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Deutsch: Quadratur des Kreises, Näherungskonstruktion nach Ramanujan von 1914, mit Weiterführung der Konstruktrion
English: Squaring the circle, approximitiy construction according Ramanujan of 1914, with continuation of the construction
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Author Petrus3743
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Quadratur des Kreises, Näherungskonstruktion nach Ramanujan von 1914, mit Weiterführung der Konstruktrion, Animation
Squaring the circle, approximitiy construction according Ramanujan of 1914, with continuation of the construction, animation
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Im Jahr 1914 ermittelte Ramanujan für eine noch genauere Quadratur als die von 1913, den folgenden Näherungswert für die Kreiszahl

[1]

in dem acht Nachkommastellen mit denen von gleich sind.

Ramanujan konstruierte in dieser Quadratur nicht die Seitenlänge des gesuchten Quadrates, es genügte ihm die Strecke OS darzustellen.[2] In der obigen Weiterführung der Konstruktion, wird die Strecke OS zusammen mit der Strecke OB zur Darstellung der mittleren Proportionalen (rote Strecke OG) herangezogen.[3]

Fehler

Bei einem Kreis mit Radius r = 1 [LE]:

  • Konstruierte Seite des Quadrates a = 1,77245385062141... [LE]
  • Soll-Seite des Quadrates as = = 1,772453850905516... [LE]
  • Absoluter Fehler = a - as = -0,00000000028411... = -2,841...E-10 [LE]
  • Fläche des konstruierten Quadrates A = a2 = 3,14159265258265... [FE]
  • Soll-Fläche des Quadrates As = = 3,141592653589793... [FE]
  • Absoluter Fehler = A - As = -0,000000001007143... = -1,007...E-9 [FE]

Beispiele zur Veranschaulichung der Fehlers

  • Bei einem Kreis mit dem Radius r = 10.000 km wäre der Fehler der Seite a ≈ -2,8 mm
  • Bei einem Kreis mit dem Radius r = 10 m wäre der Fehler der Fläche A ≈ -0,1 mm2

Error

In a circle of radius r = 1 [unit length, ul]:

  • Constructed side of the square a = 1.77245385062141... [ul]
  • Target side of the square as = = 1.772453850905516... [ul]
  • Absolute error = a - as = -0.00000000028411... = -2.841...E-10 [ul]
  • Surface of the constructed square A = a2 = 3.14159265258265... [unit area, ua]
  • Target area of the square As = = 3.141592653589793... [ua]
  • Absolute error = A - As = -0,000000001007143... = -1.007...E-9 [ua]

Examples to illustrate the errors:

  • In a circle of radius r = 10,000 km would be the fault of the side length a ≈ -2.8 mm
  • In the case of a circle with the radius r = 10 m would be the error of the surface A ≈ -0.1 mm2

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  1. S. A. Ramanujan: Modular Equations and Approximations to π In: Quarterly Journal of Mathematics. 12. Another curious approximation to π is, 43, (1914), S. 350–372. Aufgelistet in: Published works of Srinivasa Ramanujan Abgerufen am 21. November 2016
  2. Modular Equations and Approximations to π In: Quarterly Journal of Mathematics. 12. Another curious approximation to π is ... Fig. 2, 44, (1914), S. 350–372. Aufgelistet in: Published works of Srinivasa Ramanujan Abgerufen am 21. November 2016
  3. Universität Magdeburg A.14 Mittelwerte. Mittlere Proportionale (PDF-Datei) Abgerufen am 21. November 2016

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20 November 2016

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Date/TimeThumbnailDimensionsUserComment
current13:07, 25 December 2016Thumbnail for version as of 13:07, 25 December 2016973 × 894 (191 KB)Petrus3743≈ π ergänzt
11:07, 25 December 2016Thumbnail for version as of 11:07, 25 December 2016973 × 894 (183 KB)Petrus3743Konstruktion vereinfacht
10:27, 9 December 2016Thumbnail for version as of 10:27, 9 December 20161,104 × 1,034 (171 KB)Petrus3743Kurzbeschreibung korrigiert
17:18, 21 November 2016Thumbnail for version as of 17:18, 21 November 20161,104 × 1,034 (171 KB)Petrus3743Bezeichnungen für Punkte korrigiert
16:52, 20 November 2016Thumbnail for version as of 16:52, 20 November 20161,104 × 1,034 (171 KB)Petrus3743User created page with UploadWizard

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