File:Squaring the circle-Ramanujan-1914.svg
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editDescriptionSquaring the circle-Ramanujan-1914.svg |
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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Source | Own work |
Author | Petrus3743 |
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Im Jahr 1914 ermittelte Ramanujan für eine noch genauere Quadratur als die von 1913, den folgenden Näherungswert für die Kreiszahl
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
editBei 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
editIn 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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- ↑ 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
- ↑ 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
- ↑ Universität Magdeburg A.14 Mittelwerte. Mittlere Proportionale (PDF-Datei) Abgerufen am 21. November 2016
File history
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Date/Time | Thumbnail | Dimensions | User | Comment | |
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current | 13:07, 25 December 2016 | 973 × 894 (191 KB) | Petrus3743 (talk | contribs) | ≈ π ergänzt | |
11:07, 25 December 2016 | 973 × 894 (183 KB) | Petrus3743 (talk | contribs) | Konstruktion vereinfacht | ||
10:27, 9 December 2016 | 1,104 × 1,034 (171 KB) | Petrus3743 (talk | contribs) | Kurzbeschreibung korrigiert | ||
17:18, 21 November 2016 | 1,104 × 1,034 (171 KB) | Petrus3743 (talk | contribs) | Bezeichnungen für Punkte korrigiert | ||
16:52, 20 November 2016 | 1,104 × 1,034 (171 KB) | Petrus3743 (talk | contribs) | User created page with UploadWizard |
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