Vigesimal Numerals on Ifẹ̀ (Togo) and Ifẹ̀ (Nigeria) Dialects of Yorùbá
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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. -
The Hexadecimal Number System and Memory Addressing
C5537_App C_1107_03/16/2005 APPENDIX C The Hexadecimal Number System and Memory Addressing nderstanding the number system and the coding system that computers use to U store data and communicate with each other is fundamental to understanding how computers work. Early attempts to invent an electronic computing device met with disappointing results as long as inventors tried to use the decimal number sys- tem, with the digits 0–9. Then John Atanasoff proposed using a coding system that expressed everything in terms of different sequences of only two numerals: one repre- sented by the presence of a charge and one represented by the absence of a charge. The numbering system that can be supported by the expression of only two numerals is called base 2, or binary; it was invented by Ada Lovelace many years before, using the numerals 0 and 1. Under Atanasoff’s design, all numbers and other characters would be converted to this binary number system, and all storage, comparisons, and arithmetic would be done using it. Even today, this is one of the basic principles of computers. Every character or number entered into a computer is first converted into a series of 0s and 1s. Many coding schemes and techniques have been invented to manipulate these 0s and 1s, called bits for binary digits. The most widespread binary coding scheme for microcomputers, which is recog- nized as the microcomputer standard, is called ASCII (American Standard Code for Information Interchange). (Appendix B lists the binary code for the basic 127- character set.) In ASCII, each character is assigned an 8-bit code called a byte. -
Mathematics in African History and Cultures
Paulus Gerdes & Ahmed Djebbar MATHEMATICS IN AFRICAN HISTORY AND CULTURES: AN ANNOTATED BIBLIOGRAPHY African Mathematical Union Commission on the History of Mathematics in Africa (AMUCHMA) Mathematics in African History and Cultures Second edition, 2007 First edition: African Mathematical Union, Cape Town, South Africa, 2004 ISBN: 978-1-4303-1537-7 Published by Lulu. Copyright © 2007 by Paulus Gerdes & Ahmed Djebbar Authors Paulus Gerdes Research Centre for Mathematics, Culture and Education, C.P. 915, Maputo, Mozambique E-mail: [email protected] Ahmed Djebbar Département de mathématiques, Bt. M 2, Université de Lille 1, 59655 Villeneuve D’Asq Cedex, France E-mail: [email protected], [email protected] Cover design inspired by a pattern on a mat woven in the 19th century by a Yombe woman from the Lower Congo area (Cf. GER-04b, p. 96). 2 Table of contents page Preface by the President of the African 7 Mathematical Union (Prof. Jan Persens) Introduction 9 Introduction to the new edition 14 Bibliography A 15 B 43 C 65 D 77 E 105 F 115 G 121 H 162 I 173 J 179 K 182 L 194 M 207 N 223 O 228 P 234 R 241 S 252 T 274 U 281 V 283 3 Mathematics in African History and Cultures page W 290 Y 296 Z 298 Appendices 1 On mathematicians of African descent / 307 Diaspora 2 Publications by Africans on the History of 313 Mathematics outside Africa (including reviews of these publications) 3 On Time-reckoning and Astronomy in 317 African History and Cultures 4 String figures in Africa 338 5 Examples of other Mathematical Books and 343 -
The Chinese Rod Numeral Legacy and Its Impact on Mathematics* Lam Lay Yong Mathematics Department National University of Singapore
The Chinese Rod Numeral Legacy and its Impact on Mathematics* Lam Lay Yong Mathematics Department National University of Singapore First, let me explain the Chinese rod numeral system. Since the Warring States period {480 B.C. to 221 B.C.) to the 17th century A.D. the Chinese used a bundle of straight rods for computation. These rods, usually made from bamboo though they could be made from other materials such as bone, wood, iron, ivory and jade, were used to form the numerals 1 to 9 as follows: 1 2 3 4 5 6 7 8 9 II Ill Ill I IIIII T II Note that for numerals 6 to 9, a horizontal rod represents the quantity five. A numeral system which uses place values with ten as base requires only nine signs. Any numeral of such a system is formed from among these nine signs which are placed in specific place positions relative to each other. Our present numeral system, commonly known as the Hindu-Arabic numeral system, is based on this concept; the value of each numeral determines the choice of digits from the nine signs 1, 2, ... , 9 anq their place positions. The place positions are called units, tens, hundreds, thousands, and so on, and each is occupied by at most one digit. The Chinese rod system employs the same concept. However, since its nine signs are formed from rod tallies, if a number such as 34 were repre sented as Jll\IU , this would inevitably lead to ambiguity and confusion. To * Text of Presidential Address delivered at the Society's Annual General Meeting on 20 March 1987. -
Outlining the Following Shows How Outline Format Is Set up and Used
Outlining The following shows how outline format is set up and used: I. Main point follows a Roman numeral. A. Minor points follow behind capital letters. B. Each minor point must refer to or be a part of the major point above 1. If there are subpoints below the minor point, use Arabic numerals. 2. Notice that each point is indented from according to its importance. 3. Each subpoint must be related to or a part of the minor point it follows. a. If there are points below subpoints, they use lower case letters. b. They are indented below the subpoint and are related to the subpoint or part of the subpoint above it. 1.) Sometimes, there are even smaller subdivisions, 2.) And these use Arabic numerals with a parenthesis. C. The next minor point below the major point II. Next Major point follows Roman numeral II. A. Minor point B. Minor point 1. Subpoint 2. Subpoint Then this same pattern and format continues until the outline is complete. Often when preparing to write an essay, the major points match the key ideas that would be in the topic sentences for the paragraphs of the essay. The first Roman numeral would be for the introductory paragraph, the next for the first body paragraph, the third for the second body paragraph, with the total number of Roman Numerals equal to the number of paragraphs in the essay. Outlines can also be used as a guide for oral presentations or for other kinds of reporting, and sometimes outlines serve as the final communication. -
Roman Numeral Chart
Roman Numeral Chart Rule: When Roman Numerals are written as capital letters, a single capital sign should me used before a single letter, and a double capital sign should be used before numerals containing two or more letters. Uncapitalized Roman numerals of one or more letters should be preceded by the letter sign. I = 1 V = 5 X = 10 L = 50 C = 100 D = 500 M = 1000 1 I ,I 36 XXXVI ,,XXXVI 71 LXXI ,,LXXI 2 II ,,II 37 XXXVII ,,XXXVII 72 LXXII ,,LXXII 3 III ,,III 38 XXXVIII ,,XXXVIII 73 LXXIII ,,LXXIII 4 IV ,,IV 39 XXXIX ,,XXXIX 74 LXXIV ,,LXXIV 5 V ,V 40 XL ,,XL 75 LXXV ,,LXXV 6 VI ,,VI 41 XLI ,,XLI 76 LXXVI ,,LXXVI 7 VII ,,VII 42 XLII ,,XLII 77 LXXVII ,,LXXVII 8 VIII ,,VIII 43 XLIII ,,XLIII 78 LXXVIII ,,LXXVIII 9 IX ,,IX 44 XLIV ,,XLIV 79 LXXIX ,,LXXIX 10 X ,X 45 XLV ,,XLV 80 LXXX ,,LXXX 11 XI ,,XI 46 XLVI ,,XLVI 81 LXXXI ,,LXXXI 12 XII ,,XII 47 XLVII ,,XLVII 82 LXXXII ,,LXXXII 13 XIII ,,XIII 48 XLVIII ,,XLVIII 83 LXXXIII ,,LXXXIII 14 XIV ,,XIV 49 XLIX ,,XLIX 84 LXXXIV ,,LXXXIV 15 XV ,,XV 50 L ,,L 85 LXXXV ,,LXXXV 16 XVI ,,XVI 51 LI ,,LI 86 LXXXVI ,,LXXXVI 17 XVII ,,XVII 52 LII ,,LII 87 LXXXVII ,,LXXXVII 18 XVIII ,,XVIII 53 LIII ,,LIII 88 LXXXVIII ,,LXXXVIII 19 XIX ,,XIX 54 LIV ,,LIV 89 LXXXIX ,,LXXXIX 20 XX ,,XX 55 LV ,,LV 90 XC ,,XC 21 XXI ,,XXI 56 LVI ,,LVI 91 XCI ,,XCI 22 XXII ,,XXII 57 LVII ,,LVII 92 XCII ,XCII 23 XXIII ,,XXIII 58 LVIII ,,LVIII 93 XCIII ,XCIII 24 XXIV ,,XXIV 59 LIX ,,LIX 94 XCIV ,XCIV 25 XXV ,,XXV 60 LX ,,LX 95 XCV ,XCV 26 XXVI ,,XXVI 61 LXI ,,LXI 96 XCVI ,XCVI 27 XXVII ,,XXVII 62 LXII ,,LXII 97 XCVII ,XCVII 28 XXVIII ,,XXVIII 63 LXIII ,,LXIII 98 XCVIII ,XCVIII 29 XXIX ,,XXIX 64 LXIV ,,LXIV 99 XCIX ,XCIX 30 XXX ,,XXX 65 LXV ,,LXV 100 C ,C 31 XXXI ,,XXXI 66 LXVI ,,LXVI 101 CI ,CI 32 XXXII ,,XXXII 67 LXVII ,,LXVII 150 CL ,CL 33 XXXIII ,,XXXIII 68 LXVIII ,,LXVIII 200 CC ,CC 34 XXXIV ,,XXXIV 69 LXIX ,,LXIX 501 DI ,DI 35 XXXV ,,XXXV 70 LXX ,,LXX 530 DXXX ,DXXX . -
Writing Numerals 0 to 5 PK.CC.2
New York State Common Core PK GRADE Mathematics Curriculum KK GRADE PK • MODULE 5 Topic A Writing Numerals 0 to 5 PK.CC.2 Focus Standard: PK.CC.2 Represent a number of objects with a written numeral 0–5 (with 0 representing a count of no objects). Instructional Days: 5 Coherence -Links to: GK–M1 Numbers to 10 Children have been working with the numerals 1–5 since Module 1, as well as the number 0 since Module 3. Throughout the year, they have had many opportunities to match a number of objects with a prewritten numeral and develop their fine motor skills. In Topic A, children build on this experience and begin to write numerals 0–5 (PK.CC.2). Lesson 1 begins with writing 0 and 1. The number 1 is particularly easy to write; it is a simple stroke from top to bottom. Often, certain numerals are already familiar because of their similarities to letters. For example, 0 looks like the letter O. During this first lesson, students focus on starting all of their numbers from the top because that has a significant impact on their ability to write efficiently in later grades. One numeral is introduced in each of the subsequent lessons in the topic (Lesson 2 focuses on 2, and so on). Numeral formation is taught using simple rhymes. Children may use tactile materials, such as clay, dough, sand, or rice, to remember the strokes for each numeral. They write using personal white boards and crayons or pencils on paper. Students first trace the numeral and then practice writing it independently using a writing rectangle and a marked starting point to help prevent reversals. -
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. -
Yorcjba Journal of Yoruba Studies Association of Nigeria
yOrCjbA Journal of Yoruba Studies Association of Nigeria UNIVERSITY OF IBADAN LIBRARY June 2013 Vol. 7 No. 2 ISSN: 1595-1324 Yoruba Journal of the Yoruba Studies Association of Nigeria UNIVERSITY OF IBADAN LIBRARY Vol. 7 No. 2 ISSN 1595-1324 June 2013 Yoruba Studies Association of Nigeria 2013 (Egbe Ommo-Ede Yoruba, Naijma) AH rights reserved. No part of this publication may be reproduced or transmitted in any form or by any mear permission. without First published June 2013 Direct all enquiries to: The Editor in-Chief Prof. Arinpe G. Adejumo Head, Department of Linguistics and African Languages University of Ibadan b 8 Ibadan UNIVERSITY OF IBADAN LIBRARY Printed in Nigeria by Agora Company, Ibadan Akoonu (Contents) Bisi Ogunsina Ojuuwoye Yoruba Lori Oro Aabo Ati ifarada Esin Nmu idagbasoke Orile-Ede - Pg. 1 Olajuyin Oyebanji Ipa Ti Ede Abinibi N Ko Lawujo - Pg. 25 Goke Alamu Ipo Asa Nmu Iselu ati Aabo Orile-ede Naijma - Pg. 35 Okewande Oluwole Tewogboye Iha Ti Yoruba Ko Si ifarada Esin Nmu Ero Ati Igbagbo Won - Pg 54 Philip Adedotun Ogundeji Ire And Ibi: The Yoruba Concepts of Good and Bad - Pg. 79 UNIVERSITY OFTaiwo IBADAN Adekemi LIBRARY Kd Rin Kd Po: A Novelist’s Perspective On Security - Pg. Ill IV Layo Ogunlola Yoruba Playwrights And The Spirit of Omoluabi: Discourse in Archetypal Theory - Pg 139 Taiwo Soneye A Review of T.M. Ilesanmi's Obinrin: A Cultural Assessment of Yoruba Women - Pg 184 Dr. Adeyemi Adeyinka A Review of Fehintola Mosadomi's Yoruba Ye Mb. A Beginning Yoruba Textbook - Pg 206 UNIVERSITY OF IBADAN LIBRARY V A Review of Fehintola Mosadomi’s Yorubd Ye Mi: A Beginning Yorubd Textbook Publisher- The University of Texas at Austin Year of Publication- 2011 Page Extent- 328 pages Place of Publication- United States of America Reviewer- Dr. -
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.