A Smoother Pebble Mathematical Explorations

by Benson, Donald C.

• ISBN13: 9780195144369
• ISBN10: 0195144368
• Format: Hardcover
• Copyright: 2003-10-30
• Publisher: Oxford University Press

Summary

This book takes a novel look at the topics of school mathematics--arithmetic, geometry, algebra, and calculus. In this stroll on the mathematical seashore we hope to find, quoting Newton, "...a smoother pebble or a prettier shell than ordinary..." This book assembles a collection ofmathematical pebbles that are important as well as beautiful.

Author Biography

Donald C. Benson is Emeritus Professor of Mathematics at the University of California, Davis.

Table of Contents

Introduction	p. 1
Bridging the Gap
Ancient Fractions	p. 5
The Egyptian Unit Fractions	p. 6
Egyptian arithmetic	p. 8
The greedy algorithm	p. 13
The Babylonians and the Sexagesimal System	p. 15
Sexagesimal fractions	p. 16
Greek Gifts	p. 18
The Heresy	p. 20
Magnitudes, Ratio, and Proportion	p. 22
Proportion according to Eudoxus	p. 24
Attributed to Theaetetus	p. 26
The Music of the Ratios	p. 33
Acoustics	p. 35
The rotating circle	p. 36
Waveforms and spectra	p. 39
Psychoacoustics	p. 44
Consonance versus dissonance	p. 45
Critical bandwidth	p. 46
Intervals, Scales, and Tuning	p. 49
Pythagorean tuning	p. 49
Approximating m octaves with n fifths	p. 51
Equal-tempered tuning	p. 54
The Shape of Things
Tubeland	p. 61
Curvature of Smooth Curves	p. 62
The inner world of the curve-bound inchworm	p. 62
Curves embedded in two dimensions	p. 63
Curves embedded in three dimensions	p. 64
Curvature of Smooth Surfaces	p. 65
Gaussian curvature--Extrinsic definition	p. 66
Tubeland--A fantasy	p. 68
Triangular excess	p. 71
Euclidean Geometry	p. 73
The parallels axiom	p. 75
Non-Euclidean Geometry	p. 75
Models of non-Euclidean geometries	p. 77
The Calculating Eye	p. 82
Graphs	p. 84
The need for graphs	p. 85
"Materials" for graphs	p. 86
Clever people invented graphs	p. 89
Coordinate Geometry	p. 93
Synthetic versus analytic	p. 94
Synthetic and analytic proofs	p. 95
Straight lines	p. 99
Conic sections	p. 101
The Great Art
Algebra Rules	p. 111
Algebra Anxiety	p. 112
Arithmetic by Other Means	p. 115
Symbolic algebra	p. 116
Algebra and Geometry	p. 120
Al-jabr	p. 121
Square root algorithms	p. 122
The Root of the Problem	p. 128
Graphical Solutions	p. 129
Quadratic Equations	p. 130
Secrecy, Jealousy, Rivalry, Pugnacity, and Guile	p. 135
Solving a cubic equation	p. 138
Symmetry Without Fear	p. 142
Symmetries of a Square	p. 145
The Group Axioms	p. 148
Isometries of the Plane	p. 150
Patterns for Plane Ornaments	p. 151
Catalog of border and wallpaper patterns	p. 151
Wallpaper watching	p. 158
The Magic Mirror	p. 160
Undecidability	p. 160
The Magic Writing	p. 162
A Smoother Pebble
On the Shoulders of Giants	p. 167
Integration Before Newton and Leibniz	p. 168
Archimedes' method for estimating pi	p. 168
Circular reasoning	p. 170
Completing the estimate of pi	p. 171
Differentiation Before Newton and Leibniz	p. 172
Descartes's discriminant method	p. 173
Fermat's difference quotient method	p. 176
Galileo's Lute	p. 177
Falling bodies	p. 177
The inclined plane	p. 179
Six-Minute Calculus	p. 184
Preliminaries	p. 185
Functions	p. 186
Limits	p. 188
Continuity	p. 189
The Damaged Dashboard	p. 191
The broken speedometer	p. 193
The derivative	p. 194
The broken odometer	p. 199
The definite integral	p. 201
Roller Coasters	p. 206
The length of a curve	p. 206
Time of descent	p. 208
Roller-Coaster Science	p. 212
The Simplest Extremum Problems	p. 213
The rectangle of maximum area with fixed perimeter	p. 213
The lifeguard's calculation	p. 215
A faster track	p. 217
A road-building project for three towns	p. 219
Inequalities	p. 220
The inequality of the arithmetic and geometric means	p. 221
Cauchy's inequality	p. 223
The Brachistochrone	p. 223
The geometry of the cycloid	p. 227
A differential equation	p. 228
The restricted brachistochrone	p. 230
The unrestricted brachistochrone	p. 235
Glossary	p. 243
Notes	p. 249
References	p. 257
Index	p. 261
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