Mathematical Reflections

by Hilton, Peter John; Holton, Derek Allan; Pedersen, Jean

• ISBN13: 9780387947709
• ISBN10: 0387947701
• Format: Hardcover
• Copyright: 1996-12-01
• Publisher: Springer Verlag

Summary

The purpose of this book is to show what mathematics is about, how it is done, and what it is good for. The relaxed and informal presentation conveys the joy of mathematical discovery and insight and makes it clear that mathematics can be an exciting and engrossing activity. Frequent questions lead the reader to see mathematics as an accessible world of thought, where understanding can turn opaque formulae into beautiful and meaningful ideas. The text presents eight topics that serve to illustrate the unity of mathematical thought as well as the diversity of mathematical ideas. Drawn from both "pure" and "applied" mathematics, they include: spirals in nature and in mathematics; the modern topic of fractals and the ancient topic of Fibonacci numbers; Pascal's Triangle and paper folding -- two topics where geometry, number theory, and algebra meet and interact; modular arithmetic and the arithmetic of the infinite. The final chapter presents some ideas about how mathematics should be done, and hence, how it should be taught; these ideas are referred to throughout the text, whenever mathematical strategy and technique are at issue. Presenting many recent discoveries that lead to interesting open questions, the book can serve as the main text in courses dealing with contemporary mathematical topics (for mathematics students or for prospective or in-service mathematics teachers) or as enrichment for other courses. It can also be read with pleasure on its own by anyone interested in the intellectually intriguing aspects of mathematics.

Table of Contents

Preface: Focusing Your Attention	vii
Going Down the Drain	1	(24)
Constructions	1	(8)
Cobwebs	9	(6)
Consolidation	15	(5)
Fibonacci Strikes	20	(2)
Denouement	22	(3)
Final Break	23	(1)
References	23	(1)
Answers for Final Break	24	(1)
A Far Nicer Arithmetic	25	(36)
General Background: What You Already Know	25	(8)
Some Special Moduli: Getting Ready for the Fun	33	(4)
Arithmetic mod p: Some Beautiful Mathematics	37	(7)
Arithmetic mod Non-primers: The Same But Different	44	(5)
Primes, Codes, and Security	49	(5)
Casting Out 9's and 11's: Tricks of the Trade	54	(7)
Final Break	58	(1)
Answers for Final Break	59	(2)
Fibonacci and Lucas Numbers	61	(26)
A Number Trick	61	(2)
The Explanation Begins	63	(9)
Divisibility Properties	72	(4)
The Number Trick Finally Explained	76	(2)
More About Divisibility	78	(3)
A Little Geometry!	81	(6)
Final Break	85	(1)
References	85	(1)
Answers for Final Break	86	(1)
Paper-Folding and Number Theory	87	(56)
Introduction: What You Can Do With--and Without--Euclidean Tools	87	(4)
Simple Paper-Folding
Going Beyond Euclid: Folding 2-Period Regular Polygons	91	(12)
Folding Numbers	103	(14)
Some Mathematical Tidbits	117	(6)
General Paper-Folding
General Folding Procedures	123	(5)
The Quasi-Order Theorem	128	(10)
Appendix: A Little Solid Geometry	138	(5)
Final Break	141	(1)
References	141	(2)
Quilts and Other Real-World Decorative Geometry	143	(42)
Quilts	143	(10)
Variations	153	(9)
Round and Round	162	(4)
Up the Wall	166	(19)
Final Break	178	(4)
References	182	(2)
Answers for Final Break	184	(1)
Pascal, Euler, Triangles, Windmills,...	185	(64)
Introduction: A Chance to Experiment	185	(2)
Pascal Sets the Scene
The Binomial Theorem	187	(9)
The Pascal Triangle and Windmill	196	(15)
The Pascal Flower and the Generalized Star of David	211	(6)
Euler Takes the Stage
Eulerian Numbers and Weighted Sums	217	(21)
Even Deeper Mysteries	238	(11)
References	247	(2)
Hair and Beyond	249	(28)
A Problem with Pigeons, and Related Ideas	249	(4)
The Biggest Number	253	(2)
The Big Infinity	255	(6)
Other Sets of Cardinality α0	261	(7)
Schroder and Bernstein	268	(1)
Cardinal Arithmetic	269	(1)
Even More Infinities?	270	(7)
Final Break	273	(1)
References	274	(1)
Answers for Final Break	275	(2)
An Introduction to the Mathematics of Fractal Geometry	277	(46)
Introduction to the Introduction: What's Different About Our Approach	277	(3)
Intuitive Notion of Self-Similarity	280	(10)
The Tent Map and the Logistic Map	290	(8)
Some More Sophisticated Material	298	(25)
Final Break	314	(3)
References	317	(2)
Answers for Final Break	319	(4)
Some of Our Own Reflections	323	(16)
General Principles	324	(5)
Specific Principles	329	(7)
Appendix: Principles of Mathematical Pedagogy	336	(3)
References	338	(1)
Index	339