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—Gödel'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
