\mnb150ÿ{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fswiss\fprq2 System;}{\f3\fswiss\fprq2 Arial;}{\f4\fmodern\fprq1 Courier New;}{\f5\froman\fprq2\fcharset2 Symbol;}{\f6\fswiss\fprq2 Helvetica;}{\f7\fswiss\fprq2\fcharset2 Arial;}{\f8\froman\fprq2 Symbol;}} {\colortbl\red0\green0\blue0;\red255\green0\blue0;\red0\green0\blue255;\red0\green128\blue0;} \deflang1031\pard\ri4\plain\f4\fs20\cf0\b ________________________________________________________________________________ \par \par \plain\f4\fs20\cf0 Inhalt....: Mathematik entdecken - Iterative Berechnung von PI mit MuPAD \par Kategorie.: Unterrichtsmaterial \par Mathematik: Analysis, Numerik, Geometrie R^2 \par MuPAD.....: 3.1.0 \par Datum.....: 2002-01-17 \par Autoren...: Kai Gehrs \par Funktionen: |, ->, plot, plot::Circle2d, plot::Polygon2d, int, sum, \par Funktionen: student::riemann, student::trapezoid, student::plotRiemann, \par Funktionen: student::plotSimpson, student::plotTrapezoid, DIGITS \par \plain\f4\fs20\cf0\b ________________________________________________________________________________ \par \plain\f3\fs36\cf0\b \par \plain\f3\fs40\cf0\b Mathematik entdecken - Iterative Berechnung der \par Zahl \plain\f8\fs56\cf0\b p\plain\f3\fs40\cf0\b \plain\f3\fs22\cf0 \par \par \plain\f3\fs24\cf3 Die Berechnung des Umfangs des Einheitskreises, das bedeutet die Bestimmung \par von 2\plain\f5\fs24\cf3 p\plain\f3\fs24\cf3 , ist ein sehr altes Problem, das vor \'fcber 4000 Jahre erstmals in Erscheinung \par getreten ist und schon im Alten Testament zu finden ist. Dort verwendete man \plain\f5\fs24\cf3 p\plain\f3\fs24\cf3 =3. \par \par In China, 5. Jahrhundert, fand man eine erstaunliche N\'e4herung von \plain\f5\fs24\cf3 p\plain\f3\fs24\cf3 durch die \par rationale Zahl 355/113, deren ersten 10 Dezimalziffern 3,141592920 lauten und \par von denen die ersten 6 Nachkommstellen korrekt sind (im Vergleich: \plain\f5\fs24\cf3 p\plain\f3\fs24\cf3 = 3,\plain\f3\fs24\cf1 141592\plain\f3\fs24\cf3 654...). \par \par Lambert bewies 1761, da\'df \plain\f5\fs24\cf3 p\plain\f3\fs24\cf3 eine \plain\f3\fs24\cf1 irrationale\plain\f3\fs24\cf3 Zahl ist, und Lindemann zeigte 1882, \par da\'df \plain\f5\fs24\cf3 p\plain\f3\fs24\cf3 \plain\f3\fs24\cf1 transzendent\plain\f3\fs24\cf3 \'fcber den rationalen Zahlen ist. \par \plain\f3\fs22\cf0 \par \plain\f3\fs24\cf3 Wir werden in diesem Notebook einige iterative Verfahren zur Berechnung von \plain\f5\fs24\cf3 p\plain\f3\fs24\cf3 \par kennlernen. \plain\f3\fs22\cf0 \par \par \plain\f3\fs28\cf0 \par \plain\f3\fs32\cf0 Die Methode von Archimedes \par \plain\f4\fs20\cf0\b ________________________________________________________________________________\plain\f3\fs32\cf0 \par \par \plain\f3\fs28\cf0 Archimedes war der erste, der eine systematische L\'f6sung des Problems \par gefunden hat. \par \par Er legte in den Einheitskreis ein gleichseitiges Dreieck und verdoppelte \par dann fortlaufend die Zahl der Ecken. 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then sqrt(2-sqrt(4-Seitenlaenge(n/2)^2)) \par else sqrt(3) \par end \par ) \par \pard\ri4\plain\f3\fs28\cf0 \par Archimedes verdoppelte die Eckenzahl mit jedem N\'e4herungsschritt: \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf1 {\pntext\f1\'b7\tab}Eckenzahl:= j -> ( \par \pard\li600\ri1\fi-300\plain\f4\fs28\cf1 if j > 1 then 2*Eckenzahl(j-1) \par else 3 \par end \par ) \par \pard\ri4\plain\f3\fs28\cf0 \par Dann ergibt sich der Umfang des n-Ecks zu: \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf1 {\pntext\f1\'b7\tab}nEckUmfang:= j->Eckenzahl(j)*Seitenlaenge(Eckenzahl(j)) \par \pard\ri4\plain\f3\fs28\cf0 \par Diesen nehmen wir nun als Ann\'e4herung von 2\plain\f5\fs28\cf0 p\plain\f3\fs28\cf0 . Wir f\'fchren die ersten 10 \par Schritte der Methode von Archimedes durch: \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf1 {\pntext\f1\'b7\tab}n:= 10: \par {\pntext\f1\'b7\tab}for j from 1 to n do \par \pard\li600\ri1\fi-300\plain\f4\fs28\cf1 print(Unquoted, "Umfang des ".Eckenzahl(j)."-Ecks: ". \par expr2text(float(nEckUmfang(j))/2) \par ) \par end: \par \par \pard\ri4\plain\f3\fs28\cf0 Erh\'f6ht man sukzessive die Eckenzahl, so l\'e4\'dft sich ein interessanter \par Vorgang beobachten. Nehmen wir beispielsweise: \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf1 {\pntext\f1\'b7\tab}n:= 40: \par \pard\li600\ri1\fi-300\plain\f4\fs28\cf1 for j from 25 to n do \par print(Unquoted, "Umfang des ".Eckenzahl(j)."-Ecks: ". \par expr2text(float(nEckUmfang(j))/2) \par ) \par end: \par \par \pard\ri4\plain\f3\fs28\cf0 Man erkennt, da\'df die N\'e4herungsfolge zu divergieren scheint! Der Grund \par liegt daran, da\'df obige Formel zur Berechnung desUmfangs des n-Ecks \par numerisch instabil ist. Es tritt Ausl\'f6schung auf, was man sich leicht \par \'fcberlegen kann. Erh\'f6ht man beispielsweise die Genauigkeit der numerischen \par Rechnungen, beispielsweise auf 50 Dezimalstellen, so erh\'e4lt man f\'fcr den 50 \par Iterationsschritt noch ein "vern\'fcnftiges" Ergebnis: \par \plain\f4\fs28\cf1 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf1 {\pntext\f1\'b7\tab}DIGITS:= 50: \par {\pntext\f1\'b7\tab}float( nEckUmfang(50) ) / 2 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf1 {\pntext\f1\'b7\tab}delete DIGITS: \par \pard\ri4\plain\f3\fs28\cf0 \par Dieses "Ph\'e4nomen" bietet interessanten Diskussionsstoff f\'fcr den Unterricht \par und motiviert die Suche nach einer numerisch besser geeigneten Formel f\'fcr \par den Umfang des n-Ecks, was uns auf die Formel \par \pard\li3000\ri4\plain\f3\fs28\cf0 {\pict\wmetafile8\picw3740\pich1598\picscalex99\picscaley99\picwgoal2138\pichgoal911 010009000003140A00000B001C0000000000050000000B0200000000050000000C023E069C0E03 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\plain\f3\fs28\cf0 \par \par Abschlie\'dfend sollen die visuellen F\'e4higkeiten von MuPAD genutzt werden, \par um die Methode von Archimedes grafisch darzustellen: \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf1 {\pntext\f1\'b7\tab}Kreis:= plot::Circle2d(1, [0,0], Color = RGB::Black): \par {\pntext\f1\'b7\tab}pi:= float(PI): \par {\pntext\f1\'b7\tab}n:= 10: \par {\pntext\f1\'b7\tab}nEck:= n -> plot::Polygon2d([[sin((2*PI*k)/n), cos((2*PI*k)/n)] \par \pard\li600\ri1\fi-300\plain\f4\fs28\cf1 $ k=0..n]): \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf1 {\pntext\f1\'b7\tab}plot( Kreis,nEck(n), Scaling = Constrained ) \par \pard\ri4\plain\f3\fs28\cf0 \par \plain\f3\fs22\cf0 \par \plain\f3\fs32\cf0 Die Formel von John Machin \par \plain\f4\fs20\cf0\b ________________________________________________________________________________\plain\f3\fs32\cf0 \par \par \plain\f3\fs28\cf0 Mitte des 17. 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000008000000320A400240000100000078000A00000026060F000A00FFFFFFFF0100000000001C 000000FB021000070000000000BC02000000000102022253797374656D0000C5120AFC14ED1200 D89FF177E19FF1772020F3774C11660C040000002D01000004000000F0010100040000002701FF FF030000000000 }}}\plain\f3\fs28\cf0 und erhielt: \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf1 {\pntext\f1\'b7\tab}G2:= G | x=x^2 \par \pard\ri4\plain\f3\fs28\cf0 \par Integriert man nun beide Seiten, so erh\'e4lt man unmittelbar die folgende \par Gleichung: \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf1 {\pntext\f1\'b7\tab}G3:= int( lhs(G2),x ) = int( rhs(G2),x ) \par \pard\ri4\plain\f3\fs28\cf0 \par Bekanntlich ist arctan(1)=\plain\f7\fs28\cf0 p/4\plain\f3\fs28\cf0 , d.h. die rechte Seite liefert uns eine beliebig \par genaue N\'e4herung von \plain\f7\fs28\cf0 p/4\plain\f3\fs28\cf0 , wenn wir f\'fcr x den Wert 1 einsetzen und n \par hinreichend gro\'df w\'e4hlen. \par \par Wir nehmen uns die rechte Seite dieser Gleichung also vor und fassen \par sie als Funktion in x und n auf: \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf1 {\pntext\f1\'b7\tab}Pi:= fp::unapply( rhs(G3),x,n ): \par \pard\ri4\plain\f3\fs28\cf0 \par Damit erhalten wir eine erste N\'e4herung von \plain\f7\fs28\cf0 p\plain\f3\fs28\cf0 \'fcber: \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf1 {\pntext\f1\'b7\tab}4.0*Pi(1,2) \par \pard\ri4\plain\f3\fs28\cf0 \par Die Konvergenz ist sehr langsam, wie folgende N\'e4herungen zeigen: \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf1 {\pntext\f1\'b7\tab}[4.0*Pi(1,n) $ n=1..50] \par \pard\ri4\plain\f3\fs28\cf0 \par \par Eine numerisch wesentlich g\'fcnstigere Formel ist \par \par \tab \tab \plain\f7\fs28\cf0 p\plain\f3\fs28\cf0 = 16 arctan(1/5) - 4 arctan(1/239), \par \par die wir aus unseren Berechnungen von oben mit n Schritten direkt wie folgt \par ausdr\'fccken k\'f6nnen: \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf1 {\pntext\f1\'b7\tab}Pi2:= n -> 16*Pi(1/5,n) + (-4)*Pi(1/239,n) \par \pard\ri4\plain\f3\fs28\cf0 \par Sie stammt von John Machin aus dem Jahr 1706. \par \par Im Vergleich zu \plain\f3\fs28\cf1\b Pi(50) \plain\f3\fs28\cf0 - der letzte Eintrag der Liste - sind bereits \par 10 Stellen erf\'fcllt: \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf1 {\pntext\f1\'b7\tab}float( Pi2(50) ) \par \pard\ri4\plain\f3\fs28\cf0 \par Damit rechnete John Machin als erster \plain\f7\fs28\cf0 p\plain\f3\fs28\cf0 auf 100 Dezimalen aus. \par Wir k\'f6nnen in wenigen Sekunden herausfinden, wie viele Schritte er \par rechnen mu\'dfte, um diesen Wert zu erhalten: \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf1 {\pntext\f1\'b7\tab}DIGITS:= 100: pi:= float(PI): n:= 1: \par \pard\li600\ri1\fi-300\plain\f4\fs28\cf1 eps:= 10^(-DIGITS): \par while abs(pi-Pi2(n)) > eps do n:= n+1 end: \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf1 {\pntext\f1\'b7\tab}n \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf1 {\pntext\f1\'b7\tab}Pi2(n) \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf1 {\pntext\f1\'b7\tab}float(%) \par \pard\ri4\plain\f3\fs28\cf0 \par Neumann rechnete mit der Eniac, dem ersten elektronischen Rechenautomaten, \par \plain\f7\fs28\cf0 p\plain\f3\fs28\cf0 auf 2035 Stellen aus (er wollte die Ziffernh\'e4ufigkeiten mit statistischen Mitteln \par untersuchen). \par \par Das dauerte 70 Stunden. Soviel Zeit w\'fcrden Sie uns sicher nicht zubilligen. \par \par Mit MuPAD ist das alles einfacher und schneller: Das Ergebnis lautet \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf1 {\pntext\f1\'b7\tab}DIGITS:= 2035: float( PI ) \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf1 {\pntext\f1\'b7\tab}delete DIGITS: \par \pard\ri4\plain\f3\fs22\cf0 \par \par \plain\f3\fs32\cf0 Numerische Integration \par \plain\f4\fs20\cf0\b ________________________________________________________________________________\plain\f3\fs32\cf0 \par \par \plain\f3\fs28\cf0 Ein weiteres Verfahren zur Approximation von \plain\f7\fs28\cf0 p\plain\f3\fs28\cf0 ist \'fcber das Integral \par \pard\li3000\ri4\plain\f3\fs28\cf0 {\object\objemb{\*\objclass Equation.3}{\*\objname Object1}\objw219\objh419\objscalex36\objscaley36{\*\objdata 01050000 02000000 0B000000 4571756174696F6E2E3300 00000000 00000000 000E0000 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}\plain\f3\fs28\cf0 \par \pard\ri4\plain\f3\fs28\cf0 mit Hilfe numerische Verfahren wie beispielsweise \'fcber Riemannsche \par Summen, die Trapez- oder die Simpsonregel. \par \par Auch hier treten symbolische und numerische Rechnungen sowie die \par Visualisierungsm\'f6glichkeiten von CAS in Erscheinung. \par \par Wir geben obige Funktion in MuPAD ein: \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf1 {\pntext\f1\'b7\tab}f:= x -> sqrt(1-x^2): \par \pard\ri4\plain\f3\fs28\cf0 \par und erhalten die Riemann-Summen mit n Iterationen \'fcber die von MuPAD \par mitgelieferte Funktion\plain\f3\fs28\cf0\b \plain\f3\fs28\cf1 riemann( f(x),x=a..b,n )\plain\f3\fs28\cf0 :\plain\f3\fs28\cf1 \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf1 {\pntext\f1\'b7\tab}export(student): \par \pard\li600\ri1\fi-300\plain\f4\fs28\cf1 F:= riemann( f,x=0..1,4 ) \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf1 {\pntext\f1\'b7\tab}unfreeze( F ) \par \pard\ri4\plain\f3\fs28\cf0 \par In Gleitkommadarstellung: \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf1 {\pntext\f1\'b7\tab}4 * float( F ) \par \pard\ri4\plain\f3\fs28\cf0 \par Entsprechend bestimmen wir eine Approximation von \plain\f7\fs28\cf0 p\plain\f3\fs28\cf0 \'fcber die \par Trapezregel: \par \plain\f3\fs28\cf1 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf1 {\pntext\f1\'b7\tab}F:= trapezoid( f,x=0..1,4 ) \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf1 {\pntext\f1\'b7\tab}4 * float( F ) \par \pard\ri4\plain\f3\fs28\cf0 \par und abschlie\'dfend im Vergleich die Simpsonregel: \par \plain\f3\fs28\cf1 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf1 {\pntext\f1\'b7\tab}F:= simpson( f,x=0..1,4 ) \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf1 {\pntext\f1\'b7\tab}4 * float( F ) \par \pard\ri4\plain\f3\fs28\cf0 \par Nat\'fcrlich k\'f6nnen diese Formeln erst im Unterricht hergeleitet, um sie dann \par in MuPAD in Form von kleinen Prozeduren zu implementieren. Aufgrund der \par K\'fcrze der Zeit habe ich die von MuPAD bereits implementierten Prozeduren \par verwendet. \par \par Solche numerischen Berechnungen geben Aufschlu\'df \'fcber die G\'fcte der \par Integrationsverfahren: \par \plain\f3\fs28\cf1 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf1 {\pntext\f1\'b7\tab}for n in [6,10,20,50,100] do \par \pard\li600\ri1\fi-300\plain\f4\fs28\cf1 print(Unquoted, \par "n = ".n.":\\n". \par " Rechteck: ".expr2text(4*float(riemann(f,x=0..1,n)))."\\n". \par " Trapez : ".expr2text(4*float(trapezoid(f,x=0..1,n)))."\\n". \par " Simpson : ".expr2text(4*float(simpson(f,x=0..1,n)))."\\n". \par "--------------------------" \par ) \par end: \par \pard\ri4\plain\f3\fs28\cf0 \par Stoff, der zum Experimentieren und zu Diskussionen anregt. \par \par Zum Abschlu\'df verwenden wir die Visualisierungsm\'f6glichkeiten von MuPAD, \par um den Proze\'df der numerischen Approximation anschaulich zu verdeutlichen: \par \plain\f3\fs28\cf1 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf1 {\pntext\f1\'b7\tab}plot( student::plotRiemann( f(x),x=0..1,8 ), \par \pard\li600\ri1\fi-300\plain\f4\fs28\cf1 Scaling = Constrained ) \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf1 {\pntext\f1\'b7\tab}plot( student::plotTrapezoid( f(x),x=0..1,8 ), \par \pard\li600\ri1\fi-300\plain\f4\fs28\cf1 Scaling = Constrained ) \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf1 {\pntext\f1\'b7\tab}plot( student::plotSimpson( f(x),x=0..1,8 ), \par \pard\li600\ri1\fi-300\plain\f4\fs28\cf1 Scaling = Constrained ) \par \par \pard\ri4\plain\f3\fs32\cf0 \par Moderne Methoden der Computer Algebra \par \plain\f4\fs20\cf0\b ________________________________________________________________________________\plain\f3\fs32\cf0 \par \par \plain\f3\fs28\cf0 Mit modernen Methoden der Mathematik aus dem Bereich der Computer- \par algebra und schnellen Algorithmen f\'fcr Ganzzahl- und Gleitkommaarithmetik \par mit hoher Genauigkeit, basierend auf der "Fast Fourier Transformation" und \par "Schneller Division" wird heute die Jagd nach dem Weltrekord f\'fcr die Nach- \par kommastellen von \plain\f7\fs28\cf0 p\plain\f3\fs28\cf0 betrieben. \par \par Derzeit liegt der Weltrekord bei mehreren \plain\f3\fs28\cf2 Billionen Ziffern\plain\f3\fs28\cf0 , der aber wohl bald \par wieder \'fcbertroffen werden d\'fcrfte. \par \par Solche Berechnungen dienen als gute Tests f\'fcr Hardware von Supercomputern, \par bevor sie zum Verkauf angeboten werden. \par \par Mit Computer-Algebra-Systemen wie MuPAD lassen sich bereits tausende von \par Nachkommstellen von \plain\f7\fs28\cf0 p\plain\f3\fs28\cf0 in sekundenschnelle bestimmen. \par \par In 200 Millisekunden berechnet MuPAD \plain\f7\fs28\cf0 p\plain\f3\fs28\cf0 beispielsweise auf 10.000 Stellen genau, \par wobei bei folgender Eingabe die meiste Zeit f\'fcr die Ausgabe des Ergebnisses \par ben\'f6tigt wird: \par \plain\f4\fs22\cf2 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf1 {\pntext\f1\'b7\tab}DIGITS:= 10000: \par {\pntext\f1\'b7\tab}time(float(PI)); \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf1 {\pntext\f1\'b7\tab}delete DIGITS: \par \pard\ri4\plain\f3\fs28\cf0 \par Bis heute ist unklar, ob die Dezimalstellen systematisch angeordnet sind oder \par aber zufallsverteilt sind. \par \par Wir wissen ebenso wenig, ob beispielsweise die Ziffer 1 unendlich oft in der \par Dezimalentwicklung von \plain\f7\fs28\cf0 p\plain\f3\fs28\cf0 vorkommt. \par \plain\f3\fs22\cf0 \par \plain\f4\fs22\cf1 \par \plain\f4\fs20\cf0\b ________________________________________________________________________________ \par \plain\f3\fs22\cf0 \par \plain\f3\fs22\cf3\b Anmerkungen:\plain\f3\fs22\cf3 \par \plain\f3\fs20\cf3\b 1\plain\f3\fs20\cf3 . Weitere Anregungen finden Sie in der Buchreihe \plain\f3\fs20\cf1 Mathematik 1 x anders\plain\f3\fs20\cf3 . In dieser Reihe \par wird eine Vielzahl unterschiedlichster mathematischer Probleme mit MuPAD gel\'f6st. Die \par B\'fccher k\'f6nnen unter \plain\f6\fs20\cf2 www.schule.mupad.de/literatur\plain\f3\fs20\cf3 kostenfrei kopiert werden. \par \plain\f3\fs20\cf3\b 2\plain\f3\fs20\cf3 . Ein herzliches Dankesch\'f6n an dieser Stelle an Frank Postel, von dem dieses sch\'f6ne Notebook \par stammt. \par \plain\f4\fs20\cf0\b _______________________________________________________________________________ \par \par \par }