Arithmetic with Binary-Encoded Balanced Ternary Numbers
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Balanced Ternary Arithmetic on the Abacus
Balanced Ternary Arithmetic on the Abacus What is balanced ternary? It is a ternary number system where the digits of a number are powers of three rather than powers of ten, but instead of adding multiples of one or two times various powers of three to make up the desired number, in balanced ternary some powers of three are added and others are subtracted. Each power of three comprisising the number is represented by either a +, a -, or 0, to show that the power of three is either added, subtracted, or not present in the number. There are no multiples of a power of three greater than +1 or -1, which greatly simplifies multiplication and division. Another advantage of balanced ternary is that all integers can be represented, both positive and negative, without the need for a separate symbol to indicate plus or minus. The most significant ternary digit (trit) of any positive balanced ternary number is + and the most significant trit of any negative number is -. Here is a list of a few numbers in decimal, ternary, and balanced ternary: Decimal Ternary Balanced ternary ------------------------------------------------- -6 - 20 - + 0 (-9+3) -5 - 12 - + + (-9+3+1) -4 - 11 - - ( -3-1) -3 - 10 - 0 etc. -2 - 2 - + -1 - 1 - 0 0 0 1 1 + 2 2 + - 3 10 + 0 4 11 + + 5 12 + - - 6 20 + - 0 Notice that the negative of a balanced ternary number is formed simply by inverting all the + and - signs in the number. Thomas Fowler in England, about 1840, built a balanced ternary calculating machine capable of multiplying and dividing balanced ternary numbers. -
A Ternary Arithmetic and Logic
Proceedings of the World Congress on Engineering 2010 Vol I WCE 2010, June 30 - July 2, 2010, London, U.K. A Ternary Arithmetic and Logic Ion Profeanu which supports radical ontological dualism between the two Abstract—This paper is only a chapter, not very detailed, of eternal principles, Good and Evil, which oppose each other a larger work aimed at developing a theoretical tool to in the course of history, in an endless confrontation. A key investigate first electromagnetic fields but not only that, (an element of Manichean doctrine is the non-omnipotence of imaginative researcher might use the same tool in very unusual the power of God, denying the infinite perfection of divinity areas of research) with the stated aim of providing a new perspective in understanding older or recent research in "free that has they say a dual nature, consisting of two equal but energy". I read somewhere that devices which generate "free opposite sides (Good-Bad). I confess that this kind of energy" works by laws and principles that can not be explained dualism, which I think is harmful, made me to seek another within the framework of classical physics, and that is why they numeral system and another logic by means of which I will are kept far away from public eye. So in the absence of an praise and bring honor owed to God's name; and because adequate theory to explain these phenomena, these devices can God is One Being in three personal dimensions (the Father, not reach the design tables of some plants in order to produce them in greater number. -
VSI Openvms C Language Reference Manual
VSI OpenVMS C Language Reference Manual Document Number: DO-VIBHAA-008 Publication Date: May 2020 This document is the language reference manual for the VSI C language. Revision Update Information: This is a new manual. Operating System and Version: VSI OpenVMS I64 Version 8.4-1H1 VSI OpenVMS Alpha Version 8.4-2L1 Software Version: VSI C Version 7.4-1 for OpenVMS VMS Software, Inc., (VSI) Bolton, Massachusetts, USA C Language Reference Manual Copyright © 2020 VMS Software, Inc. (VSI), Bolton, Massachusetts, USA Legal Notice Confidential computer software. Valid license from VSI required for possession, use or copying. Consistent with FAR 12.211 and 12.212, Commercial Computer Software, Computer Software Documentation, and Technical Data for Commercial Items are licensed to the U.S. Government under vendor's standard commercial license. The information contained herein is subject to change without notice. The only warranties for VSI products and services are set forth in the express warranty statements accompanying such products and services. Nothing herein should be construed as constituting an additional warranty. VSI shall not be liable for technical or editorial errors or omissions contained herein. HPE, HPE Integrity, HPE Alpha, and HPE Proliant are trademarks or registered trademarks of Hewlett Packard Enterprise. Intel, Itanium and IA64 are trademarks or registered trademarks of Intel Corporation or its subsidiaries in the United States and other countries. Java, the coffee cup logo, and all Java based marks are trademarks or registered trademarks of Oracle Corporation in the United States or other countries. Kerberos is a trademark of the Massachusetts Institute of Technology. -
Fibonacci P-Codes and Codes of the “Golden” P-Proportions: New Informational and Arithmetical Foundations of Computer Scienc
British Journal of Mathematics & Computer Science 17(1): 1-49, 2016, Article no.BJMCS.25969 ISSN: 2231-0851 SCIENCEDOMAIN international www.sciencedomain.org Fibonacci p-codes and Codes of the “Golden” p-proportions: New Informational and Arithmetical Foundations of Computer Science and Digital Metrology for Mission-Critical Applications Alexey Stakhov1* 1International Club of the Golden Section, Canada. Author’s contribution The sole author designed, analyzed and interpreted and prepared the manuscript. Article Information DOI: 10.9734/BJMCS/2016/25969 Editor(s): (1) Wei-Shih Du, Department of Mathematics, National Kaohsiung Normal University, Taiwan. Reviewers: (1) Santiago Tello-Mijares, Instituto Tecnológico Superior de Lerdo, Mexico. (2) Danyal Soybas, Erciyes University, Turkey. (3) Manuel Alberto M. Ferreira, ISCTE-IUL, Portugal. Complete Peer review History: http://sciencedomain.org/review-history/14921 Received: 28th March 2016 Accepted: 2nd June 2016 Original Research Article Published: 7th June 2016 _______________________________________________________________________________ Abstract In the 70s and 80s years of the past century, the new positional numeral systems, called Fibonaссi p-codes and codes of the golden p-proportions, as new informational and arithmetical foundations of computer science and digital metrology, was considered as one of the most important directions of Soviet computer science and digital metrology for mission-critical applications. The main advantage of this direction was to improve the information reliability of computer and measuring systems. This direction has been patented abroad widely (more than 60 patents of US, Japan, England, France, Germany, Canada and other countries). Theoretical basis of this direction have been described in author’s books "Introduction into the Algorithmic Measurement Theory" (1977), and "Codes of the Golden Proportion" (1984). -
Basic Digit Sets for Radix Representation
LrBRAJ7T LIBRARY r TECHNICAL REPORT SECTICTJ^ Kkl B7-CBT SSrTK*? BAVAI PC "T^BADUATB SCSOp POSTGilADUAIS SCSftQl NPS52-78-002 . NAVAL POSTGRADUATE SCHOOL Monterey, California BASIC DIGIT SETS FOR RADIX REPRESENTATION David W. Matula June 1978 a^oroved for public release; distribution unlimited. FEDDOCS D 208.14/2:NPS-52-78-002 NAVAL POSTGRADUATE SCHOOL Monterey, California Rear Admiral Tyler Dedman Jack R. Bor sting Superintendent Provost The work reported herein was supported in part by the National Science Foundation under Grant GJ-36658 and by the Deutsche Forschungsgemeinschaft under Grant KU 155/5. Reproduction of all or part of this report is authorized. This report was prepared by: UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered) READ INSTRUCTIONS REPORT DOCUMENTATION PAGE BEFORE COMPLETING FORM 1. REPORT NUMBER 2. GOVT ACCESSION NO. 3. RECIPIENT'S CATALOG NUMBER NPS52-78-002 4. TITLE (and Subtitle) 5. TYPE OF REPORT ft PERIOD COVERED BASIC DIGIT SETS FOR RADIX REPRESENTATION Final report 6. PERFORMING ORG. REPORT NUMBER 7. AUTHORS 8. CONTRACT OR GRANT NUM8ERfa; David W. Matula 9. PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT, PROJECT, TASK AREA ft WORK UNIT NUMBERS Naval Postgraduate School Monterey, CA 93940 11. CONTROLLING OFFICE NAME AND ADDRESS 12. REPORT DATE Naval Postgraduate School June 1978 Monterey, CA 93940 13. NUMBER OF PAGES 33 14. MONITORING AGENCY NAME ft AODRESSf// different from Controlling Olllce) 15. SECURITY CLASS, (ot thta report) Unclassified 15«. DECLASSIFI CATION/ DOWN GRADING SCHEDULE 16. DISTRIBUTION ST ATEMEN T (of thta Report) Approved for public release; distribution unlimited. 17. DISTRIBUTION STATEMENT (of the mbatract entered In Block 20, if different from Report) 18. -
Part I Number Representation
Part I Number Representation Parts Chapters 1. Numbers and Arithmetic 2. Representing Signed Numbers I. Number Representation 3. Redundant Number Systems 4. Residue Number Systems 5. Basic Addition and Counting 6. Carry-Lookahead Adders II. Addition / Subtraction 7. Variations in Fast Adders 8. Multioperand Addition 9. Basic Multiplication Schemes 10. High-Radix Multipliers III. Multiplication 11. Tree and Array Multipliers 12. Variations in Multipliers 13. Basic Division Schemes 14. High-Radix Dividers IV . Division Elementary Operations Elementary 15. Variations in Dividers 16. Division by Convergence 17. Floating-Point Reperesentations 18. Floating-Point Operations V. Real Arithmetic 19. Errors and Error Control 20. Precise and Certifiable Arithmetic 21. Square-Rooting Methods 22. The CORDIC Algorithms VI. Function Evaluation 23. Variations in Function Evaluation 24. Arithmetic by Table Lookup 25. High-Throughput Arithmetic 26. Low-Power Arithmetic VII. Implementation Topics 27. Fault-Tolerant Arithmetic 28. Past,Reconfigurable Present, and Arithmetic Future Appendix: Past, Present, and Future Mar. 2015 Computer Arithmetic, Number Representation Slide 1 About This Presentation This presentation is intended to support the use of the textbook Computer Arithmetic: Algorithms and Hardware Designs (Oxford U. Press, 2nd ed., 2010, ISBN 978-0-19-532848-6). It is updated regularly by the author as part of his teaching of the graduate course ECE 252B, Computer Arithmetic, at the University of California, Santa Barbara. Instructors can use these slides freely in classroom teaching and for other educational purposes. Unauthorized uses are strictly prohibited. © Behrooz Parhami Edition Released Revised Revised Revised Revised First Jan. 2000 Sep. 2001 Sep. 2003 Sep. 2005 Apr. 2007 Apr. -
ARRAY SET ADDRESSING: ENABLING EFFICIENT HEXAGONALLY SAMPLED IMAGE PROCESSING by NICHOLAS I. RUMMELT a DISSERTATION PRESENTED TO
ARRAY SET ADDRESSING: ENABLING EFFICIENT HEXAGONALLY SAMPLED IMAGE PROCESSING By NICHOLAS I. RUMMELT A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2010 1 °c 2010 Nicholas I. Rummelt 2 To my beautiful wife and our three wonderful children 3 ACKNOWLEDGMENTS Thanks go out to my family for their support, understanding, and encouragement. I especially want to thank my advisor and committee chair, Joseph N. Wilson, for his keen insight, encouragement, and excellent guidance. I would also like to thank the other members of my committee: Paul Gader, Arunava Banerjee, Jeffery Ho, and Warren Dixon. I would like to thank the Air Force Research Laboratory (AFRL) for generously providing the opportunity, time, and funding. There are many people at AFRL that played some role in my success in this endeavor to whom I owe a debt of gratitude. I would like to specifically thank T.J. Klausutis, Ric Wehling, James Moore, Buddy Goldsmith, Clark Furlong, David Gray, Paul McCarley, Jimmy Touma, Tony Thompson, Marta Fackler, Mike Miller, Rob Murphy, John Pletcher, and Bob Sierakowski. 4 TABLE OF CONTENTS page ACKNOWLEDGMENTS .................................. 4 LIST OF TABLES ...................................... 7 LIST OF FIGURES ..................................... 8 ABSTRACT ......................................... 10 CHAPTER 1 INTRODUCTION AND BACKGROUND ...................... 11 1.1 Introduction ................................... 11 1.2 Background ................................... 11 1.3 Recent Related Research ........................... 15 1.4 Recent Related Academic Research ..................... 16 1.5 Hexagonal Image Formation and Display Considerations ......... 17 1.5.1 Converting from Rectangularly Sampled Images .......... 17 1.5.2 Hexagonal Imagers .......................... -
Math Book from Wikipedia
Math book From Wikipedia PDF generated using the open source mwlib toolkit. See http://code.pediapress.com/ for more information. PDF generated at: Mon, 25 Jul 2011 10:39:12 UTC Contents Articles 0.999... 1 1 (number) 20 Portal:Mathematics 24 Signed zero 29 Integer 32 Real number 36 References Article Sources and Contributors 44 Image Sources, Licenses and Contributors 46 Article Licenses License 48 0.999... 1 0.999... In mathematics, the repeating decimal 0.999... (which may also be written as 0.9, , 0.(9), or as 0. followed by any number of 9s in the repeating decimal) denotes a real number that can be shown to be the number one. In other words, the symbols 0.999... and 1 represent the same number. Proofs of this equality have been formulated with varying degrees of mathematical rigour, taking into account preferred development of the real numbers, background assumptions, historical context, and target audience. That certain real numbers can be represented by more than one digit string is not limited to the decimal system. The same phenomenon occurs in all integer bases, and mathematicians have also quantified the ways of writing 1 in non-integer bases. Nor is this phenomenon unique to 1: every nonzero, terminating decimal has a twin with trailing 9s, such as 8.32 and 8.31999... The terminating decimal is simpler and is almost always the preferred representation, contributing to a misconception that it is the only representation. The non-terminating form is more convenient for understanding the decimal expansions of certain fractions and, in base three, for the structure of the ternary Cantor set, a simple fractal. -
A Combinatorial Approach to Binary Positional Number Systems
Acta Math. Hungar. DOI: 10.1007/s10474-013-0387-8 A COMBINATORIAL APPROACH TO BINARY POSITIONAL NUMBER SYSTEMS A. VINCE Department of Mathematics, University of Florida, Gainesville, FL, USA e-mail: avince@ufl.edu (Received June 14, 2013; revised August 13, 2013; accepted August 22, 2013) Abstract. Although the representation of the real numbers in terms of a base and a set of digits has a long history, new questions arise even in the binary case – digits 0 and 1. A binary positional√ number system (binary radix system) with base equal to the golden ratio (1 + 5)/2 is fairly well known. The main result of this paper is a construction of infinitely many binary radix systems, each one constructed combinatorially from a single pair of binary strings. Every binary radix system that satisfies even a minimal set of conditions that would be expected of a positional number system, can be constructed in this way. 1. Introduction The terms positional number system, radix system,andβ-expansion that appear in the literature all refer to the representation of real numbers in terms of a given base or radix B and a given finite set D of digits. Histori- cally the base is 10 and the digit set is {0, 1, 2,...,9} or, in the binary case, the base is 2 and the digit set is {0, 1}. Alternative choices for the set of dig- its goes back at least to Cauchy, who suggested the use of negative digits in base 10, for example D = {−4, −3,...,4, 5}.Thebalanced ternary system is a base 3 system with digit set D = {−1, 0, 1} discussed by Knuth in [10]. -
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. The user may make one back-up copy of this publication to avoid losing it. The user may not give copies of this publication to others, or make it available for others to copy or download. For a multi-user license contact [email protected] 2. All rights reserved: All content in this publication is copyrighted and Bentham Science Publishers own the copyright. You may not copy, reproduce, modify, remove, delete, augment, add to, publish, transmit, sell, resell, create derivative works from, or in any way exploit any of this publication’s content, in any form by any means, in whole or in part, without the prior written permission from Bentham Science Publishers. 3. The user may print one or more copies/pages of this eBook/chapter for their personal use. -
Dirac's Principle of Mathematical Beauty, Mathematics of Harmony
1 Dirac’s Principle of Mathematical Beauty, Mathematics of Harmony and “Golden” Scientific Revolution Alexey Stakhov The International Club of the Golden Section 6 McCreary Trail, Bolton, ON, L7E 2C8, Canada [email protected] • www.goldenmuseum.com Abstract. In this study we develop the Mathematics of Harmony as a new interdisciplinary direction of modern science. The newest discoveries in different fields of modern science based on the Mathematics of Harmony, namely, mathematics (a general theory of hyperbolic functions and a solution to Hilbert’s Fourth Problem, algorithmic measurement theory and “golden” number theory), computer science (the “golden information technology), crystallography (quasi-crystals), chemistry (fullerenes), theoretical physics and cosmology (Fibonacci-Lorentz transformations, the “golden” interpretation of special theory of relativity and “golden” interpretation of the Universe evolution), botany (new geometric theory of phyllotaxis), genetics (“golden” genomatrices) and so on, are creating a general picture of the “Golden” Scientific Revolution, which can influence fundamentally on the development of modern science and education. Contents Preface 1. Introduction: Dirac’s Principle of Mathematical Beauty and “beautiful” mathematical objects 2. A new approach to the mathematics origins 3. The Mathematics of Harmony as a “beautiful” mathematical theory 4. The “Golden” Fibonacci goniometry: a revolution in the theory of hyperbolic functions 5. The “Golden” Fibonacci goniometry and Hilbert’s Fourth Problem: revolution in hyperbolic geometry 6. Fibonacci and “golden” matrices: a unique class of square matrices 7. New scientific principles based on the Golden Section 8. The Mathematics of Harmony: a renaissance of the oldest mathematical theories 9. The “Golden” information technology: a revolution in computer science 10. -
College of Basic Sciences & Humanities
SECTION X FACULTY OF BASIC SCIENCES AND HUMANITIES General Information Disciplines • Biochemistry • Botany • Business Studies • Chemistry • Economics and Sociology • Journalism, Languagues and Culture • Mathematics, Statistics and Physics • Microbiology • Zoology • Course curriculum for Award of 3-year B.Sc. Degree on opting out of 5-year integrated M.Sc. (Hons) Programme in Biochemistry • Semester-wise Programme for 5-year Integrated M.Sc. (Hons) in Biochemistry • Course curriculum for Award of 3-year B.Sc. Degree on opting out of 5-year integrated M.Sc. (Hons) Programme in Botany • Semester-wise Programme for 5-year Integrated M.Sc. (Hons) in Botany • Course curriculum for Award of 3-year B.Sc. Degree on opting out of 5-year integrated M.Sc. (Hons) Programme in Chemistry • Semester-wise Programme for 5-year Integrated M.Sc. (Hons) in Chemistry • Course curriculum for Award of 3-year B.Sc. Degree on opting out of 5-year integrated M.Sc. (Hons) Programme in Microbiology • Semester-wise Programme for 5-year Integrated M.Sc. (Hons) in Microbiology • Course curriculum for Award of 3-year B.Sc. Degree on opting out of 5-year integrated M.Sc. (Hons) Programme in Zoology • Semester-wise Programme for 5-year Integrated M.Sc. (Hons) in Zoology 371 COLLEGE OF BASIC SCIENCES AND HUMANITIES Basic Sciences provide scientific capital from which practical application of knowledge is drawn. Keeping in view the significance of basic sciences and humanities for proper understanding and development of different areas of agriculture and allied fields, the College of Basic Sciences and Humanities was established in October, 1965. Dr A S Kahlon was the founder Dean of the College and he continued in this position up to October, 1978.