Mathematics: The Man-Made Universe / Edition 3

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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. 
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Product Details

  • ISBN-13: 9780486404509
  • Publisher: Dover Publications
  • Publication date: 11/18/2010
  • Edition description: Unabridged
  • Edition number: 3
  • Pages: 592
  • Product dimensions: 6.47 (w) x 9.13 (h) x 1.10 (d)

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—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
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  • Anonymous

    Posted June 21, 2005

    great book for homeschoolers and the self-taught

    This is a gentle introduction to math that's worth learning (as opposed to high school math). You won't get this in school, perhaps because your teachers don't know it. What makes this book stand out is that it helps you to see what a mathematican actually does; it helps you understand the process of studying examples and generalizing (again, unlike high school math).

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  • Anonymous

    Posted July 4, 2000

    Amazing book for the young and hungry.

    This book presents some very fundamental yet deep mathematical concepts in a friendly way. It strongly presents both practical and theoretical aspects of mathematics. I first read this book in my early teens and I fell in love with it. I read it twice more over the next decade. I would strongly recommend this book to anyone who has a hunger for mathematics.

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