Basic Digit Sets for Radix Representation
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Zero Displacement Ternary Number System: the Most Economical Way of Representing Numbers
Revista de Ciências da Computação, Volume III, Ano III, 2008, nº3 Zero Displacement Ternary Number System: the most economical way of representing numbers Fernando Guilherme Silvano Lobo Pimentel , Bank of Portugal, Email: [email protected] Abstract This paper concerns the efficiency of number systems. Following the identification of the most economical conventional integer number system, from a solid criteria, an improvement to such system’s representation economy is proposed which combines the representation efficiency of positional number systems without 0 with the possibility of representing the number 0. A modification to base 3 without 0 makes it possible to obtain a new number system which, according to the identified optimization criteria, becomes the most economic among all integer ones. Key Words: Positional Number Systems, Efficiency, Zero Resumo Este artigo aborda a questão da eficiência de sistemas de números. Partindo da identificação da mais económica base inteira de números de acordo com um critério preestabelecido, propõe-se um melhoramento à economia de representação nessa mesma base através da combinação da eficiência de representação de sistemas de números posicionais sem o zero com a possibilidade de representar o número zero. Uma modificação à base 3 sem zero permite a obtenção de um novo sistema de números que, de acordo com o critério de optimização identificado, é o sistema de representação mais económico entre os sistemas de números inteiros. Palavras-Chave: Sistemas de Números Posicionais, Eficiência, Zero 1 Introduction Counting systems are an indispensable tool in Computing Science. For reasons that are both technological and user friendliness, the performance of information processing depends heavily on the adopted numbering system. -
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. -
Number Systems and Radix Conversion
Number Systems and Radix Conversion Sanjay Rajopadhye, Colorado State University 1 Introduction These notes for CS 270 describe polynomial number systems. The material is not in the textbook, but will be required for PA1. We humans are comfortable with numbers in the decimal system where each po- sition has a weight which is a power of 10: units have a weight of 1 (100), ten’s have 10 (101), etc. Even the fractional apart after the decimal point have a weight that is a (negative) power of ten. So the number 143.25 has a value that is 1 ∗ 102 + 4 ∗ 101 + 3 ∗ 0 −1 −2 2 5 10 + 2 ∗ 10 + 5 ∗ 10 , i.e., 100 + 40 + 3 + 10 + 100 . You can think of this as a polyno- mial with coefficients 1, 4, 3, 2, and 5 (i.e., the polynomial 1x2 + 4x + 3 + 2x−1 + 5x−2+ evaluated at x = 10. There is nothing special about 10 (just that humans evolved with ten fingers). We can use any radix, r, and write a number system with digits that range from 0 to r − 1. If our radix is larger than 10, we will need to invent “new digits”. We will use the letters of the alphabet: the digit A represents 10, B is 11, K is 20, etc. 2 What is the value of a radix-r number? Mathematically, a sequence of digits (the dot to the right of d0 is called the radix point rather than the decimal point), : : : d2d1d0:d−1d−2 ::: represents a number x defined as follows X i x = dir i 2 1 0 −1 −2 = ::: + d2r + d1r + d0r + d−1r + d−2r + ::: 1 Example What is 3 in radix 3? Answer: 0.1. -
Scalability of Algorithms for Arithmetic Operations in Radix Notation∗
Scalability of Algorithms for Arithmetic Operations in Radix Notation∗ Anatoly V. Panyukov y Department of Computational Mathematics and Informatics, South Ural State University, Chelyabinsk, Russia [email protected] Abstract We consider precise rational-fractional calculations for distributed com- puting environments with an MPI interface for the algorithmic analysis of large-scale problems sensitive to rounding errors in their software imple- mentation. We can achieve additional software efficacy through applying heterogeneous computer systems that execute, in parallel, local arithmetic operations with large numbers on several threads. Also, we investigate scalability of the algorithms for basic arithmetic operations and methods for increasing their speed. We demonstrate that increased efficacy can be achieved of software for integer arithmetic operations by applying mass parallelism in het- erogeneous computational environments. We propose a redundant radix notation for the construction of well-scaled algorithms for executing ba- sic integer arithmetic operations. Scalability of the algorithms for integer arithmetic operations in the radix notation is easily extended to rational- fractional arithmetic. Keywords: integer computer arithmetic, heterogeneous computer system, radix notation, massive parallelism AMS subject classifications: 68W10 1 Introduction Verified computations have become indispensable tools for algorithmic analysis of large scale unstable problems (see e.g., [1, 4, 5, 7, 8, 9, 10]). Such computations require spe- cific software tools; in this connection, we mention that our library \Exact computa- tion" [11] provides appropriate instruments for implementation of such computations ∗Submitted: January 27, 2013; Revised: August 17, 2014; Accepted: November 7, 2014. yThe author was supported by the Federal special-purpose program "Scientific and scientific-pedagogical personnel of innovation Russia", project 14.B37.21.0395. -
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. -
Number Systems Radix Number Systems Radix Number Systems
Number Systems 4/26/2010 Number Systems Radix Number Systems binary, octal, and hexadecimal numbers basic idea of a radix number system ‐‐ why used how do we count: conversions, including to/from decimal didecimal : 10 dis tinc t symblbols, 0‐9 negative binary numbers when we reach 9, we repeat 0‐9 with 1 in front: 0,1,2,3,4,5,6,7,8,9, 10,11,12,13, …, 19 floating point numbers then with a 2 in front, etc: 20, 21, 22, 23, …, 29 character codes until we reach 99 then we repeat everything with a 1 in the next position: 100, 101, …, 109, 110, …, 119, …, 199, 200, … other number systems follow the same principle with a different base (radix) Lubomir Bic 1 Lubomir Bic 2 Radix Number Systems Radix Number Systems octal: we have only 8 distinct symbols: 0‐7 binary: we have only 2 distinct symbols: 0, 1 when we reach 7, we repeat 0‐7 with 1 in front why?: digital computer can only represent 0 and 1 012345670,1,2,3,4,5,6,7, 10,11,12,13, …, 17 when we reach 1, we repeat 0‐1 with 1 in front then with a 2 in front, etc: 20, 21, 22, 23, …, 27 0,1, 10,11 until we reach 77 then we repeat everything with a 1 in the next position: then we repeat everything with a 1 in the next position: 100, 101, 110, 111, 1000, 1001, 1010, 1011, 1100, … 100, 101, …, 107, 110, …, 117, …, 177, 200, … decimal to binary correspondence: decimal to octal correspondence: D 01234 5 6 7 8 9 101112… D 0 1 … 7 8 9 10 11 … 15 16 17 … 23 24 … 63 64 … B 0 1 10 11 100 101 110 111 1000 1001 1010 1011 1100 … O 0 1 … 7 10 11 12 13 … 17 20 21 … 27 30 … 77 100 … -
United States Patent 15 3,700,872 May [45] Oct
United States Patent 15 3,700,872 May [45] Oct. 24, 1972 54 RADX CONVERSON CRCUTS Primary Examiner - Maynard R. Wilbur Assistant Examiner- Leo H. Boudreau (72) Inventor: Frederick T. May, Austin,Tex. Attorney-Hanifin and Clark and D. Kendall Cooper (73) Assignee: International Business Corporation, Armonk, N.Y. 57) ABSTRACT 22 Filed: Aug. 22, 1969 Data conversion circuits with optimized common hardware convert numbers expressed in a first radix C (21) Appl. No.: 852,272 to other radices n1, m2, etc., with the mode of opera Related U.S. Application Data tion being controlled to establish a radix C to radix ml conversion in one mode, a radix C to radix n2 conver 63 Continuation-in-part of Ser. No. 517,764, Dec. sion in another mode, etc., on a selective basis, as 30, 1965, abandoned. desired. In a first embodiment, numbers represented in a binary (base 2) radix Care converted to a base 10 52 U.S. Ct.............. 235/155,340/347 DD, 235/165 (n1) or base 12(m2) representation. In a second em 51 int. Cl.............................................. HO3k 13/24 bodiment, numbers stored in a ternary (base 3) radix 58 Field of Search......... 340/347 DD; 235/155, 165 C representation are converted to a base 12 (n1) or base 10 (m2) representation. The circuits are 56 References Cited predicated upon recognition of the fact that a number UNITED STATES PATENTS represented in a first radix can be converted readily to a second radix using shared hardware if the following 2,620,974 12/1952 Waltat.................... 23.5/155 X equation is satisfied: 2,831,179 4/1958 Wright et al.......... -
A Useful Tool for Rhinoplasty to Correct High Radix
Brazilian Journal of Otorhinolaryngology 87 (2021) 59---65 Brazilian Journal of OTORHINOLARYNGOLOGY www.bjorl.org ORIGINAL ARTICLE Radix saw: a useful tool for rhinoplasty to correct high radix a b,∗ c Süreyya ¸eneldirS , Denizhan Dizdar , Altu˘g Tuna a Süreyya ¸eneldirS Clinic, Istanbul, Turkey b ˙Istinye University, Faculty of Medicine, Bahc¸elievler Medical Park Hospital, Department of Otolaryngology, ˙Istanbul, Turkey c Private Office, Frankfurt, Germany Received 13 February 2019; accepted 26 June 2019 Available online 7 August 2019 KEYWORDS Abstract Rhinoplasty; Introduction: The most difficult aspect of radix lowering is determining the maximum amount Radix; of bone that can be removed with osteotomes; here, we describe use of a radix saw, which is Nose a new tool for determining this amount. Objective: In this study, we describe use of a radix saw, which is a new tool to reduce the radix. Methods: The medical charts of 96 patients undergoing surgery to lower a high radix between 2016 and 2017 were assessed retrospectively. All operations were performed by the senior surgeon. Outcomes were assessed by comparing preoperative photographs with the most recent follow-up photographs (minimum of 6 months postoperatively). The photographs were all taken using the same imaging settings, and with consistent subject distance and angulation. The photographs were subsequently analysed by authors. Results: The study population consisted of 96 patients (70 women, 26 men) who underwent rhinoplasty between 2016 and 2017. The mean age of the patients was 28.8 years (range: 18---50 years) and the mean clinical follow-up period was 1.8 years. No patient required revision surgery due to radix problems, and there were no cases with unwanted bone fragments or radix asymmetry. -
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. -
About Numbers How These Basic Tools Appeared and Evolved in Diverse Cultures by Allen Klinger, Ph.D., New York Iota ’57
About Numbers How these Basic Tools Appeared and Evolved in Diverse Cultures By Allen Klinger, Ph.D., New York Iota ’57 ANY BIRDS AND Representation of quantity by the AUTHOR’S NOTE insects possess a The original version of this article principle of one-to-one correspondence 1 “number sense.” “If is on the web at http://web.cs.ucla. was undoubtedly accompanied, and per- … a bird’s nest con- edu/~klinger/number.pdf haps preceded, by creation of number- mtains four eggs, one may be safely taken; words. These can be divided into two It was written when I was a fresh- but if two are removed, the bird becomes man. The humanities course had an main categories: those that arose before aware of the fact and generally deserts.”2 assignment to write a paper on an- the concept of number unrelated to The fact that many forms of life “sense” thropology. The instructor approved concrete objects, and those that arose number or symmetry may connect to the topic “number in early man.” after it. historic evolution of quantity in differ- At a reunion in 1997, I met a An extreme instance of the devel- classmate from 1954, who remem- ent human societies. We begin with the bered my paper from the same year. opment of number-words before the distinction between cardinal (counting) As a pack rat, somehow I found the abstract concept of number is that of the numbers and ordinal ones (that show original. Tsimshian language of a tribe in British position as in 1st or 2nd). -
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 .......................... -
Curiosities Regarding the Babylonian Number System
Curiosities Regarding the Babylonian Number System Sherwin Doroudi April 12, 2007 By 2000 bce the Babylonians were already making significant progress in the areas of mathematics and astronomy and had produced many elaborate mathematical tables on clay tablets. Their sexagecimal (base-60) number system was originally inherited from the Sumerian number system dating back to 3500 bce. 1 However, what made the Babylonian number system superior to its predecessors, such as the Egyptian number system, was the fact that it was a positional system not unlike the decimal system we use today. This innovation allowed for the representation of numbers of virtually any size by reusing the same set of symbols, while also making it easier to carry out arithmetic operations on larger numbers. Its superiority was even clear to later Greek astronomers, who would use the sexagecimal number system as opposed to their own native Attic system (the direct predecessor to Roman numerals) when making calculations with large numbers. 2 Most other number systems throughout history have made use of a much smaller base, such as five (quinary systems), ten (decimal systems), or in the case of the Mayans, twenty (vigesimal systems), and such choices for these bases are all clearly related to the fingers on the hands. The choice (though strictly speaking, it's highly unlikely that any number system was directly \chosen") of sixty is not immediately apparent, and at first, it may even seem that sixty is a somewhat large and unwieldy number. Another curious fact regarding the ancient number system of the Babylonians is that early records do not show examples of a \zero" or null place holder, which is an integral part of our own positional number system.