The Quipu Is Most Commonly Known As the Numeric Recording System Of
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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. -
Reflections and Observations on Peru's Past and Present Ernesto Silva Kennesaw State University, [email protected]
Journal of Global Initiatives: Policy, Pedagogy, Perspective Volume 7 Number 2 Pervuvian Trajectories of Sociocultural Article 13 Transformation December 2013 Epilogue: Reflections and Observations on Peru's Past and Present Ernesto Silva Kennesaw State University, [email protected] Follow this and additional works at: https://digitalcommons.kennesaw.edu/jgi Part of the International and Area Studies Commons, and the Social and Cultural Anthropology Commons This work is licensed under a Creative Commons Attribution 4.0 License. Recommended Citation Silva, Ernesto (2013) "Epilogue: Reflections and Observations on Peru's Past and Present," Journal of Global Initiatives: Policy, Pedagogy, Perspective: Vol. 7 : No. 2 , Article 13. Available at: https://digitalcommons.kennesaw.edu/jgi/vol7/iss2/13 This Article is brought to you for free and open access by DigitalCommons@Kennesaw State University. It has been accepted for inclusion in Journal of Global Initiatives: Policy, Pedagogy, Perspective by an authorized editor of DigitalCommons@Kennesaw State University. For more information, please contact [email protected]. Emesto Silva Journal of Global Initiatives Volume 7, umber 2, 2012, pp. l83-197 Epilogue: Reflections and Observations on Peru's Past and Present Ernesto Silva 1 The aim of this essay is to provide a panoramic socio-historical overview of Peru by focusing on two periods: before and after independence from Spain. The approach emphasizes two cultural phenomena: how the indigenous peo ple related to the Conquistadors in forging a new society, as well as how im migration, particularly to Lima, has shaped contemporary Peru. This contribu tion also aims at providing a bibliographical resource to those who would like to conduct research on Peru. -
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). -
"Computers" Abacus—The First Calculator
Component 4: Introduction to Information and Computer Science Unit 1: Basic Computing Concepts, Including History Lecture 4 BMI540/640 Week 1 This material was developed by Oregon Health & Science University, funded by the Department of Health and Human Services, Office of the National Coordinator for Health Information Technology under Award Number IU24OC000015. The First "Computers" • The word "computer" was first recorded in 1613 • Referred to a person who performed calculations • Evidence of counting is traced to at least 35,000 BC Ishango Bone Tally Stick: Science Museum of Brussels Component 4/Unit 1-4 Health IT Workforce Curriculum 2 Version 2.0/Spring 2011 Abacus—The First Calculator • Invented by Babylonians in 2400 BC — many subsequent versions • Used for counting before there were written numbers • Still used today The Chinese Lee Abacus http://www.ee.ryerson.ca/~elf/abacus/ Component 4/Unit 1-4 Health IT Workforce Curriculum 3 Version 2.0/Spring 2011 1 Slide Rules John Napier William Oughtred • By the Middle Ages, number systems were developed • John Napier discovered/developed logarithms at the turn of the 17 th century • William Oughtred used logarithms to invent the slide rude in 1621 in England • Used for multiplication, division, logarithms, roots, trigonometric functions • Used until early 70s when electronic calculators became available Component 4/Unit 1-4 Health IT Workforce Curriculum 4 Version 2.0/Spring 2011 Mechanical Computers • Use mechanical parts to automate calculations • Limited operations • First one was the ancient Antikythera computer from 150 BC Used gears to calculate position of sun and moon Fragment of Antikythera mechanism Component 4/Unit 1-4 Health IT Workforce Curriculum 5 Version 2.0/Spring 2011 Leonardo da Vinci 1452-1519, Italy Leonardo da Vinci • Two notebooks discovered in 1967 showed drawings for a mechanical calculator • A replica was built soon after Leonardo da Vinci's notes and the replica The Controversial Replica of Leonardo da Vinci's Adding Machine . -
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. -
Suanpan” in Chinese)
Math Exercise on the Abacus (“Suanpan” in Chinese) • Teachers’ Introduction • Student Materials Introduction Cards 1-7 Practicing Basics Cards 8-11 Exercises Cards 12, 14, 16 Answer keys Cards 13, 15, 17 Learning: Card 18 “Up,” “Down,” “Rid,” “Advance” Exercises: Addition (the numbers 1-9) Cards 18-28 Advanced Addition Cards 29-30 Exercises: Subtraction Cards 31-39 (the numbers 1-9) Acknowledgment: This unit is adapted from A Children’s Palace, by Michele Shoresman and Roberta Gumport, with illustrations by Elizabeth Chang (University of Illinois Urbana-Champagne, Center for Asian Studies, Outreach Office, 3rd ed., 1986. Print edition, now out of print.) 1 Teachers’ Introduction: Level: This unit is designed for students who understand p1ace value and know the basic addition and subtraction facts. Goals: 1. The students will learn to manipulate one form of ca1cu1ator used in many Asian countries. 2. The concept of p1ace value will be reinforced. 3. The students will learn another method of adding and subtracting. Instructions • The following student sheets may be copied so that your students have individual sets. • Individual suanpan for your students can be ordered from China Sprout: http://www.chinasprout.com/shop/ Product # A948 or ATG022 Evaluation The students will be able to manipulate a suanpan to set numbers, and to do simple addition and subtraction problems. Vocabulary suanpan set beam rod c1ear ones rod tens rod hundreds rod 2 Card 1 Suanpan – Abacus The abacus is an ancient calculator still used in China and other Asian countries. In Chinese it is called a “Suanpan.” It is a frame divided into an upper and lower section by a bar called the “beam.” The abacus can be used for addition, subtraction, multiplication, and division. -
An EM Construal of the Abacus
An EM Construal of the Abacus 1019358 Abstract The abacus is a simple yet powerful tool for calculations, only simple rules are used and yet the same outcome can be derived through various processes depending on the human operator. This paper explores how a construal of the abacus can be used as an educational tool but also how ‘human computing’ is illustrated. The user can gain understanding regarding how to operate the abacus for addition through a constructivist approach but can also use the more informative material elaborated in the EMPE context. 1 Introduction encapsulates all of these components together. The abacus can, not only be used for addition and 1.1 The Abacus subtraction, but also for multiplication, division, and even for calculating the square and cube roots of The abacus is a tool engineered to assist with numbers [3]. calculations, to improve the accuracy and the speed The Japanese also have a variation of the to complete such calculations at the same time to abacus known as the Soroban where the only minimise the mental calculations involved. In a way, difference from the Suanpan is each column an abacus can be thought of as one’s pen and paper contains one less Heaven and Earth bead [4]. used to note down the relevant numerals for a calculation or the modern day electronic calculator we are familiar with. However, the latter analogy may be less accurate as an electronic calculator would compute an answer when a series of buttons are pressed and requires less human input to perform the calculation compared to an abacus. -
Ophir De España & Fernando De Montesinos's Divine
OPHIR DE ESPAÑA & FERNANDO DE MONTESINOS’S DIVINE DEFENSE OF THE SPANISH COLONIAL EMPIRE: A MYSTERIOUS ANCESTRAL MERGING OF PRE-INCA AND CHRISTIAN HISTORIES by NATHAN JAMES GORDON A.A., Mt. San Jacinto College, 2006 B.A., University of Colorado, 2010 M.A., University of Colorado, 2012 A thesis submitted to the Faculty of the Graduate School of the University of Colorado in partial fulfillment of the requirment for the degree of Doctor of Philosophy Department of Spanish and Portuguese 2017 This thesis entitled: Ophir de España & Fernando de Montesinos’s Divine Defense of the Spanish Colonial Empire: A Mysterious Ancestral Merging of pre-Inca and Christian Histories written by Nathan James Gordon has been approved for the Department of Spanish and Portuguese Andrés Prieto Leila Gómez Gerardo Gutiérrez Núria Silleras-Fernández Juan Dabove Date The final copy of this thesis has been examined by the signatories, and we find that both the content and the form meet acceptable presentation standards of scholarly work in the above mentioned discipline. ii ABSTRACT Gordon, Nathan James (Ph.D., Spanish Literature, Department of Spanish and Portuguese) Ophir de España & Fernando de Montesinos’s Divine Defense of the Spanish Colonial Empire: A Mysterious Ancestral Merging of pre-Inca and Christian Histories Thesis directed by Associate Professor Andrés Prieto Over the last two centuries, Books I and III of Ophir de España: Memorias historiales y políticas del Perú (1644) by Fernando de Montesinos have been generally overlooked. The cause of this inattention is associated with the mysterious and unique pre-Columbian historical account from Book II, which affords the most extensive version of Andean genealogy. -
The What and Why of Whole Number Arithmetic: Foundational Ideas from History, Language and Societal Changes
Portland State University PDXScholar Mathematics and Statistics Faculty Fariborz Maseeh Department of Mathematics Publications and Presentations and Statistics 3-2018 The What and Why of Whole Number Arithmetic: Foundational Ideas from History, Language and Societal Changes Xu Hu Sun University of Macau Christine Chambris Université de Cergy-Pontoise Judy Sayers Stockholm University Man Keung Siu University of Hong Kong Jason Cooper Weizmann Institute of Science SeeFollow next this page and for additional additional works authors at: https:/ /pdxscholar.library.pdx.edu/mth_fac Part of the Science and Mathematics Education Commons Let us know how access to this document benefits ou.y Citation Details Sun X.H. et al. (2018) The What and Why of Whole Number Arithmetic: Foundational Ideas from History, Language and Societal Changes. In: Bartolini Bussi M., Sun X. (eds) Building the Foundation: Whole Numbers in the Primary Grades. New ICMI Study Series. Springer, Cham This Book Chapter is brought to you for free and open access. It has been accepted for inclusion in Mathematics and Statistics Faculty Publications and Presentations by an authorized administrator of PDXScholar. Please contact us if we can make this document more accessible: [email protected]. Authors Xu Hu Sun, Christine Chambris, Judy Sayers, Man Keung Siu, Jason Cooper, Jean-Luc Dorier, Sarah Inés González de Lora Sued, Eva Thanheiser, Nadia Azrou, Lynn McGarvey, Catherine Houdement, and Lisser Rye Ejersbo This book chapter is available at PDXScholar: https://pdxscholar.library.pdx.edu/mth_fac/253 Chapter 5 The What and Why of Whole Number Arithmetic: Foundational Ideas from History, Language and Societal Changes Xu Hua Sun , Christine Chambris Judy Sayers, Man Keung Siu, Jason Cooper , Jean-Luc Dorier , Sarah Inés González de Lora Sued , Eva Thanheiser , Nadia Azrou , Lynn McGarvey , Catherine Houdement , and Lisser Rye Ejersbo 5.1 Introduction Mathematics learning and teaching are deeply embedded in history, language and culture (e.g. -
Directions for Knots: Reef, Bowline, and the Figure Eight
Directions for Knots: Reef, Bowline, and the Figure Eight Materials Two ropes, each with a blue end and a red end (try masking tape around the ends and coloring them with markers, or using red and blue electrical tape around the ends.) Reef Knot (square knot) 1. Hold the red end of the rope in your left hand and the blue end in your right. 2. Cross the red end over the blue end to create a loop. 3. Pass the red end under the blue end and up through the loop. 4. Pull, but not too tight (leave a small loop at the base of your knot). 5. Hold the red end in your right hand and the blue end in your left. 6. Cross the red end over and under the blue end and up through the loop (here, you are repeating steps 2 and 3) 7. Pull Tight! Bowline The bowline knot (pronounced “bow-lin”) is a loop knot, which means that it is tied around an object or tied when a temporary loop is needed. On USS Constitution, sailors used bowlines to haul heavy loads onto the ship. 1. Hold the blue end of the rope in your left hand and the red end in your right. 2. Cross the red end over the blue end to make a loop. 3. Tuck the red end up and through the loop (pull, but not too tight!). 4. Keep the blue end of the rope in your left hand and the red in your right. 5. Pass the red end behind and around the blue end. -
Knot Masters Troop 90
Knot Masters Troop 90 1. Every Scout and Scouter joining Knot Masters will be given a test by a Knot Master and will be assigned the appropriate starting rank and rope. Ropes shall be worn on the left side of scout belt secured with an appropriate Knot Master knot. 2. When a Scout or Scouter proves he is ready for advancement by tying all the knots of the next rank as witnessed by a Scout or Scouter of that rank or higher, he shall trade in his old rope for a rope of the color of the next rank. KNOTTER (White Rope) 1. Overhand Knot Perhaps the most basic knot, useful as an end knot, the beginning of many knots, multiple knots make grips along a lifeline. It can be difficult to untie when wet. 2. Loop Knot The loop knot is simply the overhand knot tied on a bight. It has many uses, including isolation of an unreliable portion of rope. 3. Square Knot The square or reef knot is the most common knot for joining two ropes. It is easily tied and untied, and is secure and reliable except when joining ropes of different sizes. 4. Two Half Hitches Two half hitches are often used to join a rope end to a post, spar or ring. 5. Clove Hitch The clove hitch is a simple, convenient and secure method of fastening ropes to an object. 6. Taut-Line Hitch Used by Scouts for adjustable tent guy lines, the taut line hitch can be employed to attach a second rope, reinforcing a failing one 7. -
Viracocha Christ Among the Ancient Peruvians?
Viracocha Christ among the Ancient Peruvians? Scott Hoyt There came from a southern direction a white man of great stature, who, by his aspect and presence, called forth great veneration and obedience. This man who thus appeared had great power, insomuch that he could change plains into mountains, and great hills into valleys, and make water flow out of stones. As soon as such power was beheld, the people called him the Maker of created things, the Prince of all things, Father of the Sun. For they say that he performed other wonders, giving life to men and animals, so that by his hand marvellous great benefits were conferred on the people. In many places he gave orders to men how they should live, and he spoke lovingly to them . admonishing them that they should do good . and that they should be loving and charitable to all. In most parts he is generally called Ticiviracocha. [And] that wherever [he] . came and there were sick, he healed them, and where there were blind he gave them sight by only uttering words.1 —Pedro de Cieza de León, Catholic Historian, 1550 iracocha was the principal deity of ancient Peru, and according to Vthe cronistas (Catholic historians, mostly priests, arriving in Peru 1. Pedro de Cieza de León, The Second Part of the Chronicle of Peru (El Seno- rio de los Incas), trans. and ed. Clements R. Markham (1874; London: Hakluyt Society, 1883; repr. Boston: Elibron Classics, 1999), 5–6. Senorio was originally handwritten in 1550. Citations are to the 1999 edition. BYU Studies Quarterly 54, no.