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Number Theory and Mirror-Symmetrical Arithmetic

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Number Theory and Mirror-Symmetrical Arithmetic 10671_9789813228610_tp.indd 1 28/9/17 2:51 PM b2530 International Strategic Relations and China’s National Security: World at the Crossroads This page intentionally left blank b2530_FM.indd 6 01-Sep-16 11:03:06 AM K E 10671_9789813228610_tp.indd 2 28/9/17 2:51 PM Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 SA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 K office: 57 Shelton Street, Covent Garden, London WC2H 9HE Library of Congress Cataloging-in-Publication Data Names: Stakhov, A. P. (Alexey Petrovich), author. Title: Numeral systems with irrational bases for mission-critical applications / Alexey Stakhov (International Club of the Golden Section, Canada). Description: New Jersey : World Scientific, 2018. | Series: Series on knots and everything ; vol. 61 | Includes bibliographical references and index. Identifiers: LCCN 2017030467 | ISBN 9789813228610 (hardcover : alk. paper) Subjects: LCSH: Golden section. | Fibonacci numbers. Classification: LCC QA466 .S7825 2018 | DDC 512.7/2--dc23 LC record available at https://lccn.loc.gov/2017030467 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Copyright © 2018 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, SA. In this case permission to photocopy is not required from the publisher. Printed in Singapore Eh - 10671 - Numeral Systems with Irrational Bases.indd 5 25-09-17 12:15:25 PM Contents Preface ............................................................................................................................. vii Introduction ................................................................................................................... xvii Acknowledgements .......................................................................................................... xxi Chapter 1. Preliminary Historical and Mathematical Information ............................. 1 1.1. The idea of harmony in its historical development ................................................... 1 1.2. Proclus hypothesis: A new view on Euclid’s “Elements” and the history of mathematics .............................................................................................................. 6 1.3. The golden ratio in Euclid’s Elements .................................................................... 10 1.4. The algebraic identities of the golden ratio ............................................................. 17 1.5. Fibonacci numbers.................................................................................................. 19 1.6. Lucas numbers ........................................................................................................ 29 1.7. Binet’s formulas ..................................................................................................... 31 1.8. Pascal’s triangle and Fibonacci p-numbers ............................................................. 33 1.9. The golden p-proportions ....................................................................................... 39 1.10. Phyllotaxis as the main reason for the creation of Fibonacci number theory in modern mathematics ........................................................................................... 42 Chapter 2. A New View on Numeral Systems: Unusual Hypotheses, Surprising Properties and Applications ....................................................................... 49 2.1. Dodecahedron, solar system, Egyptian calendar and Babylonian numeral system ..................................................................................................................... 49 2.2. Icosahedron and its connection with solar system, Mayan’s calendar and Mayan’s numeral system ........................................................................................ 53 2.3. Surprising properties of the Egyptian decimal arithmetic ....................................... 55 2.4. Decimal system ...................................................................................................... 57 2.5. History of binary system: since ancient time to John von Neumann’s Principles ................................................................................................................ 60 2.6. Canonical and symmetrical numeral systems ......................................................... 65 2.7. Ternary-symmetrical system and ternary arithmetic ............................................... 68 2.8. Brusentsov’s ternary principle and ternary technology........................................... 81 v vi Numeral Systems with Irrational Bases Chapter 3. Bergman’s System, “Golden” Number Theory and Mirror-Symmetrical Arithmetic ................................................................ 89 3.1. George Bergman and Bergman’s system ................................................................ 89 3.2. The “Golden” number theory and new properties of natural numbers ................... 91 3.3. The “golden” ternary mirror-symmetrical representation ..................................... 109 3.4. The ternary mirror-symmetrical summation and subtraction ................................ 121 3.5. The ternary mirror-symmetrical multiplication and division ................................ 128 3.6. Technical realizations of the ternary mirror-symmetrical arithmetical devices .... 131 3.7. Matrix and pipeline mirror-symmetrical arithmetical unit .................................... 133 3.8. Conclusions to Chapter 3 ...................................................................................... 138 Chapter 4. Fibonacci p-Codes and Concept of Fibonacci Computers ..................... 145 4.1. Fibonacci p-codes ................................................................................................. 145 4.2. The minimal form and redundancy of Fibonacci p-codes ..................................... 150 4.3. Fibonacci arithmetic ............................................................................................. 158 4.4. Counting and subtracting of the binary 1’s ........................................................... 163 4.5. Fibonacci summation and subtraction .................................................................. 166 4.6. Fibonacci multiplication and division ................................................................... 169 4.7. Useful applications of the Fibonacci code ............................................................ 173 4.8. Concept of the Fibonacci arithmetical processor for noise-immune computations ........................................................................................................ 179 4.9. Boolean realization of the original Fibonacci element basis ................................. 185 4.10. Fibonacci counter based on the minimal form ...................................................... 190 4.11. USA researches in Fibonacci computer field ....................................................... 202 Chapter 5. Codes of the Golden p-Proportions and Their Applications in Computer Science and “Golden” Metrology .......................................... 203 5.1 Definition of the codes of the golden p-proportions ............................................. 203 5.2 Partial cases of the codes of the golden p-proportions .......................................... 204 5.3 Conversion of numbers from traditional numeral systems to Bergman’s system ................................................................................................................... 205 5.4 The “golden” arithmetic ....................................................................................... 210 5.5 Application of the codes of the golden p-proportions in digital metrology .......... 222 5.6. Digit-to-analog (DAC) and analog-to-digit converters (ADC) based on the “golden” resistive divisors .................................................................................... 228 5.7 Conclusion ........................................................................................................... 238 Bibliography .................................................................................................................. 241 Index .............................................................................................................................. 247 Preface 1. The most important applied problems, which stimulated the development of mathematics on the stage of its origin The great Russian geometer Nikolay Lobachevsky begins his famous “Geometric study on the theory of parallel lines” by the following words: “In geometry, I found some imperfections, which, in my opinion, are the reason why this science until now is in the state, in which it had come to us from Euclid. I attribute to these imperfections the following: the vagueness in the first definitions of geometric quantities, the methods of measurement of these quantities, and, finally, the important gap in the theory of parallel lines.” It is important to emphasize that in this quote Lobachevsky posed
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