Mathematics: The Man-Made Universe(Dover Books on Mathematics)

Anyone can appreciate the beauty, depth, and vitality of mathematics with the help of this highly readable text, specially developed from a college course designed to appeal to students in a variety of fields. Readers with little mathematical background are exposed to a broad range of subjects chosen from number theory, topology, set theory, geometry, algebra, and analysis. Starting with a survey of questions on weight, the text discusses the primes, the fundamental theorem of arithmetic, rationals and irrationals, tiling, tiling and electricity, probability, infinite sets, and many other topics. Each subject illustrates a significant idea and lends itself easily to experiments and problems. Useful appendices offer an overview of the basic ideas of arithmetic, the rudiments of algebra, suggestions on teaching mathematics, and much more, including answers and comments for selected exercises.

Table of Contents:
Map; Guide; Preface 1. Questions on weighing Weighing with a two-pan balance and two measures-Problems raised-Their algebraic phrasing 2. The primes The Greek prime-manufacturing machine-Gaps between primes-Average gap and 1/1 + 1/2 + 1/3 + ... + 1/N-Twin primes 3. The Fundamental Theorem of Arithmetic Special natural numbers-Every special number is prime-"Unique factorization" and "every prime is special" compared-Euclidean algorithm-Every prime number is special-The concealed theorem 4. Rationals and Irrationals The Pythagorean Theorem--he square root of 2-Natural numbers whose square root is irrational-Rational numbers and repeating decimals 5. Tiling The rationals and tiling a rectangle with equal squares-Tiles of various shapes-use of algebra-Filling a box with cubes 6. Tiling and electricity Current-The role of the rationals-Applications to tiling-Isomorphic structures 7. The highway inspector and the salesman A problem in topology-Routes passing once over each section of highway-Routes passing once through each town 8. Memory Wheels A problem raised by an ancient word-Overlapping n-tuplets-Solution-History and applications 9. The Representation of numbers Representing natural numbers-The decimal system (base ten)-Base two-Base three-Representing numbers between 0 and 1-Arithmetic in base three-The Egyptian system-The decimal system and the metric system 10. Congruence Two integers congruent modulo a natural number-Relation to earlier chapters-Congruence and remainders-Properties of congruence-Casting out nines-Theorems for later use 11. Strange algebras Miniature algebras-Tables satisfying rules-Commutative and idempotent tables-Associativity and parentheses-Groups 12. Orthogonal tables Problem of the 36 officers-Some experiments-A conjecture generalized-Its fate-Tournaments-Application to magic squares 13. Chance Probability-Dice-The multiplication rule-The addition rule-The subtraction rule-Roulette-Expectation-Odds-Baseball-Risk in making decisions 14. The fifteen puzzle The fifteen puzzle-A problem in switching cords-Even and odd arrangements-Explanation of the Fifteen puzzle-Clockwise and counterclockwise 15. Map coloring The two-color theorem-Two three-color theorems-The five-color theorem-The four-color conjecture 16. Types of numbers Equations-Roots-Arithmetic of polynomials-Algebraic and transcendental numbers-Root r and factor X-r-Complex numbers-Complex numbers applied to alternating current-The limits of number systems 17. Construction by straightedge and compass Bisection of line segment-Bisection of angle-Trisection of line segment-Trisection of 90 angle-Construction of regular pentagon-Impossibility of constructing regular 9-gon and trisecting 60 angle 18. Infinite sets A conversation from the year 1638-Sets and one-to-one correspondence-Contrast of the finite with the infinite-Three letters of Cantor-Cantor's Theorem-Existence of transcendentals 19. A general view The branches of mathematics-Topology and set theory as geometries-The four "shadow" geometries-Combinatorics-Algebra-Analysis-Probability-Types of proof-Cohen's theorem-Truth and proof-Godel's theorem Appendix A. Review of arithmetic A quick tour of the basic ideas of arithmetic Appendix B. Writing mathematics Some words of advice and caution Appendix C. The rudiments of algebra A review of algebra, which is reduced to eleven rules Appendix D. Teaching mathematics Suggestions to prospective and practicing teachers Appendix E. The geometric and harmonic series Their properties-Applications of geometric series to probability Appendix F. Space of any dimension Definition of space of any dimension Appendix G. Update Answers and comments for selected exercises Index

About the Author :
An emeritus Professor of Mathematics at the University of California, Davis, Sherman K. Stein is the author of a longtime bestselling calculus book for McGraw-Hill.