Number Systems
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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. -
Introduction to Numerical Computing Number Systems in the Decimal System We Use the 10 Numeric Symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 to Represent Numbers
Statistics 580 Introduction to Numerical Computing Number Systems In the decimal system we use the 10 numeric symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 to represent numbers. The relative position of each symbol determines the magnitude or the value of the number. Example: 6594 is expressible as 6594 = 6 103 + 5 102 + 9 101 + 4 100 Example: · · · · 436.578 is expressible as 436:578 = 4 102 + 3 101 + 6 100 + 5 10−1 + 7 10−2 + 8 10−3 · · · · · · We call the number 10 the base of the system. In general we can represent numbers in any system in the polynomial form: z = ( akak− a a : b b bm )B · · · 1 · · · 1 0 0 1 · · · · · · where B is the base of the system and the period in the middle is the radix point. In com- puters, numbers are represented in the binary system. In the binary system, the base is 2 and the only numeric symbols we need to represent numbers in binary are 0 and 1. Example: 0100110 is a binary number whose value in the decimal system is calculated as follows: 0 26 + 1 25 + 0 24 + 0 23 + 1 22 + 1 21 + 0 20 = 32 + 4 + 2 · · · · · · · = (38)10 The hexadecimal system is another useful number system in computer work. In this sys- tem, the base is 16 and the 16 numeric and arabic symbols used to represent numbers are: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F Example: Let 26 be a hexadecimal number. -
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. -
Data Representation in Computer Systems
© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION There are 10 kinds of people in the world—those who understand binary and those who don’t. © Jones & Bartlett Learning, LLC © Jones—Anonymous & Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION © Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION CHAPTER Data Representation © Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC NOT FOR SALE 2OR DISTRIBUTIONin ComputerNOT FOR Systems SALE OR DISTRIBUTION © Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION 2.1 INTRODUCTION he organization of any computer depends considerably on how it represents T numbers, characters, and control information. The converse is also true: Stan- © Jones & Bartlettdards Learning, and conventions LLC established over the years© Jones have determined& Bartlett certain Learning, aspects LLC of computer organization. This chapter describes the various ways in which com- NOT FOR SALE putersOR DISTRIBUTION can store and manipulate numbers andNOT characters. FOR SALE The ideas OR presented DISTRIBUTION in the following sections form the basis for understanding the organization and func- tion of all types of digital systems. The most basic unit of information in a digital computer is called a bit, which is a contraction of binary digit. In the concrete sense, a bit is nothing more than © Jones & Bartlett Learning, aLLC state of “on” or “off ” (or “high”© Jones and “low”)& Bartlett within Learning, a computer LLCcircuit. In 1964, NOT FOR SALE OR DISTRIBUTIONthe designers of the IBM System/360NOT FOR mainframe SALE OR computer DISTRIBUTION established a conven- tion of using groups of 8 bits as the basic unit of addressable computer storage. -
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.......... -
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. -
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. -
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). -
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. -
A Recursive Radix Conversion Formula and Its Application to Multiplication and Division
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Camp. R Morhs. bith Appls., Vol. 2. pp. ZJS-265 Pergamon Press 1976. Pnnted in Great Britain A RECURSIVE RADIX CONVERSION FORMULA AND ITS APPLICATION TO MULTIPLICATION AND DIVISION H. ASAI Lockheed Electronics Company, Houston, Texas 77058,U.S.A. Communicated by E. Y. Rodin (Received March 1976) Abstract-A recursive formula for number conversion from one radix representation to another radix representation is presented. This formula differs from the existing ones in two major aspects. First, it utilizes a digit shift technique which provides faster accumulation of higher significant digits in the final result. Second, it is suitable for parallel computation, so that the length of time necessary for number conversion can be shortened. Thus the longer the digit number, the more appreciation in conversion time-saving will result. Applications of the recursive formula are studied in multiplication and division for negative radix numbers as well as for positive radix numbers. The multiplication and division presented here are especially useful for computations of 7 word precision, since successive bit-carrying propagation to the most significant digit hardly ever occurs. 1. INTRODUCTION Internal number representation in a digital computer is usually different from decimal representation which has been utilized in our daily lives for centuries. Thus, a conversion process from decimal to internal representation and vice versa is always necessary in order to perform a computation within a digital computer. Binary representation is employed in most digital computers for hardware simplicity, so that the conversions from decimal to binary (or its power representation 2’ for i = 2, 3,.