A Mathematical Orchard: Problems and Solutions

(MAA Problem Book Series)

Mark I. Krusemeyer, George T. Gilbert e Loren C. Larson

The Mathematical Association of America | 2012 | 410 páginas | rar - pdf | 1,9 Mb

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This volume is a republication and expansion of the much-loved Wohascum County Problem Book, published in 1993. The original 130 problems have been retained and supplemented by an additional 78 problems. The puzzles contained within, which are accessible but never routine, have been specially selected for their mathematical appeal, and detailed solutions are provided. The reader will encounter puzzles involving calculus, algebra, discrete mathematics, geometry and number theory, and the volume includes an appendix identifying the prerequisite knowledge for each problem. A second appendix organises the problems by subject matter so that readers can focus their attention on particular types of problems if they wish. This collection will provide enjoyment for seasoned problem solvers and for those who wish to hone their skills.

• An entertaining collection of problems, tried and tested by experienced educators
• Detailed solutions are included and some problems are solved in multiple ways
• Accessible to those of advanced secondary school/high school level, through to undergraduate and above

Table of Contents
Preface
1. The problems
2. The solutions
Appendix 1. Prerequisites by problem number
Appendix 2. Problem numbers by subject
Index.

Journey through Mathematics Creative Episodes in Its History

Enrique A. González-Velasco

Springer | 2011| 478 páginas | pdf | 4,4 Mb

link direto
link

This book offers an accessible and in-depth look at some of the most  important episodes of two thousand years of mathematical history. Beginning with trigonometry and moving on through logarithms, complex numbers, infinite series, and calculus, this book profiles some of the lesser known but crucial contributors to modern day mathematics. It is unique in its use of primary sources as well as its accessibility; a knowledge of first-year calculus is the only prerequisite. But undergraduate and graduate students alike will appreciate this glimpse into the fascinating process of mathematical creation.

The history of math is an intercontinental journey, and this book showcases brilliant mathematicians from Greece, Egypt, and India, as well as Europe and the Islamic world. Several of the primary sources have never before been translated into English. Their interpretation is thorough and readable, and offers an excellent background for teachers of high school mathematics as well as anyone interested in the history of math.

TABLE OF CONTENTS
Preface ix
1 TRIGONOMETRY 1
1.1 The Hellenic Period 1
1.2 Ptolemy’s Table of Chords 10
1.3 The Indian Contribution 25
1.4 Trigonometry in the Islamic World 34
1.5 Trigonometry in Europe 55
1.6 From Viète to Pitiscus 65
2 LOGARITHMS 78
2.1 Napier’s First Three Tables 78
2.2 Napier’s Logarithms 88
2.3 Briggs’ Logarithms 101
2.4 Hyperbolic Logarithms 117
2.5 Newton’s Binomial Series 122
2.6 The Logarithm According to Euler 136
3 COMPLEX NUMBERS 148
3.1 The Depressed Cubic 148
3.2 Cardano’s Contribution 150
3.3 The Birth of Complex Numbers 160
3.4 Higher-Order Roots of Complex Numbers 173
3.5 The Logarithms of Complex Numbers 181
3.6 Caspar Wessel’s Breakthrough 185
3.7 Gauss and Hamilton Have the Final Word 190
4 INFINITE SERIES 195
4.1 The Origins 195
4.2 The Summation of Series 203
4.3 The Expansion of Functions 212
4.4 The Taylor and Maclaurin Series 220
5 THE CALCULUS 230
5.1 The Origins 230
5.2 Fermat’s Method of Maxima and Minima 234
5.3 Fermat’s Treatise on Quadratures 248
5.4 Gregory’s Contributions 258
5.5 Barrow’s Geometric Calculus 275
5.6 From Tangents to Quadratures 283
5.7 Newton’s Method of Infinit Series 289
5.8 Newton’s Method of Fluxions 294
5.9 Was Newton’s Tangent Method Original? 302
5.10 Newton’s First and Last Ratios 306
5.11 Newton’s Last Version of the Calculus 312
5.12 Leibniz’ Calculus: 1673–1675 318
5.13 Leibniz’ Calculus: 1676–1680 329
5.14 The Arithmetical Quadrature 340
5.15 Leibniz’ Publications 349
5.16 The Aftermath 358
6 CONVERGENCE 368
6.1 To the Limit 368
6.2 The Vibrating String Makes Waves 369
6.3 Fourier Puts on the Heat 373
6.4 The Convergence of Series 380
6.5 The Difference Quotient 394
6.6 The Derivative 401
6.7 Cauchy’s Integral Calculus 405
6.8 Uniform Convergence 407
BIBLIOGRAPHY 412

The Mathematics of Games and Gambling

Edward Packel

The Mathematical Association of America | 2006 - 2.ª edição| 190 páginas | rar - pdf | 880 kb

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password: matav


The first edition of this book was reprinted eight times! This book introduces and develops some of the important and beautiful elementary mathematics needed for rational analysis of various gambling and game activities. Most of the standard casino games (roulette, , blackjack, keno), some social games (backgammon, poker, bridge) and various other activities (state lotteries, horse racing, etc.) are treated in ways that bring out their mathematical aspects. The mathematics developed ranges from the predictable concepts of probability, expectation, and binomial coefficients to some less well-known ideas of elementary game theory. The Second Edition includes new material on: sports betting and the mathematics behind it; Game theory applied to bluffing in poker and related to the “Texas Holdem phenomenon”: The Nash equilibrium concept and its emergence in the popular culture: Internet links to games and to Java applets for practice and classroom use. The only formal mathematics background the reader needs is some facility with high school algebra. Game-related exercises are included at the end of most chapters for readers interested in working with and expanding ideas treated in the text. Solutions to some of the exercises appear at the end of the book.

Contents
Preface to the First Edition ix
Preface to the Second Edition xiii
1 The Phenomenon of Gambling 1
1.1 A selective history . . . 1
1.2 The gambler in fact and fiction . . . 5
2 Finite Probabilities and Great Expectations 13
2.1 The probability concept and its origins . . 13
2.2 Dice, cards, and probabilities  . . 15
2.3 Roulette, probability and odds. . 17
2.4 Compound probabilities: The rules of the game  . . 20
2.5 Mathematical expectation and its application .. . 22
2.6 Exercises  . . 26
3 Backgammon and Other Dice Diversions 29
3.1 Backgammon oversimplified  . . 29
3.2 Rolling spots and hitting blots  . . 32
3.3 Enteringand bearingoff .  . 34
3.4 The doubling cube  . . 36
3.5 Craps  . . 40
3.6 Chuck-a-Luck . . . 45
3.7 Exercises . . 47
4 Permutations, Combinations, and Applications 51
4.1 Careful counting: Is order important? . . 51
4.2 Factorials and other notation  . . 53
4.3 Probabilities in poker  . . 55
4.4 Betting in poker: A simple model  . . 59
4.5 Distributions in bridge  . . 67
4.6 Keno type games . . 71
4.7 Exercises . . 73
5 Play it Again Sam: The Binomial Distribution 79
5.1 Games and repeatedtrials . . 79
5.2 The binomial distribution . . 79
5.3 Beating the odds and the “law” of averages  . . 83
5.4 Betting systems  . . 90
5.5 A brief blackjack breakthrough. . 93
5.6 Exercises  . . 95
6 Elementary Game Theory 99
6.1 What isgame theory?  . . 99
6.2 Games in extensive form . . 100
6.3 Two-persongames in normal form  . . 105
6.4 Zero-sumgames . . 107
6.5 Nonzero-sum games, Nash equilibria and the prisoners’ dilemma  . 113
6.6 Simple n-persongames . . 118
6.7 Power indices. . 120
6.8 Games computers play  . . . 123
6.9 Exercises . . . 129
7 Odds and Ends 135
7.1 The mathematics of bluffing and the Texas Holdem invasion . . 135
7.2 Off to the races  . . 141
7.3 Lotteries and your expectation . . . 147
7.4 The gambler’s ruin  . 158

The Logic of Chance: An Essay on the Foundations and Province of the Theory of Probability

John Venn

Macmillan And Company | 1888

online:
archive.org
books.google

CHELSEA PUBLISHING | 1962 - 4ª edição 

online:
catalog.hathitrust.org

No mathematical background is necessary to appreciate this classic of probability theory, which remains unsurpassed in its clarity, readability, and sheer charm. Its author, British logician John Venn (1834-1923), popularized the famous Venn Diagrams that are commonly used for teaching elementary mathematics. In The Logic of Chance, he employs the same directness that makes his diagrams so effective.
The three-part treatment commences with an overview of the physical foundations of the science of probability, including surveys of the arrangement and formation of the series of probability; the origin or process of causation of the series; how to discover and prove the series; and the conception of randomness. The second part examines the logical superstructure on the basis of physical foundations, encompassing the measurement of belief; the rules of inference in probability; the rule of succession; induction; chance, causation, and design; material and formal logic; modality; and fallacies. The final section explores various applications of the theory of probability, including such intriguing aspects as insurance and gambling, the credibility of extraordinary stories, and approximating the truth by means of the theory of averages.