Mathematics: The Man-Made Universe (eBook)

Map; Guide; Preface1. Questions on weighing Weighing with a two-pan balance and two measures—Problems raised—Their algebraic phrasing2. The primes The Greek prime-manufacturing machine—Gaps between primes—Average gap and 1/1 + 1/2 + 1/3 + . . . + 1/N—Twin primes3. 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 theorem4. Rationals and Irrationals The Pythagorean Theorem-—he square root of 2—Natural numbers whose square root is irrational—Rational numbers and repeating decimals5. Tiling The rationals and tiling a rectangle with equal squares—Tiles of various shapes—use of algebra—Filling a box with cubes6. Tiling and electricity Current—The role of the rationals—Applications to tiling—Isomorphic structures7. The highway inspector and the salesman A problem in topology—Routes passing once over each section of highway—Routes passing once through each town8. Memory Wheels A problem raised by an ancient word—Overlapping n-tuplets—Solution—History and applications9. 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 system10. Congruence Two integers congruent modulo a natural number—Relation to earlier chapters—Congruence and remainders—Properties of congruence—Casting out nines—Theorems for later use11. Strange algebras Miniature algebras—Tables satisfying rules—Commutative and idempotent tables—Associativity and parentheses—Groups12. Orthogonal tables Problem of the 36 officers—Some experiments—A conjecture generalized—Its fate—Tournaments—Application to magic squares13. Chance Probability—Dice—The multiplication rule—The addition rule—The subtraction rule—Roulette—Expectation—Odds—Baseball—Risk in making decisions14. The fifteen puzzle The fifteen puzzle—A problem in switching cords—Even and odd arrangements—Explanation of the Fifteen puzzle—Clockwise and counterclockwise15. Map coloring The two-color theorem—Two three-color theorems—The five-color theorem—The four-color conjecture16. 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 systems17. 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° angle18. 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 transcendentals19. 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—Gödel's theoremAppendix A. Review of arithmetic A quick tour of the basic ideas of arithmeticAppendix B. Writing mathematics Some words of advice and cautionAppendix C. The rudiments of algebra A review of algebra, which is reduced to eleven rulesAppendix D. Teaching mathematics Suggestions to prospective and practicing teachersAppendix E. The geometric and harmonic series Their properties—Applications of geometric series to probabilityAppendix F. Space of any dimension Definition of space of any dimensionAppendix G. Update Answers and comments for selected exercises Index