3 \par \par \pard\ri4\plain\f4\fs28\cf0 Da gelten muss: y2<{\pict\wmetafile8\picw1323\pich1027\picscalex99\picscaley99\picwgoal754\pichgoal586 0100090000030508000006001C0000000000050000000B0200000000050000000C0203042B0503 0000001E00050000000C020A043205050000000B0200000000030000001E00050000000C021004 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