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the Surprising Truths of Mathematics, Second Edition
"Yearning for the Impossible: The Surprising Truth of Mathematics, Second Edition explores the history of mathematics from the perspective of the creative tension between common sense and the "impossible" as the author follows the discovery or invention of new concepts that have marked mathematical progress. The author puts these creations into a broader context involving related "impossibilities" from art, literature, philosophy, and physics. This new edition contains many new exercises and commentaries, clearly discussing a wide range of challenging subjects."--Provided by publisher.
Cover; Half title; Title; Copyright; Dedication; Preface to the Second Edition; Preface; Contents; Chapter 1 The Irrational; 1.1 The Pythagorean Dream; 1.2 The Pythagorean Theorem; 1.3 Irrational Triangles; 1.4 The Pythagorean Nightmare; 1.5 Explaining the Irrational; 1.6 The Continued Fraction for 2; 1.7 Equal Temperament; Chapter 2 The Imaginary; 2.1 Negative Numbers; 2.2 Imaginary Numbers; 2.3 Solving Cubic Equations; 2.4 Real Solutions via Imaginary Numbers; 2.5 WhereWere Imaginary Numbers before 1572?; 2.6 Geometry ofMultiplication; 2.7 Complex Numbers GiveMore thanWe Asked for.
2.8 Why Call Them "Complex" Numbers?Chapter 3 The Horizon; 3.1 Parallel Lines; 3.2 Coordinates; 3.3 Parallel Lines and Vision; 3.4 Drawing withoutMeasurement; 3.5 The Theorems of Pappus and Desargues; 3.6 The Little Desargues Theorem; 3.7 What Are the Laws of Algebra?; 3.8 Projective Addition andMultiplication; Chapter 4 The Infinitesimal; 4.1 Length and Area; 4.2 Volume; 4.3 Volume of a Tetrahedron; 4.4 The Circle; 4.5 The Parabola; 4.6 The Slopes of Other Curves; 4.7 Slope and Area; 4.8 The Value of ơ; 4.9 Ghosts of Departed Quantities; Chapter 5 Curved Space.
5.1 Flat Space andMedieval Space5.2 The 2-Sphere and the 3-Sphere; 5.3 Flat Surfaces and the Parallel Axiom; 5.4 The Sphere and the Parallel Axiom; 5.5 Non-Euclidean Geometry; 5.6 Negative Curvature; 5.7 The Hyperbolic Plane; 5.8 Hyperbolic Space; 5.9 Mathematical Space and Actual Space; Chapter 6 The Fourth Dimension; 6.1 Arithmetic of Pairs; 6.2 Searching for an Arithmetic of Triples; 6.3 Why n-tuples Are Unlike Numbers when n ı 3; 6.4 Quaternions; 6.5 The Four-Square Theorem; 6.6 Quaternions and Space Rotations; 6.7 Symmetry in Three Dimensions; 6.8 Tetrahedral Symmetry and the 24-Cell.
6.9 The Regular PolytopesChapter 7 The Ideal; 7.1 Discovery and Invention; 7.2 Division with Remainder; 7.3 The Euclidean Algorithm; 7.4 Unique Prime Factorization; 7.5 Gaussian Integers; 7.6 Gaussian Primes; 7.7 Rational Slopes and Rational Angles; 7.8 Unique Prime Factorization Lost; 7.9 Ideals-Unique Prime Factorization Regained; Chapter 8 Periodic Space; 8.1 The Impossible Tribar; 8.2 The Cylinder and the Plane; 8.3 Where theWild Things Are; 8.4 PeriodicWorlds; 8.5 Periodicity and Topology; 8.6 A Brief History of Periodicity; 8.7 Non-Euclidean Periodicity; Chapter 9 The Infinite.
9.1 Finite and Infinite9.2 Potential and Actual Infinity; 9.3 The Uncountable; 9.4 The Diagonal Argument; 9.5 The Transcendental; 9.6 Yearning for Completeness; Epilogue; References; Index.
Includes bibliographical references and index.