Irrational NumbersCambridge University Press, 18.08.2005 - 164 Seiten In this monograph, Ivan Niven provides a masterful exposition 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 technique. The last third of the monograph treats normal and transcendental numbers, including the Lindemann theorem, and the Gelfond-Schneider theorem. The book is wholly self-contained. The results needed from analysis and algebra are central. Well-known theorems, and complete references to standard works are given to help the beginner. The chapters are for the most part independent. There are notes at the end of each chapter citing the main sources used by the author and suggesting further reading. |
Inhalt
CHAPTER | 1 |
SIMPLE IRRATIONALITIES | 15 |
The hyperbolic exponential and logarithmic functions | 22 |
CERTAIN ALGEBRAIC NUMBERS | 28 |
THE APPROXIMATION OF IRRATIONALS BY RATIONALS | 42 |
CONTINUED FRACTIONS | 50 |
FURTHER DIOPHANTINE APPROXIMATIONS | 68 |
ALGEBRAIC AND TRANSCENDENTAL NUMBERS | 83 |
Equivalent definitions | 105 |
The measure of the set of normal numbers | 114 |
Proof of the theorem | 125 |
Squaring the circle | 132 |
Two lemmas | 139 |
LIST OF NOTATION | 151 |
| 157 | |
| 163 | |
Häufige Begriffe und Wortgruppen
algebraic numbers Amer apply approximation arbitrarily argument assume base blocks Chapter choose close complete conclude constant contradiction convergents COROLLARY decimal defined definition denote digits divisible divisor elements equation establish example exist expansion extension factor field finite Fn(x follows functions give given Hence implies induction inequality infinite integral coefficients interval irrational number irrationality irreducible Jp[x lattice point least Lemma length less linear Math mathematical measure zero minimal polynomial monic multiple non-zero normal notation nth root obtain points polynomial positive integer prime primitive Proof prove quadratic irrational R(cos rational integral rational numbers real numbers relation replaced result root of unity satisfies sequence side simple continued fraction sufficiently large Suppose Theorem Theory tion tional unique values write zero