Stillwell J. Yearning for the impossible: the surprising truth of mathematics

AK Peters, Ltd., 2006. — xiv, 230 p. — ISBN 1-56881-254-X, 978-1-56881-254-0.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: - Irrational and Imaginary Numbers - The Fourth Dimension - Curved Space - Infinity and others The author puts these creations into a broader context involving related "impossibilities" from art, literature, philosophy, and physics. By imbedding mathematics into a broader cultural context and through his clever and enthusiastic explication of mathematical ideas the author broadens the horizon of students beyond the narrow confines of rote memorization and engages those who are curious about the place of mathematics in our intellectual landscape.The irrational
The Pythagorean Dream
The Pythagorean Theorem
Irrational Triangles
The Pythagorean Nightmare
Explaining the Irrational
The Continued Fraction for √2
Equal TemperamentThe imaginary
Negative Numbers
Imaginary Numbers
Solving Cubic Equations
Real Solutions via Imaginary Numbers
Where Were Imaginary Numbers before 1572?
Geometry of Multiplication
Complex Numbers Give More than We Aked For
Why Call Them "Complex" Numbers?The Horizon
Parallel Lines
Coordinates
Parallel Lines and Vision
Drawing without Measurement
The Theorems of Pappus and Desargues
The Little Desargues Theorem
What Are the Laws of Algebra?
Projective Addition and MultiplicationThe Infinitesimal
Length and Area
Volume
Volume of a Tetrahedron
The Circle
The Parabola
The Slopes of Other Curves
Slope and Area
The Value of π
Ghosts of Departed Quantities.Curved Space
Flat Space and Medieval Space
The 2-Sphere and the 3-Sphere
Flat Surfaces and the Parallel Axiom
The Sphere and the Parallel Axiom
Non-Euclidean Geometry
Negative Curvature
The Hyperbolic Plane
Hyperbolic Space
Mathematical Space and Actual SpaceThe Fourth Dimension
Arithmetic of Pairs
Searching for an Arithmetic of Triples
Why n-tuples Are Unlike Numbers when n > 3
Quaternions
The Four-Square Theorem
Quaternions and Space Rotations
Symmetry in Three Dimensions
Tetrahedral Symmetry and the 24-Cell
The Regular PolytopesThe Ideal
Discovery and Invention
Division with Remainder
Unique Prime Factorization
Gaussian Integers
Gaussian Primes
Rational Slopes and Rational Angles
Unique Prime Factorization Lost
Ideals, or Unique Prime Factorization Regained.Periodic Space
The Impossible Tribar
The Cylinder and the Plane
Where the Wild Things Are
Periodic Worlds
Periodicity and Topology
A Brief History of PeriodicityThe Infinite
Finite and Infinite
Potential and Actual Infinity
The Uncountable
The Diagonal Argument
The Transcendental
Yearning for CompletenessEpilogueФайл: отскан. стр. (b/w 300 dpi) + OCR + закладки