Mathematical Reflections: In a Room with Many Mirrors

ISBN-13: 9781461273455 / Angielski / Miękka / 2012 / 352 str.

Peter Hilton; Derek Holton; Jean Pedersen

Opis

Focusing Your Attention The purpose of this book is Cat least) twofold. First, we want to show you what mathematics is, what it is about, and how it is done-by those who do it successfully. We are, in fact, trying to give effect to what we call, in Section 9.3, our basic principle of mathematical instruction, asserting that "mathematics must be taught so that students comprehend how and why mathematics is qone by those who do it successfully./I However, our second purpose is quite as important. We want to attract you-and, through you, future readers-to mathematics. There is general agreement in the (so-called) civilized world that mathematics is important, but only a very small minority of those who make contact with mathematics in their early education would describe it as delightful. We want to correct the false impression of mathematics as a combination of skill and drudgery, and to re- inforce for our readers a picture of mathematics as an exciting, stimulating and engrossing activity; as a world of accessible ideas rather than a world of incomprehensible techniques; as an area of continued interest and investigation and not a set of procedures set in stone.

Spis treści

1 Going Down the Drain.- 1.1 Constructions.- 1.2 Cobwebs.- 1.3 Consolidation.- 1.4 Fibonacci Strikes.- 1.5 Dénouement.- Final Break.- References.- Answers for Final Break.- 2 A Far Nicer Arithmetic.- 2.1 General Background: What You Already Know.- 2.2 Some Special Moduli: Getting Ready for the Fun.- 2.3 Arithmetic mod p: Some Beautiful Mathematics.- 2.4 Arithmetic mod Non-primes: The Same But Different.- 2.5 Primes, Codes, and Security.- 2.6 Casting Out 9’s and 11’s: Tricks of the Trade.- Final Break.- Answers for Final Break.- 3 Fibonacci and Lucas Numbers.- 3.1 A Number Trick.- 3.2 The Explanation Begins.- 3.3 Divisibility Properties.- 3.4 The Number Trick Finally Explained.- 3.5 More About Divisibility.- 3.6 A Little Geometry!.- Final Break.- References.- Answers for Final Break.- 4 Paper-Folding and Number Theory.- 4.1 Introduction: What You Can Do With—and Without—Euclidean Tools.- I Simple Paper-Folding.- 4.2 Going Beyond Euclid: Folding 2-Period Regular Polygons.- 4.3 Folding Numbers.- 4.4 Some Mathematical Tidbits.- II General Paper-Folding.- 4.5 General Folding Procedures.- 4.6 The Quasi-Order Theorem.- 4.7 Appendix: A Little Solid Geometry.- Final Break.- References.- 5 Quilts and Other Real-World Decorative Geometry.- 5.1 Quilts.- 5.2 Variations.- 5.3 Round and Round.- 5.4 Up the Wall.- Final Break.- References.- Answers for Final Break.- 6 Pascal, Euler, Triangles, Windmills.- 6.1 Introduction: A Chance to Experiment.- I Pascals Set the Scene.- 6.2 The Binomial Theorem.- 6.3 The Pascal Triangle and Windmill.- 6.4 The Pascal Flower and the Generalized Star of David.- II Euler Takes the Stage.- 6.5 Eulerian Numbers and Weighted Sums.- 6.6 Even Deeper Mysteries.- References.- 7 Hair and Beyond.- 7.1 A Problem with Pigeons, and Related Ideas.- 7.2 The Biggest Number.- 7.3 The Big Infinity.- 7.4 Other Sets of Cardinality ?0.- 7.5 Schröder and Bernstein.- 7.6 Cardinal Arithmetic.- 7.7 Even More Infinities?.- Final Break.- References.- Answers for Final Break.- 8 An Introduction to the Mathematics of Fractal Geometry.- 8.1 Introduction to the Introduction: What’s Different About Our Approach.- 8.2 Intuitive Notion of Self-Similarity.- 8.3 The lént Map and the Logistic Map.- 8.4 Some More Sophisticated Material.- Final Break.- References.- Answers for Final Break.- An Introduction to the Mathematics of Fractal Geometry.- 8.1 Introduction to the Introduction: What’s Different About Our Approach.- 8.2 Intuitive Notion of Self-Similarity.- 8.3 The tent Map and and the Logistic Map.- 8.4 Some more Sophisticated Material.- Final Break.- References.- Answer for Final Break.- 9 Some of Our Own Reflections.- 9.1 General Principles.- 9.2 Specific Principles.- 9.3 Appendix: Principles of Mathematical Pedagogy.- References.

O autorze

The purpose of this book is to show what mathematics is about, how it is done, and what it is good for. The eight topics covered illustrate the unity of mathematical thought as well as the diversity of mathematical ideas; the relaxed and informal discussion makes it clear that mathematics can be exciting and engrossing. The book, which includes recent discoveries that lead to open questions, will give pleasure to anyone interested in the intellectually intriguing aspects of mathematics.