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Irrational Numbers AMS / MAA THE CARUS MATHEMATICAL MONOGRAPHS VOL 11 Irrational Numbers Ivan Niven 10.1090/car/011 CARUS MATHEMATICAL MONOGRAPHS 11 Irrational Numbers Ivan Niven Published and Distributed by The Mathematical Association of America Sixth Printing 2006 Copyright ©1956, 1985 by The Mathematical Association of America Paperbound edition issued March 2005 Library of Congress Number: 56009936 Paperbound ISBN 978-0-88385-038-1 eISBN 978-1-61444-011-6 Hardcover (out of print) ISBN 978-0-88385-011-4 Printed in the United States of America THE CARUS MATHEMATICAL MONOGRAPHS Published by THE MATHEMATICAL ASSOCIATION OF AMERICA Editorial Committee Tibor Rado, Chairman Samuel Eilenberg P. R. Halmos C. G. Latimer Ν. H. McCoy I. S. Sokolnikoff The following Monographs have been published: 1. Calculus of Variations, by G. A. Bliss (out of print) 2. Analytic Functions of a Complex Variable, by D. R. Curtiss (out of print) 3. Mathematical Statistics, by H. L. Rietz (out of print) 4. Projective Geometry, by J. W. Young (out of print) 5. A History of Mathematics in America before 1900, by D. E. Smith and Jekuthiel Ginsburg (out of print) 6. Fourier Series and Orthogonal Polynomials, by Dunham Jackson (out of print) 7. Vectors and Matrices, by C. C. MacDuffee (out of print) 8. Rings and Ideals, by Ν. H. McCoy (out of print) 9. The Theory of Algebraic Numbers, second edition, by Harry Pollard and Harold G Diamond 10. The Arithmetic Theory of Quadratic Forms, by B. W. Jones (out of print) 11. Irrational Numbers, by Ivan Niven 12. Statistical Independence in Probability, Analysis and Number Theory, by Mark Kac 13. A Primer of Real Functions, fourth edition, by Ralph P. Boas, Jr. Revised and updated by Harold P. Boas 14. Combinatorial Mathematics, by Herbert J. Ryser 15. Noncommutative Rings, by I. N. Herstein 16. Dedekind Sums, by Hans Rademacher and Emil Grosswald 17. The Schwarz Function and its Applications, by Philip J. Davis 18. Celestial Mechanics, by Harry Pollard 19. Field Theory and its Classical Problems, by Charles Robert Hadlock 20. The Generalized Riemann Integral, by Robert M. McLeod 21. From Error-Correcting Codes through Sphere Packings to Simple Groups, by Thomas M. Thompson 22. Random Walks and Electric Networks, by Peter G. Doyle and J. Laurie Snell 23. Complex Analysis: The Geometric Viewpoint, second edition, by Steven G Krantz 24. Knot Theory, by Charles Livingston 25. Algebra and Tiling: Homomorphisms in the Service of Geometry, by Sherman Stein and Sandor Szabo 26. The Sensual (Quadratic) Form, by John H. Conway assisted by Francis Y. C. Fung 27. A Panorama of Harmonic Analysis, by Steven G Krantz 28. Inequalities from Complex Analysis, by John P. D'Angelo 29. Ergodic Theory of Numbers, by Karma Dajani and Cor Kraaikamp 30. A Tour through Mathematical Logic, by Robert S. Wolf PREFACE This monograph is intended as an exposition of some central results on irrational numbers, and is not aimed at providing an exhaustive treatment of the problems with which it deals. The term "irrational numbers," a usage inherited from ancient Greece which is not too felicitous in view of the everyday meaning of the word "irrational," is employed in the title in a generic sense to include such related categories as transcendental and normal num- bers. The entire subject of irrational numbers cannot of course be encompassed in a single volume. In the selec- tion of material the main emphasis has been on those as- pects of irrational numbers commonly associated with number theory and Diophantine approximations. The top- ological facets of the subject are not included, although the introductory part of Chapter I has a sketch of some of the simplest set-theoretic properties of the irrationals as a part of the continuum. The axiomatic basis for irra- tional numbers, proceeding say from the Peano postulates for the natural numbers to the construction of the real numbers, is purposely omitted, because in the first place the aim is not in the direction of the foundations of math- ematics, and in the second place there are excellent treat- ments of this topic readily available. vii viii PREFACE The customary organization of a book with related sub- jects grouped together has been modified in part by con- sideration of the degree of difficulty of the topics, proceed- ing from the easiest to the most difficult. For example, al- most all the theorems on irrational numbers in Chapter II are implied by the stronger results of Chapter IX, but, whereas Chapter II requires only calculus and the barest rudiments of number theory for understanding, Chapter IX presupposes some basic results on algebraic numbers and complex functions. The first seven chapters are dis- tinctly easier reading than the last three, with fewer pre- requisite results needed and less mathematical maturity required of the reader. The chapters are for the most part independent of one another and so can be read separately; the major exception to this statement is the use in Chapter VI of some results from Chapter V. The only knowledge required of the reader beyond quite elementary mathematics is some algebraic number theory in Chapters III, IX and X, and some function theory in Chapters VI, VIII, IX and X. Most of the results needed are well-known theorems, central to the mainstream of mathematics, and complete references are given to stand- ard works. In those few instances where the prerequisite material is at all special, it has been included in the text. The books by Hardy and Wright, Koksma, Perron, and Siegel listed on page 157 have been very helpful, and I have made free use of these excellent sources. Further source material is listed in the notes at the ends of the chapters. These references, along with the remarks in the Notes, may be taken or left alone at the reader's choice. Some further results beyond the scope of this book are also listed in the Notes; however, as any expert in the subject will readily see, I have not attempted to be either systematic or complete about this. For the convenience of the reader PREFACE ix there is appended a list of notation and a glossary on pages 151 to 156. * * * Substantial improvements in the book have resulted from discussion of many points with my colleagues at the Uni- versity of Oregon, and from bibliographic suggestions by Professor C. D. Olds. I am also indebted to the Edito- rial Committee of the Carus Monographs for help in remov- ing several errors and obscurities. But especially I wish to acknowledge my indebtedness to Professor H. S. Zucker- man who has been actively interested in this project from the start. Discussions with him during the early stages in- fluenced markedly the final versions of Chapters I and V. In addition he has read the manuscript very thoroughly and critically. However I did not invariably follow the suggestions of these friendly critics; so the responsibility for the shortcomings of the monograph is entirely mine. IVAN NIVEN University of Oregon July 1956 CONTENTS CHAPTER PAGE I. RATIONALS AND IRRATIONALS 1. The preponderance of irrationals 1 2. Countability 4 3. Dense sets 5 4. Decimal expansions 6 II. SIMPLE IRRATIONALITIES 1. Introduction 15 2. The trigonometric functions and χ 16 3. The hyperbolic, exponential, and logarithmic functions. 22 III. CERTAIN ALGEBRAIC NUMBERS 1. Introduction 28 2. Further background material 30 3. The factorization of xn - 1 33 4. Certain trigonometric values 36 5. Extension to the tangent 38 IV. THE APPROXIMATION OF IRRATIONALS BY RATIONALS 1. The problem 42 2. A generalization 44 3. Linearly dependent sets 48 V. CONTINUED FRACTIONS 1. The Euclidean algorithm 51 2. Uniqueness 53 3. Infinite continued fractions 54 4. Infinite continued fraction expansions 59 5. The convergents as approximations 61 6. Periodic continued fractions 63 VI. FURTHER DIOPHANTINE APPROXIMATIONS 1. A basic result 68 2. Best possible approximations 70 xi xii CONTENTS CHAPTER PAGE 3. Uniform distributions 71 4. A proof by Fourier analysis 75 VII. ALGEBRAIC AND TRANSCENDENTAL NUMBERS 1. Closure properties of algebraic numbers 83 2. A property of algebraic integers 85 3. Transcendental numbers 87 4. The order of approximation 88 VIII. NORMAL NUMBERS 1. Definition of a normal number 94 2. The measure of the set of normal numbers 98 3. Equivalent definitions 104 4. A normal number exhibited 112 IX. THE GENERALIZED LINDEMANN THEOREM 1. Statement of the theorem 117 2. Preliminaries 118 3. Proof of the theorem 124 4. Applications of the theorem 131 5. Squaring the circle 132 X. THE GELFOND-SCHNEIDER THEOREM 1. Hubert's seventh problem 134 2. Background material 135 3. Two lemmas 137 4. Proof of the Gelfond-Schneider theorem 142 LIST OP NOTATION 151 GLOSSARY 153 REFERENCE BOOKS 157 INDEX OF TOPICS 159 INDEX OF NAMES 163 LIST OF NOTATION The g.o.d., or greatest common divisor, of the in- tegers h and k. The same notation is used for the interval from Λ to A; on the real line, where h and k are any real numbers. The greatest integer less than or equal to x, i.e., the unique integer η satisfying the inequality η £ χ < η + 1. The fractional part of x; thus (χ) = χ — [χ]. Euler's φ function. The finite field having the ρ elements 0, 1, 2, · · ·, ρ — 1, with addition and multiplication defined modulo the prime number p. The set of all polynomials in χ with coefficients in Jp. The nth cyclotomic polynomial, i.e., the monic polynomial whose zeros are the primitive nth roots of unity. The field of rational numbers. The degree of the field Κ over the field F, it being presumed that Κ contains F.
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