Classical Notions and Definitions
In this chapter we bring together some basic definitions and facts on continued fractions. After a small introduction we prove a convergence theorem for infinite regular continued fractions. Further, we prove existence and uniqueness of continued fraction
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Oleg Karpenkov
Geometry of Continued Fractions
Oleg Karpenkov Dept. of Mathematical Sciences University of Liverpool Liverpool, UK
ISSN 1431-1550 Algorithms and Computation in Mathematics ISBN 978-3-642-39367-9 ISBN 978-3-642-39368-6 (eBook) DOI 10.1007/978-3-642-39368-6 Springer Heidelberg New York Dordrecht London Library of Congress Control Number: 2013946250 Mathematics Subject Classification (2010): 11J70, 11H06, 11P21, 52C05, 52C07 © Springer-Verlag Berlin Heidelberg 2013 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
Preface
Continued fractions appear in many different branches of mathematics: the theory of Diophantine approximations, algebraic number theory, coding theory, toric geometry, dynamical systems, ergodic theory, topology, etc. One of the metamathematical explanations of this phenomenon is based on an interesting structure of the set of real numbers endowed with two operations: addition a + b and inversion 1/b. This structure appeared for the first time in the Euclidean algorithm, which was known several thousand years ago. Similarly to the structures of fields and rings (with operations of addition a + b and multiplication a ∗ b), structures with addition and inversion can be found i
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