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. The field obtained by extending F by the adjunc- tion of a. F(a) consists of the set of all rational functions f(a)/g(a), where / and g are polynomials with coefficients in F, and g(a) ^ 0.
n] A finite continued fraction, whose value can be de-
fined recursively as [xo, x\, • · ·, xn-2, Xn-i + l/xnl- An infinite continued fraction, with value lim [χχ, xt, ·•, Xn]- Is contained in. Belongs to, is a member of. 151 LIST OF NOTATION
The union of; A U Β is the set of elements χ such that χ is a member of at least one of the sets A, B. d is a divisor of n. d is not a divisor of n. Same as ex, the exponential function. The maximum of. The maximum of the absolute values of the alge- braic number a and its conjugates. GLOSSARY
The purpose of this glossary is to supplement, rather than replace, the index. Many of the terms here are not defined elsewhere in this monograph.
algebraic integer. An algebraic number of a special type, namely one
1 that satisfies some equation χ" + αχχ"" + · —\- an = 0 with rational integral coefficients. The minimal polynomial of an al- gebraic integer is also monic with integral coefficients. algebraic number. Any complex number that satisfies some equation of the form xn + ajx""1 + · · · + a„ =0 with rational coefficients. almost all. A set S constitute a set of measure zero. Archimedean property. If a and β are any two positive real num- bers, then there exists a positive integer η such that na > β. basis of a field. See finite extension field. closed interval. See interval. convergent. See n-th convergent. countable set. A set that can be put into one-to-one correspondence with the natural numbers 1, 2, 3, · · ·. See the notes on Chapter 1 for a variation of this definition that is sometimes used. cyclotomic polynomial. The nth cyclotomic polynomial Fn(x) is the unique monic polynomial whose zeros are the nth primitive roots of unity. degree of a field. See finite extension field. degree of an algebraic number. The same as the degree of the mini- mal polynomial of the algebraic number. denumerable set. One that can be put into one-to-one correspond- ence with the natural numbers 1, 2, 3, division algorithm. Given rational integers a and 6 > 0, one obtains integers g and r such that ο = bq + r with 0 ί r < t. Eisenstein irreducibility criterion. Let ρ be a prime and f(x) = αοχ" + a\xn~l + · · · + a„ a polynomial with integral coefficients 153 154 GLOSSARY
such that ρ )f ao, p2 )( a„, p\a, for i = 1, 2, · • ·, n. Then/(x) is irreducible over the rational numbers. elementary symmetric polynomials. The elementary symmetric poly-
nomials (or functions) of X\, X2, · • ·, in are the polynomials σι, σ<ι,
• • •, ση defined by the identity
H(y- ^) = vn - «Μ"-1 + °Μη~* +••• + (-I)V„. I- 1 Euler φ function. For any positive integer η, φ(η) is the number of positive integers j £ η such that the g.c.d. (j, n) = 1. Its value is φ[η) = n(l — pf ')(1 — pi"1) · · • (1 — ρΓ1), where the distinct prime factors of η are pi, P2, · · ·, Pr- everywhere dense. A set £ of real numbers is everywhere dense in an interval / if, given any two numbers a and β in I, say with a < β, there is a number s in
denoted by Jp consisting of the numbers 0, 1, 2, · · ·, ρ — X where ρ is a prime number, addition and multiplication being defined modulo p. finite extension field. A field Κ is said to be a finite extension of a field F if Κ contains F and if there exists a finite set of elements
in K, say ki, ki, · · ·, kn, such that every element in if is expressi- ble as a linear combination of these elements, Σα,-fc,·, with coeffi- cients α,· in F. If no fewer than η elements of Κ can serve thus,
then η is the degree of Κ over F, and the elements k\, k%, • · ·, kn are said to form a basis for JC over F. Stated otherwise, the degree of Κ over F is the maximum number of elements of Κ that are linearly independent over F. finite field. A field with a finite number of elements. fundamental theorem of arithmetic. Any positive integer except 1 can be written as a product of prime factors which is unique apart from the order of the factors. GLOSSARY 155 fundamental theorem on symmetric polynomials. Any symmetric polynomial f(xi, x% • • ·, x„) with coefficients in a field F is ex- pressible as a polynomial p(
ent over a field F if there exist elements ci, C2, • · ·, cn in not all η zero, such that \\ °iai = 0· Otherwise they are linearly inde- pendent. measure zero. A set of real numbers or points is said to have meas- ure zero if it is possible to cover the points of the set with a col- lection of intervals of arbitrarily small total length. minimal polynomial of an algebraic number. The unique monic poly- nomial of least degree with rational coefficients which has the al- gebraic number as a zero. monic polynomial. One having 1 as the coefficient of the term of highest degree. n-th convergent. The nth convergent to the continued fraction [xo, xi, 32, · · · ] is the finite continued fraction [zo, Xl, * ' XrtJ. open interval. See interval. partial quotient. See simple continued fraction. primitive n-th root of unity. An nth root of unity whose powers give all the nth roots of unity. Such a number is expressible as e2r<*/n 156 GLOSSARY
= cos (2vk/ri) -J- i sin (2wk/n) for some integer k relatively prime to n. rational number. One that can be written as the quotient of two integers; thus h/k with k 0. reducible polynomial. A polynomial f(x) with coefficients in a field F is reducible over F if it can be factored non-trivially into two polynomials g(x) and h{x) with coefficients in F, f(x) = ^(x) h{x). By "non-trivially" we mean that each of g(x) and A(x) has degree at least one. simple continued fraction. The continued fraction [xo, xi, xi, · · • ] is simple if each partial quotient Χχ is an integer, positive except per- haps when i = 0. symmetric polynomial. f(x\, x% · · ·, xn) is symmetric in x\, x% · · ·,
xn if any permutation of the xy leaves / invariant. transcendental number. A complex number that is not algebraic. triangle inequality. Any two complex numbers a and β satisfy
l« +01 = |«| + |/3| and \a -β\ ^ \a\ - \β\. REFERENCE BOOKS
G. Birkhoff and S. MacLane, A Survey of Modern Algebra, Macmil- lan, 2nd edition, 1953. G. Chrystal, Algebra, vol. II, London, Adam and Charles Black, 2nd edition, 1900. H. Davenport, The Higher Arithmetic, Hutchinson's University Library, 1952. G. H. Hardy and Ε. M. Wright, The Theory of Numbers, Oxford, 3rd edition, 1954. E. W. Hobson, Squaring the Circle, Cambridge, 1913; reprinted by Chelsea, New York, 1953. E. Kamke, Theory of Sets, Dover, 1950. J. F. Koksma, Diophantische Approximationen, Ergebnisse der Malhematik, Band IV, Heft 4, Berlin, Springer, 1936; reprinted by Chelsea. E. Landau, Foundations of Analysis, Chelsea, 1951. T. Nagell, Number Theory, John Wiley, 1951. O. Ore, Number Theory and Its History, McGraw-Hill, 1948. O. Perron, Irrationalzahlen, Berlin, de Gruyter, 1910; reprinted by Chelsea, New York, 1951. H. Pollard, Algebraic Theory of Numbers, Carus Monograph No. 9, John Wiley, 1950. C. L. Siegel, Transcendental Numbers, Annals of Mathematics Studies No. 16, Princeton, 1949. R. L. Wilder, Foundations of Mathematics, John Wiley, 1952.
157
INDEX OF TOPICS
Absolutely normal number, 116 Continued fraction, periodic, 63 Algebraic integers, 29 simple, 52 everywhere dense property, Convergent to a continued frac- 85 tion, 57 Algebraic number, 28 Countability, 4 degree of, 28 alternative definition, 14 order of approximation of, 90, of algebraic numbers, 87 93 of rational numbers, 4 Algebraic numbers, closure prop- Cyclotomic polynomial, 33 erties, 84 coefficients of, 34 countability of, 87 irreducibility of, 36 form a field, 84 Approximation, best possible, 62, Decimal expansion, 6 68, 70, 81, 93 Degree of an algebraic number, of an irrational number, 42, 28 44, 61, 68, 70 Dense set, 5 of several numbers simultane- Denumerable set, 4 ously, 45, 47, 48 order of, 88 e, irrationality of, 11 to a complex irrational num- transcendence of, 25 ber, 81 Euclidean algorithm, 51 Complex irrational number, 81 Euler φ function, 30 Complex rational number, 83 Euler's constant, 150 Conjugate algebraic numbers, Everywhere dense set, 5 122 Exponential function, irrational Continued fraction, 52 values of, 23 finite, 52 linear independence of, 117 infinite, 57 transcendental values of, 131 160 INDEX OF TOPICS
Finite continued fraction, 52 Linearly dependent numbers, si- multaneous approximation Gauss's lemma, 29 to, 48 Gelfond-Schneider theorem, 134 Liouville number, 91 transcendence of, 92 Hubert's seventh problem, 134 Logarithms, irrationality of, 23, Hurwitz's theorem, 68, 70 24 Hyperbolic functions, irrational transcendence of, 131, 135 values of, 22 transcendental values of, 131 Measure, of algebraic numbers, 87 of irrational numbers, 2 Incommensurable numbers, 14 of normal numbers, 103 Infinite continued fraction, 57 of rational numbers, 2 Integral basis, 136 of transcendental numbers, 88 Irrationality, criteria for, 10, 12, Minimal polynomial, 28 15, 44, 60, 75 Monic polynomial, 28 of e, 11 of exponential function, 23 Normal field, 121 of hyperbolic functions, 22 Normal numbers, definition, 95, of logarithms, 23, 24 96 of y/m, 16 measure of, 103 of ir, 19 uniform distribution property, of trigonometric functions, 17, 110 21, 41 Norm of an algebraic number, Irrational number, 1 136 uniform distribution of mul- tiples of, 72 Order of approximation, 88 Irrational numbers, everywhere of an algebraic number, 90, dense, 5 93 measure of, 2 of a rational number, 89 not countable, 5 Irreducibility of cyclotomic Partial quotient, 52 polynomial, 36 Periodic continued fraction, 63 Periodic decimal, 12 Kronecker's theorem, 82 Pi, irrationality of, 19 transcendence of, 117 Lattice point, 44 Pigeon-hole principle, 43 Lindemann theorem, 117 Primitive roots of unity, 30 Linear independence of exponen- tial functions, 117 Quadratic irrational, 64 INDEX OF TOPICS 161
Rationality, criteria for, 10, 12, Transcendental numbers, defini- 44, 60, 75 tion, 29 Rational number, 1 measure of, 88 order of approximation, 89 Trigonometric functions, alge- Rational numbers, countability braic values of, 29, 37, 39 of, 4 irrational values of, 17, 21, 41 everywhere dense, 5 rational values of, 41 measure of, 2 transcendental values of, 131
Simple continued fraction, 52 Uniform distribution, 71 Simply normal number, 94 of multiples of an irrational Squaring the circle, 132 number, 72 property of normal numbers, 110 Transcendence, of e, 25 Weyl criterion for, 76, 82 of Liouville numbers, 92 of logarithms, 131, 135 Vandermonde determinant, 136 of r, 117 of values of various functions, Weyl criterion for uniform distri- 131 bution, 76, 82
INDEX OF NAMES
Arnold, Β. H., 41 Gauss, C. F., 15, 29 Gelfond, A. O., 134, 149 Besicovitch, A. S., 116 Good, I. J., 116 Birkhoff, G., 14, 157 Graves, L. M., 97 Bohl, P., 82 Borel, E., 95, 115, 116 Halmos, P. R., 115 Breusch, R., 27 Hamming, R. W., 41 Butlewski, Z., 27 Hanson, Η. Α., 116 Hardy, G. H., 27, 50, 67, 82, 133, Cantor, G., 6, 14, 93 157 Cassels, J. W. S., 116 Hermite, C, 27, 117, 132 Champernowne, D. G., 116 Hubert, D., 134 Chrystal, G., 27, 157 Hille, E., 149 Copeland, A. H., 116 Hobson, E. W., 133, 157 Corput, J. G. van der, 82 Hudson, H. P., 133 Courant, R., 133 Hurwitz, Α., 27, 81, 82
Iwamoto, Y., 27 Davenport, H., 67, 157 Diananda, P. H., 14 Jackson, D., 75 Dickson, L. E., 93, 133 Dietrich, V. E., 133 Kamke, E., 93, 157 Dirichlet, G. P. L., 50 Dyson, F. J., 93 Kline, M., 14 Knopp, K, 136 Koksma, J. F., 27, 50, 82, 157 Erdos, P., 116 Kronecker, L., 82 Eves, H., 41 Lambert, J. H., 27 Ford, L. R., 81, 82 Landau, E., 14, 157 163 164 INDEX OF NAMES
Lehmer, D. H., 37, 41 Sarton, G., 14 LeVeque, W. J., 133 Schneider, Th., 134, 149 Lindemann, F., 27, 117, 132, 133 Siegel, C. L., 93, 149, 150, 157 Liouville, J., 93 Sierpinski, W., 82 Skolem, Th., 132 MacLane, S., 14, 27, 157 Spiegel, M. R., 14 Maxfield, J. E., 116 Steinberg, R., 132 Nagell, T., 157 Swift, E., 41 Neugebauer, O., 14 Niven, I., 27, 116, 133 Thomas, J. M., 83 Thue, Α., 93 Olmsted, J. Η. M., 41 Oppenheim, Α., 14 Underwood, R. S., 41 Ore, 0., 157 Uspensky, J. V., 118, 136 Perron, 0., 67, 157 Pillai, S. S., 96, 116 Wall, D. D., 116 Pollard, H., 29, 39, 92, 93, 120, Webber, G. C., 150 Weierstrass, K., 132 136, 157 Weyl, H., 75, 76, 82 Popken, J., 27 Wilder, R. L., 14, 93, 157 Rado, R., 27 Wright, Ε. M., 27, 50, 67, 82, Redheffer, R. M., 132, 133 133, 157 Rees, D., 116 Robbins, H., 133 Young, J. W. Α., 133 Rosenthal, Α., 133 Roth, K. F., 93 Zuckerman, H. S., 67, 116 AMS / MAA THE CARUS MATHEMATICAL MONOGRAPHS
In this monograph, Ivan Niven, provides a masterful exposi- tion of some central results on irrational, transcendental, and normal numbers. He gives a complete treatment by elementary methods of the irrationality of the exponential, logarithmic, and trigonometric functions with rational arguments. The approximation of irrational numbers by rationals, up to such results as the best possible approximation of Hurwitz, is also given with elementary techniques. The last third of the mono- graph treats normal and transcendental numbers, including the transcendence of p and its generalization in the Lindermann theorem, and the Gelfond-Schneider theorem.
Most of the material in the fi rst two-thirds of the book presup- poses only calculus and beginning number theory. The book is almost wholly self-contained. The results needed from analysis and algebra are central, and well-known theorems, and complete references to standard works are given to help the beginner. The chapters are for the most part independent. There is a set of notes at the end of each chapter citing the main sources used by the author, and suggesting further readings.