Continued fractions C. D Olds美国新数学丛书《连分数》的英文原版Illu

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详细说明：Continued fractions C. D Olds美国新数学丛书《连分数》的英文原版Illustrations by Carl Bass Third Printing C Copyright, 1963, by Yale University All rights reserved under International and Pan-American Copyright Conventions. Published in New York by Random House, Inc, and simultaneously in Toronto, Canada, by Random House of Canada, Limited School Edition Published by The L. W. Singer Company Library of Congress Catalog Card Number: 61-12185 Manufactured in the united States of America Note to the reader is book is one of a series written by professional mathematicians in order to make some important mathematical ideas interesting and understanda ble to a large audience of high school students and laymen. Most of the volumes in the New Mathematical library cover topics not usually included in the high school curriculum; they vary in difficulty, and, even within a single book, some parts require a greater degree of concentration than others. Thus, while the reader needs little technical knowledge to understand most of these books he will have to make an intellectual effort, If the reader has so far encountered mathematics only in classroom work, he should keep in mind that a book on mathematics cannot be read quickly Nor must he expect to understand all parts of the book on first reading He should feel free to skip complicated parts and return to them later; often an argument will be clarified by a subse- quent remark. On the other hand, sections containing thoroughly familiar material may be read very quickly The best way to learn mathematics is to do mathematics, and each book includes problems, some of which may require considerable thought. The reader is urged to acquire the habit of reading with paper and pencil in hand; in this way mathema tics will become in- creasingly meaningful to him For the authors and editors this is a new venture They wish to acknowledge the generous help given them by the many high school teachers and students who assisted in the preparation of these mono graphs. The editors are interested in reactions to the books in this series and hope that readers will write to: Editorial Committee of the NML series. in care of THE INSTITUTE OF MATHEMATICAL SCIENCES NEW YORK UNIVERSITY. New york 3N.y The editors NEW MATHEMATICAL LIBRARY Ocher titles will be announced when ready 1. NUMBERS: RATIONAL AND IRRATIONAL by Ivan Niven 2. WHAT IS CALCULUS ABOUT? by W. w. Sawyer 3. INTRODUCTION TO INEQUALITIES by E. Beckenbach and R. Bellman 4. GEOMETRIC INEQUALITIES by N. D Kazarinoff 5. THE CONTEST PROBLEm BOOK, Annual High School Contests of the Mathematical Association of America, 1950-1960 compiled and with solutions by Charles T salkind 6. THE LORE OF LARGE NUMBERS bY P.J. Davis 7. USES OF INFINITY by Leo Zippin 8. GEOMETRIC TRANSFORMATIONS by I M. Yagloin, trans- lated from the Russian by allen Shields 9. CONTINUED FRACTIONS by C. D Olds 10. GRAPHS AND THEIR USES by Oystein Ore 11. HUNGARIAN PROBLEM BOOK I. based on the eotvos Competitions, 1894-1905 12. HUNGariaN PROBLEM BOoK II. based on the Eotvos Competitions, 1906-1928 13 EPISODES FROM THE EARLY HISTORY OF MATHE MATICS by Asger Aaboe 14 GROUPS AND THEIR GRaPhs by I. Grossman and W. magnus 15. MATHEMATICS oF choice by Ivan Niven 16. FROM PYTHAGORAS TO eiNSTEin by K.O. Friedrichs 17. THE MAA PROBLEM BOOK II, Annual high School Contests of the mathematical Association of America, 1961-1965. com piled and with solutions by Charles T. salkind Contents Preface Chapter 1 Expansion of Rational Fractions 1.1 Introduction 1.2 Definitions and notation 35578 1. 3 Expansion of Rational fractions 1.4 Expansion of Rational Fractions 13 (General Discussion 1.5 Convergents and Their Properties 19 1.6 Iifferences of Convergents 27 1. 7 Some Historical Comments Chapter 2 Diophantine Equations 31 2.1 Introduction 31 2.2 The Method used Extensively by euler 32 2.3 The Indeterminate Equation ac-by =+1 36 2. 4 The General Solution of ax -bi y=c,(a,b)=142 2.5 The General Solution of a.+ by =c,(a, b)=144 26 The general solution of ax±By=±C 46 2.7 Sailors, Coconuts, and monkeys 48 Chapter 3 Expansion of Irrational Numbers 51 3.1 Introduction 51 3.2 Preliminary Examples 52 3.3 Convergents 58 3.4 Additional Theorems on Convergents 61 3.5 Some Notions of a limit 63 3.6 Infinite Continued fractions 66 33.7 Approximation Theorems 70 3.8 Geometrical Interpretation of Continued fractions 77 3.9 Solution of the Equation x 2= ax 1 80 3.10 Fibonacci Numbers 81 3.11 A Method for Calculating Logarithms 84 CONTENTS Chapter 4 Periodic Continued Fractions 4.1 Introduction 88 4.2 Purely Periodic Continued fractions 4.3 Quadratic irrationals 96 4.4 Reduced Quadratic Irrationals 4.5 Converse of Theorem 4.1 104 ±6 Lagrange' s Theorem 110 4.7 The Continued Fraction for vN 112 4.8 Pells Equation, a2- Ny2=+ 113 4.9 How to Obtain Other Solutions of Pells Equation 118 Chapter 5 epilogue 123 5.1 Introduction 123 5.2 Statement of the problem 123 5.3 Hurwitz' Theorem 124 5.4 Conclusion 129 Appendix I. Proof That x2-3y2=-1 Has No Integral Solutions 131 Appendix II. Some Miscellaneous Expansions 134 Solutions to Problems 140 References 159 Index 16 Preface At first glance nothing seems simpler or less significant tha an riting a number for example g, in the form 9 l+=1+ s1+ 1 3+ 3 1+ It turns out. however, that fractions of this form, calledcontinued fractions,, provide much insight into many mathematical pro blems, particularly into the nature of numbers Continued fractions were studied by the great mathematicians of the seventeenth and eighteenth centuries and are a subject of active investigation today. Nearly all books on the theory of numbers include a chapter on continued fractions but these accounts are condensed and rather difficult for the beginner. The plan in this book is to present an casy- going discussion of simple continued fractions that can be under stood by anyone who has a minimum of mathematical training Mathematicians often think of their subject as a creative art rather than as a science and this attitude is refected in the pages that follow. Chapter 1 shows how continued fractions might be dis covered accidentally, and then, by means of examples, how rational fractions can be expanded into continued fractions. Gradually more general notation is introduced and preliminary theorems are stated and proved. In Chapter 2 these results are applied to the solution of inear Diophantine equations. This chapter should be easy to read; it is, if anything, more detailed than necessary PREFACE Chapter 3 deals with the expansion of irrational numbers into infinite continued fractions, and includes an introductory discussion of the idea of limits here one sees how continued fractions can be used to give better and better rational approximations to irrational numbers. These and later results are closely connected with and supplement similar ideas developed in Niven' s book, Numbers Rational and Irrational The periodic properties of continued fractions are discussed in Chapter 4. The reader will find this chapter more challenging than che others, but the end results are rewarding. The main part of the chapter develops a proof of lagranges theorem that the continued fraction expansion of every quadratic irrational is periodic after a certain stage; this fact is then used as the key to the solution of Pells equation Chapter 5 is designed to give the reader a look into the future, and to suggest further study of the subject. Here the famous theorem of Hurwitz is discussed, and other theorems closely related to it are mentioned It goes without saying that one should not read"a mathematics book. It is better to get out pencil and paper and rewrite the book A student of ma thematics should wrestle with every step of a proof if he does not understand it in the first round, he should plan to return to it later and tackle it once again until it is mastered. In addition he should test his grasp of the subject by working the prob- lems at the end of the sections. These are mostly of an elementary nature, closely related to the text, and should not present any difficulties. Their answers appear at the end of the book The first of the two appendices gives a proof that x2-3y2 =-1 has no solution in integers, and Appendix ii is a collection of mis cellaneous expansions designed to show how the subject has devel- oped many of these expansions are difficult to obtain. Finally, there is a short list of references. In the text " Crystal [2,, for example refers to item 2 listed in the references I wish to express my thanks to the School Mathematics Study Group for including this book in the New Mathematical library series, and to the Editorial Panel for suggestions which have im proved the book. Particular thanks are due to Dr. Anneli Lax, not only for technical advice, so freely given, but also for her critical reading of the text. I am also grateful to my wife who typed the original manuscript, and to Mrs. Ruth murray, who prepared the final ty rescript. C. D. olds Los Altos California. 1961 CHAPTER ONE Expansion of Rational fractions 1.1 Introduction Imagine that an algebra student attempts to solve the quadratic equation (1.1) 3x-1=0 as follows: He first divides through by a and writes the equation in the form x=3 The unknown quantity a is still found on the right-hand side of this equation and hence can be replaced by its equal, namely 3+l /a This gives 3 3 Repeating this replacement of x by 3+1/ a several more times he obtains the expression (1.2) 3十 3十 3十 3 3+

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