Yearning for the Impossible: The Surprising Truths of Mathematics, Second Edition (AK Peters/CRC Recreational Mathematics Series)

"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 Yearning for the Impossible: The Surprising Truths of Mathematics, 2nd Edition Preface to the Second Edition Preface Contents 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 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 of Multiplication 2.7 Complex Numbers Give More than We Asked for 2.8 Why Call Them “Complex” Numbers? 3 The Horizon 3.1 Parallel Lines 3.2 Coordinates 3.3 Parallel Lines and Vision 3.4 Drawing without Measurement 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 and Multiplication 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 π 4.9 Ghosts of Departed Quantities 5 Curved Space 5.1 Flat Space and Medieval Space 5.2 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 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≥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 Polytopes 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 8 Periodic Space 8.1 The Impossible Tribar 8.2 The Cylinder and the Plane 8.3 Where the Wild Things Are 8.4 Periodic Worlds 8.5 Periodicity and Topology 8.6 A Brief History of Periodicity 8.7 Non-Euclidean Periodicity 9 The Infinite 9.1 Finite and Infinite 9.2 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 "This book 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."-- Provided by publisher