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W HAT IS IT ? I S IT INTERESTING ? Michael Woltermann Mathematics Department Washington and Jefferson College Washington, PA 15301-4801.

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Presentation on theme: "W HAT IS IT ? I S IT INTERESTING ? Michael Woltermann Mathematics Department Washington and Jefferson College Washington, PA 15301-4801."— Presentation transcript:

1 W HAT IS IT ? I S IT INTERESTING ? Michael Woltermann Mathematics Department Washington and Jefferson College Washington, PA

2 T RIUMPH DER M ATHEMATIK 100 Great Problems of Elementary Mathematics By Heinrich Dörrie

3 S OME B ACKGROUND Heinrich Dörrie Ph. D. Georg-August-Universität Göttingen 1898 Dissertation Das quadratische Reziprozitätsgesetz im quadratischen Zahlkörper mit der Klassenzahl 1. Advisor David Hilbert Triumph der Mathematik German editions 1932, 1940 Dover (English) edition rie/DorrieContents.htm

4 F ROM THE P REFACE For a long time, I (H. Dörrie) have considered it a necessary and appealing task to write a book of celebrated problems of elementary mathematics, their origins, and above all brief, clear and understandable solutions to them. … The present work contains many pearls of mathematics from Gauss, Euler, Steiner and others. So then, let this book do its part to awaken and spread interest and pleasure in mathematical thought.

5 F ROM A R EVIEW AT A MAZON. COM The selection of problems is outstanding and lives up to the book's original title. The proofs are concise, clever, elegant, often extremely difficult and not particularly enlightening. To say that this book requires a background in college math is like saying that playing chess requires a background in how to move the pieces; it also requires a lot of thought and, preferably, a lot of experience.

6 F ROM M.W. ( SPRING 2010) A lot of things have changed since For example, terminology has changed, people are not as knowledgeable about some areas of mathematics (especially geometry) as they once were, but more knowledgeable about others (e.g. calculus). A straightforward translation would not necessarily shed more light on the problems in this book. What was required was in some cases more (or less) mathematical background, current terminology and notation to bring Triumph der Mathematik into the twenty first century.

7 55. T HE C URVATURE OF C ONIC S ECTIONS Determine the curvature of a conic section. Let the conic section be c e its eccentricity, p half the latus rectum, q = 1-e 2. An equation for c is qx 2 +y 2 -2px = 0. Let n be the length of the normal from a point P on the conic to its axis. Then The radius of the circle of curvature is

8 P ARABOLA

9 E LLIPSE

10 H YPERBOLA

11

12 I S IT INTERESTING ? Conics by Keith Kendig, MAA 2005 Goal is to see conics in a unified way. But n cubed over p squared doesn’t appear New Geometric Constructions to Determine the Radius of Curvature of Conics at any Point by Jiménez and Granero, 2007 Their approach is based on a “recently found property of conic sections”. They cite a 1999 article in Computer Aided Geometric Design, No. 16. It’s fun to implement with Geometer’s Sketchpad, Geogebra, etc.Geometer’s Sketchpad

13 T HE C ALCULUS A PPROACH The curvature at (x,y) is With y 2 =2px-qx 2,

14 A NY


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