Charming Proofs: A Journey Into Elegant Mathematics
Theorems and their proofs lie at the heart of mathematics. In speaking of the purely aesthetic qualities of theorems and proofs, G. H. Hardy wrote that in beautiful proofs 'there is a very high degree of unexpectedness, combined with inevitability and economy'. Charming Proofs presents a collection of remarkable proofs in elementary mathematics that are exceptionally elegant, full of ingenuity, and succinct. By means of a surprising argument or a powerful visual representation, the proofs in this collection will invite readers to enjoy the beauty of mathematics, and to develop the ability to create proofs themselves. The authors consider proofs from topics such as geometry, number theory, inequalities, plane tilings, origami and polyhedra. Secondary school and university teachers can use this book to introduce their students to mathematical elegance. More than 130 exercises for the reader (with solutions) are also included.
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a2 C b2 Adventures in Tiling Alsina AM-GM inequality angles calissons Cauchy-Schwarz inequality Challenges CHAPTER chessboard circle circumradius congruent construct convex quadrilateral cube cyclic quadrilateral deﬁned denote diagonals equal equilateral triangle Euclid Euler Fermat point Fibonacci Fibonacci numbers ﬁgure ﬁnd ﬁnite ﬁrst ﬁve ﬂat fold formula frieze functional equation geometric golden ratio gray triangle hence hexagon Honsberger hypotenuse illustrated in Figure inequality inﬁnite inscribed integer intersection irrational lattice points Lemma light gray line segments lunes mathematics median midpoint n-gon Nelsen parallelogram partition pentagon perpendicular pigeonhole principle polyhedron positive integers prime problem Prove Pythagorean theorem radius rectangle reﬂection regions regular result right triangle rotation Section sequence shaded triangle shown in Figure side length solution squarable star polygons tetrominoes tile the plane Tiling and Coloring triangle ABC triangular numbers trisect vertex vertices yields