Synopses & Reviews
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.
Synopsis
Developed from a course for students in a variety of fields, this highly readable volume covers a vast array of subjects, including number theory, topology, set theory, geometry, algebra, and analysis. Starting with questions on weighing, the primes, the fundamental theory of arithmetic, and rationals and irrationals, the text also surveys the representation of numbers, congruence, probability, much more. Several useful appendices, plus answers, comments for selected exercises. 1994 edition. Solution guide available upon request.
Synopsis
Highly readable volume covers a vast array of subjects--number theory, topology, set theory, geometry, algebra, and analysis, plus the primes, fundamental theory of arithmetic, rationals and irrationals, representation of numbers, congruence, probability, more. Appendices; answers for selected exercises. 1994 edition.
Synopsis
This highly readable volume covers a vast array of subjects, including number theory, topology, set theory, geometry, algebra, and analysis. Additional topics include primes, fundamental theory of arithmetic, rationals and irrationals, representation of numbers, congruence, probability, and more. A solutions manual is available upon request. 1994 edition.
Synopsis
Highly readable volume covers number theory, topology, set theory, geometry, algebra, and analysis, plus the primes, fundamental theory of arithmetic, probability, and more. Solutions manual available upon request. 1994 edition.
Table of Contents
Map; Guide; Preface
1. Questions on weighing
Weighing with a two-pan balance and two measuresProblems raisedTheir algebraic phrasing
2. The primes
The Greek prime-manufacturing machineGaps between primesAverage gap and 1/1 + 1/2 + 1/3 + . . . + 1/NTwin primes
3. The Fundamental Theorem of Arithmetic
Special natural numbersEvery special number is prime"Unique factorization" and "every prime is special" comparedEuclidean algorithmEvery prime number is specialThe concealed theorem
4. Rationals and Irrationals
The Pythagorean Theorem-he square root of 2Natural numbers whose square root is irrationalRational numbers and repeating decimals
5. Tiling
The rationals and tiling a rectangle with equal squaresTiles of various shapesuse of algebraFilling a box with cubes
6. Tiling and electricity
CurrentThe role of the rationalsApplications to tilingIsomorphic structures
7. The highway inspector and the salesman
A problem in topologyRoutes passing once over each section of highwayRoutes passing once through each town
8. Memory Wheels
A problem raised by an ancient wordOverlapping n-tupletsSolutionHistory and applications
9. The Representation of numbers
Representing natural numbersThe decimal system (base ten)Base twoBase threeRepresenting numbers between 0 and 1Arithmetic in base threeThe Egyptian systemThe decimal system and the metric system
10. Congruence
Two integers congruent modulo a natural numberRelation to earlier chaptersCongruence and remaindersProperties of congruenceCasting out ninesTheorems for later use
11. Strange algebras
Miniature algebrasTables satisfying rulesCommutative and idempotent tablesAssociativity and parenthesesGroups
12. Orthogonal tables
Problem of the 36 officersSome experimentsA conjecture generalizedIts fateTournamentsApplication to magic squares
13. Chance
ProbabilityDiceThe multiplication ruleThe addition ruleThe subtraction ruleRouletteExpectationOddsBaseballRisk in making decisions
14. The fifteen puzzle
The fifteen puzzleA problem in switching cordsEven and odd arrangementsExplanation of the Fifteen puzzleClockwise and counterclockwise
15. Map coloring
The two-color theoremTwo three-color theoremsThe five-color theoremThe four-color conjecture
16. Types of numbers
EquationsRootsArithmetic of polynomialsAlgebraic and transcendental numbersRoot r and factor XrComplex numbersComplex numbers applied to alternating currentThe limits of number systems
17. Construction by straightedge and compass
Bisection of line segment-Bisection of angle-Trisection of line segmentTrisection of 90° angleConstruction of regular pentagonImpossibility of constructing regular 9-gon and trisecting 60° angle
18. Infinite sets
A conversation from the year 1638Sets and one-to-one correspondenceContrast of the finite with the infiniteThree letters of CantorCantor's TheoremExistence of transcendentals
19. A general view
The branches of mathematicsTopology and set theory as geometriesThe four "shadow" geometriesCombinatoricsAlgebraAnalysisProbabilityTypes of proofCohen's theoremTruth and proofGödel'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 propertiesApplications 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