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Introduction to Computer Data Representation Introduction to Computer Data Representation Peter Fenwick The University of Auckland (Retired) New Zealand Bentham Science Publishers Bentham Science Publishers Bentham Science Publishers Executive Suite Y - 2 P.O. Box 446 P.O. Box 294 PO Box 7917, Saif Zone Oak Park, IL 60301-0446 1400 AG Bussum Sharjah, U.A.E. USA THE NETHERLANDS [email protected] [email protected] [email protected] Please read this license agreement carefully before using this eBook. Your use of this eBook/chapter constitutes your agreement to the terms and conditions set forth in this License Agreement. This work is protected under copyright by Bentham Science Publishers to grant the user of this eBook/chapter, a non- exclusive, nontransferable license to download and use this eBook/chapter under the following terms and conditions: 1. This eBook/chapter may be downloaded and used by one user on one computer. 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You (the user) acknowledge that you have read this Agreement, and agree to be bound by its terms and conditions. Permission for Use of Material and Reproduction Permission Information for Users Outside the USA: Bentham Science Publishers grants authorization for individuals to photocopy copyright material for private research use, on the sole basis that requests for such use are referred directly to the requestor's local Reproduction Rights Organization (RRO). The copyright fee is US $25.00 per copy per article exclusive of any charge or fee levied. In order to contact your local RRO, please contact the International Federation of Reproduction Rights Organisations (IFRRO), Rue Joseph II, 9-13 I000 Brussels, Belgium; Tel: +32 2 234 62 60; Fax: +32 2 234 62 69; E-mail: [email protected]; url: www.ifrro.org This authorization does not extend to any other kind of copying by any means, in any form, and for any purpose other than private research use. Permission Information for Users in the USA: Authorization to photocopy items for internal or personal use, or the internal or personal use of specific clients, is granted by Bentham Science Publishers for libraries and other users registered with the Copyright Clearance Center (CCC) Transactional Reporting Services, provided that the appropriate fee of US $25.00 per copy per chapter is paid directly to Copyright Clearance Center, 222 Rosewood Drive, Danvers MA 01923, USA. Refer also to www.copyright.com To Brenda CONTENTS Foreword i Preface iii CHAPTERS Introduction and Overview 1 1. Numbers and Computers 5 2. Binary and Other Representations 17 3. Signed, and Other, Representations 29 4. Basic Arithmetic and Logic 45 5. Computer Arithmetic 77 6. Floating-Point Representations 103 7. Logarithmic Representations 127 8. Characters and Text 133 9. Universal (Variable Length) Codes 159 10. Checksums and Error Control 191 11. Miscellaneous Topics 219 12. Concluding Comments 249 Bibliography 251 Index 259 i Foreword It is my great pleasure to recommend this excellent book written by my friend and colleague, Professor Peter Fenwick. During the eleven years I have known him, we have had many a discussion, often touching on topics covered here. Though this is the closest we have come to a collaboration I have little doubt that had we met earlier in our careers we would have collaborated extensively. A major contribution of this book is to bring a historical perspective to many topics that are so widely accepted that it might not be obvious there were choices to be made. The binary representation of numbers was so obvious even in the 1940s that Burks, Goldstine and von Neumann are said to have \adopted it seemingly without discussion". But Burks et al considered floating point representation, then argued against supporting it. Long ago I heard it claimed that von Neumann believed any mathematician "worth his salt" should be able to specify floating point computations using only integers. In any case, floating point only came into its own in the 1980s, with the broad acceptance of the IEEE standards. Professor Fenwick shows great insight into why it took decades to get right something as basic as the representation of numbers. A second important contribution is discussion of the introduction of redundancy to increase reliability in the presence of errors: check sums and variable-length (universal) codes. While simple check sums are frequently discussed, I know of no comparable source for a general discussion of Universal codes, an important but somewhat obscure subject. I agree with Professor Fenwick's quote, that \everybody thinks they know" about these topics, but there are big holes, even today. Surely most of us have superficial knowledge that fails us when we really need to work through the details. This book covers a huge range of material, thoroughly and concisely. I have taught a good bit of the material, but I learned much, even in areas where I claim some expertise. The book displays a deep understanding of the many and varied requirements for digital representation of information, from the obvious integers and floating point, to Zeckendorf representations and Gray codes; from 2's complement to logarithmic arithmetic; from Elias and Levenstein codes to Rice and Golomb codes and on to Ternary Comma and Fibonacci codes. ii In addition to the plethora of ways to represent numbers, it also covers represen- tation of characters and strings. While the book will serve very well as a reference, it is also fascinating reading. Many pages are devoted to obscure topics, interesting largely because of their place in history, but outside the domain of a classic textbook on computer organization or architecture. These are perhaps the most important sections, precisely because they had to be understood and discarded to get us where we are now. This book definitely does not qualify for the subtitle, \Data Representation for Dummies". While it quickly surveys common forms of representation, the pace and breadth will bewilder the true novice. On occasions, it uses terms unfamiliar (at least to an American), requiring another source.
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