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August (ap). Zwei japanische Mathematiker haben am \ Freitag einen neuen Weltrekord f\[UDoubleDot]r die Berechnung der Zahl \[Pi] \ bekanntgegeben. Wie Yasumasa Kanada und Daisuke Takahashi von der Universit\ \[ADoubleDot]t Tokio erkl\[ADoubleDot]rten, berechneten sie die Zahl, die das \ Verh\[ADoubleDot]ltnis des Kreisumfangs zum Durchmesser angibt, auf 3,22 \ Milliarden Dezimalstellen genau. Wissenschaftler der US-amerikanischen \ Columbia-Universit\[ADoubleDot]t hielten den alten Rekord mit 2,26 Milliarden \ Stellen.\n\nProfessor Kaneda sagte, seine Gruppe habe einen Gro\[SZ]computer \ benutzt, um zwei verschiedene Formeln durchrechnen zu lassen. F\[UDoubleDot]r \ die erste habe der Rechner 36 Stunden und 52 Minuten gebraucht, \ f\[UDoubleDot]r die andere 53 Stunden und 43 Minuten. Mathematiker sind der \ Meinung, da\[SZ] Pi unz\[ADoubleDot]hlig viele Dezimalstellen hat. Die pr\ \[ADoubleDot]zise Berechnung der Kreiszahl hat keinerlei praktischen Wert.\n\n\ ", FormatType->StandardForm, FontFamily->"Times", FontWeight->"Plain"], "\n", StyleBox[ButtonBox["Scientists find even more pi", ButtonData:>{ URL[ "http://www.ananova.com/news/story/sm_722745.html"], None}, ButtonStyle->"Hyperlink"], FontFamily->"Times New Roman"], StyleBox[" ", FontFamily->"Times New Roman"], StyleBox["\n", FormatType->StandardForm, FontFamily->"Times New Roman", FontWeight->"Plain"], StyleBox["\nSiehe auch:", FormatType->StandardForm, FontFamily->"Times"], StyleBox[" ", FormatType->StandardForm, FontFamily->"Times New Roman"], StyleBox[ButtonBox["Mathematik im Internet: \[Pi]", ButtonData:>{ URL[ "http://www.mathematik.uni-kassel.de/~rascha/MathWWW_Pi.htm"], None}, ButtonStyle->"Hyperlink"], FontFamily->"Times New Roman"], StyleBox[" ", FormatType->StandardForm, FontFamily->"Times New Roman"], StyleBox["\n", FormatType->StandardForm, FontFamily->"Times"] }], "Input", FontWeight->"Plain"], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Extra Tip, ", FontFamily->"Arial", FontSize->12, FontWeight->"Bold"], StyleBox["Kassel, 24. 9 1992", FontFamily->"Arial", FontSize->12] }], "Text"], Cell[TextData[{ StyleBox["Zahl Pi falsch", FormatType->StandardForm, FontFamily->"Arial", FontSize->12], StyleBox["\n", FormatType->StandardForm, FontFamily->"Arial"], StyleBox["\nHermann Wahler ist Rentner. Sein Hobby: Mathematik. Besonders \ angetan hat's ihm der Rauminhalt des Kreises. Der wird \[UDoubleDot]ber den \ Radius und die Zahl Pi berechnet. \"Ich habe das nachgerechnet, \ \[UDoubleDot]ber das Dreieck\", sagt Wahler. Ergebnis: Die Zahl Pi kann so \ nicht stimmen. Hat er sich verrechnet oder die Mathe-Professoren? Lesen Sie \ mehr auf Seite 6.\n", FormatType->StandardForm, FontFamily->"Times"] }], "Input", FontWeight->"Plain"] }, Open ]], Cell[TextData[{ StyleBox["Seite 6:\n\n", FormatType->StandardForm, FontFamily->"Times"], StyleBox["Zahl Pi neu berechnet!\n", FormatType->StandardForm, FontFamily->"Arial", FontSize->12], StyleBox["\nDr\[UDoubleDot]ckt man beim Taschenrechner auf die Taste \"\", \ erscheint der Wert 3,14159. damit errechnet man den Rauminhalt des Kreises, \ bisher. \"Diese Zahl Pi ist falsch\", meint Hermann Wahler aus Kaufungen.* \ \"Nach meinen Berechnungen ist diese Zahl kleiner als 3,14!\" Wie er drauf \ kommt? Er berechnet Pi nicht \[UDoubleDot]ber das Quadrat, sondern \ \[UDoubleDot]bers Dreieck. Normalerweise wird die Zahl Pi \[UDoubleDot]ber \ die Quadratur des Kreises berechnet. Ein Quadrat, dessen Ecken die Innenfl\ \[ADoubleDot]chen des Kreises ber\[UDoubleDot]hren; ein Quadrat, dessen \ Linien die Au\[SZ]enpunkte des Kreises ber\[UDoubleDot]hren.\n\nEine Formel \ dazu, und raus kommt die Zahl Pi: 3,14159.", FormatType->StandardForm, FontFamily->"Times"] }], "Input", FontWeight->"Plain"], Cell[TextData[StyleBox["Doch Wahler gibt sich damit nicht zufrieden. Sein \ Trick: Er zieht ein Vieleck innerhalb des Kreises. Je mehr Ecken den Kreis \ ber\[UDoubleDot]hren, desto genauer spiegelt dieses Vieleck, den Rauminhalt \ des Kreises wieder.\n\nDas Vieleck unterteilt den Kreis in viele Dreiecke \ plus die Summe der Kreisabschnitte. Wahlers Ergebnis: Die Zahl Pi ist kleiner \ als 3,14. denn beim 600-Eck liegt sie bei 3.139999998, beim 6000-Eck bei \ 3,139999976 und beim 60.000-Eck bei 3,13999981.\nHermann Wahler: \"Je mehr \ Ecken ich berechne, desto genauer wird die endg\[UDoubleDot]ltige Zahl Pi. \ Sicher schon jetzt: Die Zahl, die sich bei den bisherigen Berechnungen \ ergibt, kann so nicht stimmen.\"", FormatType->StandardForm, FontFamily->"Times"]], "Input", FontWeight->"Plain"], Cell[TextData[StyleBox["Stimmt seine Rechnung? \"Ich glaube schon, doch ich \ hoffe, da\[SZ] das jemand auf einem Gro\[SZ]rechner nachrechnet\", meint \ Wahler. Denn er selber hat nur einen Taschenrechner zur \ Verf\[UDoubleDot]gung. Es k\[ODoubleDot]nnte sein, da\[SZ] ein Rundungsfehler \ innerhalb seiner komplizierten Berechnung die Zahl Pi ver\[ADoubleDot]ndert.\n\ \"Die ganze Geschichte der Zahl habe ich verfolgt, es wurde immer um den \ Rauminhalt des Kreises gestritten\", meint Wahler. Bis eben jemand die Formel \ mit Quadratur des Kreises herausfand, doch der Kaufunger gibt sich damit \ nicht zufrieden.\n\n*Name und Ort ge\[ADoubleDot]ndert", FormatType->StandardForm, FontFamily->"Times"]], "Input", FontWeight->"Plain"], Cell[CellGroupData[{ Cell[TextData[StyleBox["Kurhessische Landeszeitung, Kassel, 7. 3. 1934\n", FontFamily->"Arial", FontSize->12, FontWeight->"Bold"]], "Text"], Cell[TextData[{ StyleBox["Eine sensationelle Ver\[ODoubleDot]ffentlichung des K.f.d.K., \ Landesleitung Kurhessen:\n\nQuadratur des Kreises entdeckt\n", FormatType->StandardForm, FontFamily->"Arial", FontSize->12], StyleBox["\nEin epochales Forschungsergebnis von unsch\[ADoubleDot]tzbarer \ Bedeutung - Der Musikforscher Willi Oberle, Kassel, l\[ODoubleDot]st ein \ Jahrhundert altes Problem. Der Konstruktion des Ton-Prismas entgegen.\nDurch \ die \"Kurhessische Landeszeitung\" gibt der Landesleiter des Kampfbundes f\ \[UDoubleDot]r deutsche Kultur, Max K\[ODoubleDot]hler, der \ \[CapitalODoubleDot]ffentlichkeit folgendes Forschungsergebnis von Willi \ Oberle, Kassel-Niederzwehren, bekannt, das besagt, da\[SZ] dieser durch die \ Musikgeometrie zur L\[ODoubleDot]sung der Quadratur des Kreises gelangt ist:\n\ \n\"Dem Musikforscher Willi Oberle in Kassel-Niederzwehren, \ fr\[UDoubleDot]her in Baden-Baden, Mitarbeiter des Kampfbundes \ f\[UDoubleDot]r deutsche Kultur, Landesleitung Kurhessen, ist es gelungen, \ aus den Ergebnissen seiner Forschungen auf dem Gebiete der Musikgeometrie die \ Quadratur des Kreises aufzudecken. [ ... ]\n\nDer Weg zur L\[ODoubleDot]sung\n\ \nDa\[SZ] in seinen Konstruktionen die Quadratur des Kreises verborgen sein k\ \[ODoubleDot]nne, vermutete Oberle schon seit 1927. Die vielen anderen \ Probleme, wie das Kristallisationsph\[ADoubleDot]nomen, das \ Schwingungsproblem (Urschwingungen), die Architektonik der Zukunft, das \ Rassenproblem (Chromosomen = Ariogermanischer Urtyp), das Urwort (Runen, \ \[ADoubleDot]yptische und griechische Inschriften), in erster Linie das \ Tonprisma erschienen Oberle aber weit wichtiger als die Quadratur des \ Kreises, denn er erkannte, da\[SZ] die L\[ODoubleDot]sung der genannten \ Probleme aus seiner Musikgeometrie so gut wie gewi\[SZ] ist. Erst seit Oberle \ Mitarbeiter des Kurhessischen Kampfbundes f\[UDoubleDot]r deutsche Kultur \ geworden war und durch ihn neuen Schaffensimpuls erhielt, wurde sein \ Interesse f\[UDoubleDot]r das Problem der Quadratur des Kreises wach.\nMit gr\ \[ODoubleDot]\[SZ]ter Intensit\[ADoubleDot]t widmete sich dann Oberle der \ Aufgabe, das Problem aus seinen Konstruktionen aufzudecken. Da\[SZ] es sich \ nur um seine Aufdeckung handelt, ist ausdr\[UDoubleDot]cklich zu betonen, \ weil die Quadratur des Kreises, wie gesagt, seit 1927 in den Konstruktionen \ Oberles bis 1933 einen Dornr\[ODoubleDot]schenschlaf hielt. [ ... ]\n\nWir \ beschr\[ADoubleDot]nken uns hier zun\[ADoubleDot]chst auf die Verk\ \[UDoubleDot]ndigung der gefundenen Grundresultate:\n\n", FormatType->StandardForm, FontFamily->"Times"], StyleBox["\"Der absolute Inhalt eines jeden Kreises ist gleich dem Quadrat \ \[UDoubleDot]ber der Hypothenuse des rechtwinckligen Dreiecks, dessen eine \ Kathede die gr\[ODoubleDot]\[SZ]te Seite des Oberleschen Trapezes darstellt, \ und dessen kleine Kathede gleich der Entfernung vom Kreismittelpunkt bis zum \ Schnittpunkte der beiden Diagonalen des das Trapez umschlie\[SZ]enden Kreises \ ist. Die beiden genannten Diagonalen schneiden sich im goldenen Schnitt und \ zugleich im rechten Winkel. Die kleinste Seite des Trapezes ist gleich dem \ Radius des Kreises (Pythagor\[ADoubleDot]ischer Lehrsatz!)\".", FormatType->StandardForm, FontFamily->"Arial"], StyleBox["\n\nForschungserfolg von unaussprechlicher Bedeutung\nDie \ Bedeutung der Entdeckung der Quadratur des Kreises l\[ADoubleDot]\[SZ]t sich \ erst daran ermessen, da\[SZ] sich die Menschheit seit Jahrhunderten mit \ diesem Problem besch\[ADoubleDot]ftigt hatte, ohne zu einer \ L\[ODoubleDot]sung zu kommen. [ ... ] \:f35f", FormatType->StandardForm, FontFamily->"Times"] }], "Input", FontWeight->"Plain"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ 1.1.2 N\[ADoubleDot]herungsweise Berechnung von \[Pi] \ \>", "Section"], Cell[TextData[{ ButtonBox["Archimedes' Approximation of Pi", ButtonData:>{ URL[ "http://itech.fgcu.edu/faculty/clindsey/mhf4404/archimedes/archimedes.\ html"], None}, ButtonStyle->"Hyperlink"], " " }], "Text", FontSize->12], Cell[TextData[{ StyleBox["Approximation des \nKreises durch regul\[ADoubleDot]re Polygone \ nach", FontFamily->"Arial", FontWeight->"Bold"], StyleBox[" ", FontFamily->"Arial", FontSize->11.875, FontWeight->"Bold"], StyleBox[ButtonBox["Archimedes", ButtonData:>{ URL[ "http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Archimedes.\ html"], None}, ButtonStyle->"Hyperlink"], FontFamily->"Arial", FontSize->11.875, FontWeight->"Bold"], StyleBox[" ", FontFamily->"Arial", FontSize->11.875, FontWeight->"Bold"], StyleBox[" (\[Dash]287 (?) \[Dash]\[Dash] \[Dash]212 )", FontFamily->"Arial", FontWeight->"Bold"] }], "Text", FontSize->12], Cell[TextData[{ "A", StyleBox["RCHIMEDES", FontSize->10], " approximierte einen Kreis von innen und au\[SZ]en erst durch ein 6-Eck, \ dann durch je ein 12-Eck, 24-, 48- und 96-Eck.\n\nDer Umfang des \ einbeschriebenen 6-Ecks ist beim Einheitskreis (also dem Kreis mit Radius \ eins) 6. Damit ist 3 die erste N\[ADoubleDot]herung von \[Pi] (wie auch in \ der ", ButtonBox["Bibel: 1. K\[ODoubleDot]nige, 7. 23", ButtonData:>{ URL[ "http://www.bibel-online.net/buch/11.1-koenige/7.html#7,23"], None}, ButtonStyle->"Hyperlink"], ").\n\nA", StyleBox["RCHIMEDES", FontSize->10], " verdoppelte bei jedem Schritt die Anzahl der Ecken der Polygone, bzw. \ halbierte die Mittelpunktswinkel. Wenn Sie eine N\[ADoubleDot]herung kennen, \ k\[ODoubleDot]nnen Sie verstehen, wie die n\[ADoubleDot]chste berechnet \ wird." }], "Text", FontSize->12], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .86603 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Times %%IncludeFont: Times /Times findfont 12 scalefont setfont % Scaling calculations 0.5 0.408852 0.433013 0.408852 [ [ 0 0 0 0 ] [ 1 .86603 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 0 m 1 0 L 1 .86603 L 0 .86603 L closepath clip newpath 0 1 0 r 1.5 Mabswid [ ] 0 setdash .90476 .43301 m .70238 .78355 L .29762 .78355 L .09524 .43301 L .29762 .08248 L .70238 .08248 L .90476 .43301 L s 1 0 0 r .97619 .43301 m .7381 .84541 L .2619 .84541 L .02381 .43301 L .2619 .02062 L .7381 .02062 L .97619 .43301 L s 0 0 1 r newpath .5 .43301 .40885 0 365.73 arc s % 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Schritt bei n = 24576 \"stehen\" und liefert \ E", FormatType->StandardForm, FontFamily->"Times", FontSlant->"Italic"], StyleBox["24576", FormatType->StandardForm, FontFamily->"Times", FontSlant->"Italic", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[" = 6,2831852 und U", FormatType->StandardForm, FontFamily->"Times", FontSlant->"Italic"], StyleBox["24576", FormatType->StandardForm, FontFamily->"Times", FontSlant->"Italic", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[" = 6,2831854, so da\[SZ] man stellengenau \[Pi] =3,1415927 erh\ \[ADoubleDot]lt.", FormatType->StandardForm, FontFamily->"Times", FontSlant->"Italic"] }], "Text", FontSize->12], Cell[TextData[{ StyleBox["Aufgabe: Rechnen Sie mit Ihrem Taschenrechner bis zum 13. Schritt \ ", FormatType->StandardForm, FontFamily->"Times"], StyleBox["E", FormatType->StandardForm, FontFamily->"Times", FontSlant->"Italic"], StyleBox["n", FormatType->StandardForm, FontFamily->"Times", FontSlant->"Italic", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[" aus und zwar nach den beiden Formeln (1) und (2). Schreiben Sie \ alle von Ihnen berechneten Zwischenwerte in eine Tabelle.\n\nZum Schlu\[SZ] \ dieses Abschnittes die ersten 999 Nachkommastellen von \[Pi] :", FormatType->StandardForm, FontFamily->"Times"] }], "Text", FontSize->12], Cell[CellGroupData[{ Cell[BoxData[ \(N[\[Pi], 1000]\)], "Input"], Cell[BoxData[ \(3.1415926535897932384626433832795028841971693993751058209749445923078164\ 062862089986280348253421170679821480865132823066470938446095505822317253594081\ 284811174502841027019385211055596446229489549303819644288109756659334461284756\ 482337867831652712019091456485669234603486104543266482133936072602491412737245\ 870066063155881748815209209628292540917153643678925903600113305305488204665213\ 841469519415116094330572703657595919530921861173819326117931051185480744623799\ 627495673518857527248912279381830119491298336733624406566430860213949463952247\ 371907021798609437027705392171762931767523846748184676694051320005681271452635\ 608277857713427577896091736371787214684409012249534301465495853710507922796892\ 589235420199561121290219608640344181598136297747713099605187072113499999983729\ 780499510597317328160963185950244594553469083026425223082533446850352619311881\ 710100031378387528865875332083814206171776691473035982534904287554687311595628\ 638823537875937519577818577805321712268066130019278766111959092164201989380952\ 57201`1000\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(NumberForm[\ N[\[Pi], 1030], DigitBlock -> 3\ , NumberSeparator -> "\< \>"]\)], "Input"], Cell[BoxData[ TagBox[ InterpretationBox["\<\"3.141 592 653 589 793 238 462 643 383 279 502 \ 884 197 169 399 375 105 820 974 944 592 307 816 406 286 208 998 628 034 825 \ 342 117 067 982 148 086 513 282 306 647 093 844 609 550 582 231 725 359 408 \ 128 481 117 450 284 102 701 938 521 105 559 644 622 948 954 930 381 964 428 \ 810 975 665 933 446 128 475 648 233 786 783 165 271 201 909 145 648 566 923 \ 460 348 610 454 326 648 213 393 607 260 249 141 273 724 587 006 606 315 588 \ 174 881 520 920 962 829 254 091 715 364 367 892 590 360 011 330 530 548 820 \ 466 521 384 146 951 941 511 609 433 057 270 365 759 591 953 092 186 117 381 \ 932 611 793 105 118 548 074 462 379 962 749 567 351 885 752 724 891 227 938 \ 183 011 949 129 833 673 362 440 656 643 086 021 394 946 395 224 737 190 702 \ 179 860 943 702 770 539 217 176 293 176 752 384 674 818 467 669 405 132 000 \ 568 127 145 263 560 827 785 771 342 757 789 609 173 637 178 721 468 440 901 \ 224 953 430 146 549 585 371 050 792 279 689 258 923 542 019 956 112 129 021 \ 960 864 034 418 159 813 629 774 771 309 960 518 707 211 349 999 998 372 978 \ 049 951 059 731 732 816 096 318 595 024 459 455 346 908 302 642 522 308 253 \ 344 685 035 261 931 188 171 010 003 137 838 752 886 587 533 208 381 420 617 \ 177 669 147 303 598 253 490 428 755 468 731 159 562 863 882 353 787 593 751 \ 957 781 857 780 532 171 226 806 613 001 927 876 611 195 909 216 420 198 938 \ 095 257 201 065 485 863 278 865 936\"\>", 3.\ 141592653589793238462643383279502884197169399375105820974944592307816406286208\ 998628034825342117067982148086513282306647093844609550582231725359408128481117\ 450284102701938521105559644622948954930381964428810975665933446128475648233786\ 783165271201909145648566923460348610454326648213393607260249141273724587006606\ 315588174881520920962829254091715364367892590360011330530548820466521384146951\ 941511609433057270365759591953092186117381932611793105118548074462379962749567\ 351885752724891227938183011949129833673362440656643086021394946395224737190702\ 179860943702770539217176293176752384674818467669405132000568127145263560827785\ 771342757789609173637178721468440901224953430146549585371050792279689258923542\ 019956112129021960864034418159813629774771309960518707211349999998372978049951\ 059731732816096318595024459455346908302642522308253344685035261931188171010003\ 137838752886587533208381420617177669147303598253490428755468731159562863882353\ 787593751957781857780532171226806613001927876611195909216420198938095257201065\ 485863278865936`1030, AutoDelete->True], (NumberForm[ #, DigitBlock -> 3, NumberSeparator -> " "]&)]], "Output"] }, Open ]] }, Open ]] }, FrontEndVersion->"4.1 for Microsoft Windows", ScreenRectangle->{{0, 1400}, {0, 975}}, CellGrouping->Manual, WindowSize->{1016, 672}, WindowMargins->{{0, Automatic}, {Automatic, 0}}, PrintingCopies->1, PrintingPageRange->{Automatic, Automatic}, PageHeaders->{{Inherited, Inherited, Inherited}, {Cell[ TextData[ "1.1 Zur Quadratur des Kreises \[Dash] eine Berechnung von \[Pi]"], "Header"], Inherited, Inherited}}, PageFooters->{{Inherited, Inherited, Inherited}, {Cell[ TextData[ "Materialien zur Vorlesung Elementare Analysis, Wintersemester 2003 / \ 4"]], Inherited, Inherited}}, PageFooterLines->{Inherited, True}, PrintingOptions->{"FirstPageHeader"->False}, Magnification->1 ] (******************************************************************* Cached data follows. 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