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Nondecimal Positional Systems
M03_LONG0847_05_AIE_C03.qxd 9/17/07 10:38 AM Page 162 Not for Sale 162 CHAPTER 3 Numeration and Computation 3.2 Nondecimal Positional Systems In the preceding section, we discussed several numeration systems, including the decimal system in common use today. As already mentioned, the decimal system is a positional system based on 10. This is probably so because we have ten fingers and historically, as now, people often counted on their fingers. In the Did You Know box at the end of this section, we will give applications of positional systems that are based on whole numbers other than 10. One very useful application of bases other than ten is as a way to deepen our understanding of number systems, arithmetic in general, and our own decimal system. Although bases other than ten were in the K–5 curriculum in the past, they are not present currently. However, nondecimal positional systems are studied in the “math methods” course that is a part of an elementary education degree and may return to elementary school in the future. We will discuss addition and subtraction in base five in the next section and multiplication in base six in the following one. Base Five Notation For our purposes, perhaps the quickest route to understanding base five notation is to reconsider the abacus of Figure 3.1. This time, however, we allow only 5 beads to be moved forward on each wire. As before, we start with the wire to the students’ right (our left) and move 1 bead forward each time as we count 1, 2, 3, and so on. -
13054-Duodecimal.Pdf
Universal Multiple-Octet Coded Character Set International Organization for Standardization Organisation Internationale de Normalisation Международная организация по стандартизации Doc Type: Working Group Document Title: Proposal to encode Duodecimal Digit Forms in the UCS Author: Karl Pentzlin Status: Individual Contribution Action: For consideration by JTC1/SC2/WG2 and UTC Date: 2013-03-30 1. Introduction The duodecimal system (also called dozenal) is a positional numbering system using 12 as its base, similar to the well-known decimal (base 10) and hexadecimal (base 16) systems. Thus, it needs 12 digits, instead of ten digits like the decimal system. It is used by teachers to explain the decimal system by comparing it to an alternative, by hobbyists (see e.g. fig. 1), and by propagators who claim it being superior to the decimal system (mostly because thirds can be expressed by a finite number of digits in a "duodecimal point" presentation). • Besides mathematical and hobbyist publications, the duodecimal system has appeared as subject in the press (see e.g. [Bellos 2012] in the English newspaper "The Guardian" from 2012-12-12, where the lack of types to represent these digits correctly is explicitly stated). Such examples emphasize the need of the encoding of the digit forms proposed here. While it is common practice to represent the extra six digits needed for the hexadecimal system by the uppercase Latin capital letters A,B.C,D,E,F, there is no such established convention regarding the duodecimal system. Some proponents use the Latin letters T and E as the first letters of the English names of "ten" and "eleven" (which obviously is directly perceivable only for English speakers). -
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. -
Assessment of Options for Handling Full Unicode Character Encodings in MARC21 a Study for the Library of Congress
1 Assessment of Options for Handling Full Unicode Character Encodings in MARC21 A Study for the Library of Congress Part 1: New Scripts Jack Cain Senior Consultant Trylus Computing, Toronto 1 Purpose This assessment intends to study the issues and make recommendations on the possible expansion of the character set repertoire for bibliographic records in MARC21 format. 1.1 “Encoding Scheme” vs. “Repertoire” An encoding scheme contains codes by which characters are represented in computer memory. These codes are organized according to a certain methodology called an encoding scheme. The list of all characters so encoded is referred to as the “repertoire” of characters in the given encoding schemes. For example, ASCII is one encoding scheme, perhaps the one best known to the average non-technical person in North America. “A”, “B”, & “C” are three characters in the repertoire of this encoding scheme. These three characters are assigned encodings 41, 42 & 43 in ASCII (expressed here in hexadecimal). 1.2 MARC8 "MARC8" is the term commonly used to refer both to the encoding scheme and its repertoire as used in MARC records up to 1998. The ‘8’ refers to the fact that, unlike Unicode which is a multi-byte per character code set, the MARC8 encoding scheme is principally made up of multiple one byte tables in which each character is encoded using a single 8 bit byte. (It also includes the EACC set which actually uses fixed length 3 bytes per character.) (For details on MARC8 and its specifications see: http://www.loc.gov/marc/.) MARC8 was introduced around 1968 and was initially limited to essentially Latin script only. -
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. -
Proposal for a Gujarati Script Root Zone Label Generation Ruleset (LGR)
Proposal for a Gujarati Root Zone LGR Neo-Brahmi Generation Panel Proposal for a Gujarati Script Root Zone Label Generation Ruleset (LGR) LGR Version: 3.0 Date: 2019-03-06 Document version: 3.6 Authors: Neo-Brahmi Generation Panel [NBGP] 1 General Information/ Overview/ Abstract The purpose of this document is to give an overview of the proposed Gujarati LGR in the XML format and the rationale behind the design decisions taken. It includes a discussion of relevant features of the script, the communities or languages using it, the process and methodology used and information on the contributors. The formal specification of the LGR can be found in the accompanying XML document: proposal-gujarati-lgr-06mar19-en.xml Labels for testing can be found in the accompanying text document: gujarati-test-labels-06mar19-en.txt 2 Script for which the LGR is proposed ISO 15924 Code: Gujr ISO 15924 Key N°: 320 ISO 15924 English Name: Gujarati Latin transliteration of native script name: gujarâtî Native name of the script: ગજુ રાતી Maximal Starting Repertoire (MSR) version: MSR-4 1 Proposal for a Gujarati Root Zone LGR Neo-Brahmi Generation Panel 3 Background on the Script and the Principal Languages Using it1 Gujarati (ગજુ રાતી) [also sometimes written as Gujerati, Gujarathi, Guzratee, Guujaratee, Gujrathi, and Gujerathi2] is an Indo-Aryan language native to the Indian state of Gujarat. It is part of the greater Indo-European language family. It is so named because Gujarati is the language of the Gujjars. Gujarati's origins can be traced back to Old Gujarati (circa 1100– 1500 AD). -
Chapter 6 MISCELLANEOUS NUMBER BASES. the QUINARY
The Number Concept: Its Origin and Development By Levi Leonard Conant Ph. D. Chapter 6 MISCELLANEOUS NUMBER BASES. THE QUINARY SYSTEM. The origin of the quinary mode of counting has been discussed with some fulness in a preceding chapter, and upon that question but little more need be said. It is the first of the natural systems. When the savage has finished his count of the fingers of a single hand, he has reached this natural number base. At this point he ceases to use simple numbers, and begins the process of compounding. By some one of the numerous methods illustrated in earlier chapters, he passes from 5 to 10, using here the fingers of his second hand. He now has two fives; and, just as we say “twenty,” i.e. two tens, he says “two hands,” “the second hand finished,” “all the fingers,” “the fingers of both hands,” “all the fingers come to an end,” or, much more rarely, “one man.” That is, he is, in one of the many ways at his command, saying “two fives.” At 15 he has “three hands” or “one foot”; and at 20 he pauses with “four hands,” “hands and feet,” “both feet,” “all the fingers of hands and feet,” “hands and feet finished,” or, more probably, “one man.” All these modes of expression are strictly natural, and all have been found in the number scales which were, and in many cases still are, in daily use among the uncivilized races of mankind. In its structure the quinary is the simplest, the most primitive, of the natural systems. -
I – Mathematics in Design – SMTA1103
SCHOOL OF SCIENCE AND HUMANITIES DEPARTMENT OF MATHEMATICS UNIT – I – Mathematics in Design – SMTA1103 UNIT1- MATHEMATICS IN DESIGN Golden Ratio The ratio of two parts of a line such that the longer part divided by the smaller part is also equal to the whole length divided by the longer part. Results: i) The ratio of the circumference of a circle to its diameter is π . ii) The ratio of the width of a rectangular picture frame to its height is not necessarily equal to he golden ratio, though some artists and architects believe a frame made using the golden ratio makes the most pleasing and beautiful shape.iii) The ratio of two successive numbers of the Fibonacci sequence give an approximate value of the golden ratio,the bigger the pair of Fibonacci Numbers, the closer the approximation. Example1: The Golden Ratio = 1.61803398874989484820... = 1.618 correct to 3 decimal places. If x n are terms of the Fibonacci sequence, then what is the least value of n for which ( x n +1 / x n = 1.618 correct to 3 decimal places. Example2: A rectangle divided into a square and a smaller rectangle. If (x + y) /y = x / y ,find the value of golden ratio. Example3: The Golden Ratio = 1.61803398874989484820... = 1.6180 correct to 4 decimal places. If xn are terms of the Fibonacci sequence, then what is the least value of n for which (x n +1) / x n = 1.618 0 correct to 4 decimal places. Example4: Obtain golden ratio in a line segment. Proportion Proportion is an equation which defines that the two given ratios are equivalent to each other. -
The Counting System of the Incas in General, There Is a Lack of Material on the History of Numbers from the Pre-Columbian Americas
History of Numbers 1b. I can determine the number represented on an Inca counting board, and a quipu cord. The Counting System of the Incas In general, there is a lack of material on the history of numbers from the pre-Columbian Americas. Most of the information available concentrates on the eastern hemisphere, with Europe as the focus. Why? First, there has been a false assumption that indigenous peoples in the Americas lacked any specialized mathematics. Second, it seems at least some of their ancient mathematics has been intentionally kept secret. And Einally, when Europeans invaded and colonized the Americas, much of that evidence was most likely destroyed. In the 1960's, two researchers attempted to discover what mathematical knowledge was known by the Incas and how they used the Peruvian quipu, a counting system using cords and knots, in their mathematics. These researchers have come to certain beliefs about the quipu that are summarized below: Counting Boards It seems the Incas did not have a complicated system of computation. Whereas other peoples in the region, such as the Mayans, were doing computations related to their rituals and calendars, the Incas seem to have been more concerned with the task of record-keeping. To do this, they used what are called “quipu”(or khipu) to record quantities of items. However, they Eirst often needed to do computations whose results would be recorded on a quipu. To do these computations, they sometimes used a counting board or Yupana (from the Quechua word "yupay": count) made from a slab of wood or stone. -
The Unicode Standard, Version 4.0--Online Edition
This PDF file is an excerpt from The Unicode Standard, Version 4.0, issued by the Unicode Consor- tium and published by Addison-Wesley. The material has been modified slightly for this online edi- tion, however the PDF files have not been modified to reflect the corrections found on the Updates and Errata page (http://www.unicode.org/errata/). For information on more recent versions of the standard, see http://www.unicode.org/standard/versions/enumeratedversions.html. Many of the designations used by manufacturers and sellers to distinguish their products are claimed as trademarks. Where those designations appear in this book, and Addison-Wesley was aware of a trademark claim, the designations have been printed in initial capital letters. However, not all words in initial capital letters are trademark designations. The Unicode® Consortium is a registered trademark, and Unicode™ is a trademark of Unicode, Inc. The Unicode logo is a trademark of Unicode, Inc., and may be registered in some jurisdictions. The authors and publisher have taken care in preparation of this book, but make no expressed or implied warranty of any kind and assume no responsibility for errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of the use of the information or programs contained herein. The Unicode Character Database and other files are provided as-is by Unicode®, Inc. No claims are made as to fitness for any particular purpose. No warranties of any kind are expressed or implied. The recipient agrees to determine applicability of information provided. Dai Kan-Wa Jiten used as the source of reference Kanji codes was written by Tetsuji Morohashi and published by Taishukan Shoten. -
An Introduction to Indic Scripts
An Introduction to Indic Scripts Richard Ishida W3C [email protected] HTML version: http://www.w3.org/2002/Talks/09-ri-indic/indic-paper.html PDF version: http://www.w3.org/2002/Talks/09-ri-indic/indic-paper.pdf Introduction This paper provides an introduction to the major Indic scripts used on the Indian mainland. Those addressed in this paper include specifically Bengali, Devanagari, Gujarati, Gurmukhi, Kannada, Malayalam, Oriya, Tamil, and Telugu. I have used XHTML encoded in UTF-8 for the base version of this paper. Most of the XHTML file can be viewed if you are running Windows XP with all associated Indic font and rendering support, and the Arial Unicode MS font. For examples that require complex rendering in scripts not yet supported by this configuration, such as Bengali, Oriya, and Malayalam, I have used non- Unicode fonts supplied with Gamma's Unitype. To view all fonts as intended without the above you can view the PDF file whose URL is given above. Although the Indic scripts are often described as similar, there is a large amount of variation at the detailed implementation level. To provide a detailed account of how each Indic script implements particular features on a letter by letter basis would require too much time and space for the task at hand. Nevertheless, despite the detail variations, the basic mechanisms are to a large extent the same, and at the general level there is a great deal of similarity between these scripts. It is certainly possible to structure a discussion of the relevant features along the same lines for each of the scripts in the set. -
Decimal Notation in the United States
Decimal Notation in the United States In the United States, we use the decimal or period (“.”) to represent the difference between whole numbers and partial numbers. We use the comma (“,”) to separate groups of three places on the whole numbers side. This might be different from the way you currently write numbers. For example, the number 1 million: Some non-U.S. Counties: 1.000.000,00 U.S.: 1,000,000.00 It is also very important to know the names for the digit places in decimal notation. Here are the names in English for the United States: Thousandths Trillions Hundred Billions Ten Billions Billions Hundred Millions Ten Millions Millions Hundred Thousands Ten Thousands Thousands Hundreds Tens Ones Tenths Hundredths Thousandths Ten Hundred Thousandths Millionths 1 , 2 3 4 , 5 6 7 , 8 9 0 , 1 2 3 . 4 5 6 7 8 9 Other than the “ones” place, the places to the right of the decimal are almost the same as the places to the left of the decimal, but with an additional “th” added at the end. How to Say Common Math Terms in U.S. English Numbers: Big Numbers: In the U.S. we separate big numbers into groups of three of the following types: Ones, Thousands, Millions, Billions, … For example: 1,234,567 has three groups Millions: 1 Thousands: 234 Ones: 567 We say each group as a separate number, followed by the group type. So, we would say: “One million, two hundred and thirty-four thousand, five hundred and sixty-seven” Decimals: When there is a decimal, we say it a few different ways: 0.4 can be expressed as “four tenths,” “zero point four,” or just