DOCSLIB.ORG

EGYPTIAN MATHEMATICS Jens Høyrup

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
EGYPTIAN MATHEMATICS Jens Høyrup 8 EGYPTIAN MATHEMATICS Jens Høyrup Ivor Grattan-Guinness in memoriam The primary references of the term “Egyptian mathematics” are the computa- tional techniques and the underlying mathematical knowledge attested in Pharaonic written sources. Secondary references are, on one hand, the corre- sponding techniques etc. as known from demotic sources and, on the other, the geometrical procedures used in Pharaonic and subsequent architecture and visual arts. Greek mathematics producedinHellenisticEgyptisthusnot included. Accordingly, all dates in the following are bce when not specified to be ce. THE SOURCES The most important written sources for Pharaonic mathematics are the Rhind Mathematical Papyrus (henceforth RMP)1 and the Moscow Mathematical 1 Two editions with ample analysis exist: T. Eric Peet, The Rhind Mathematical Papyrus, British Museum 10057 and 10058. Introduction, Transcription, Translation and Commentary (London: University Press of Liverpool, 1923); Arnold Buffum Chace, The Rhind Mathematical Papyrus. British Museum 10057 and 10058,2vols.(Oberlin,OH:MathematicalAssociationofAmerica,1927–9),vol.1(withtheassistance of Henry Parker Manning): Free Translation and Commentary and a Bibliography of Egyptian Mathematics by R. C. Archibald; vol. 2 (with Ludlow Bull and Henry Parker Manning): Photographs, Transcription, Transliteration, Literal Translation and a Bibliography of Egyptian and Babylonian Mathematics (Supplement),byR.C.ArchibaldandtheMathematical Leather Roll in the British Museum, by S. R. K. Glanville (the literal translation is indeed very literal; it is used in all quotations below from RMP). A recent facsimile edition with description and discussion and translation of large extracts is GayRobinsandCharlesShute,The Rhind Mathematical Papyrus: An Ancient Egyptian Text (London: British Museum Publications, 1987).Valuable general accounts of Egyptian mathematics which (by necessity) deal extensively with RMP are: Kurt Vogel, Vorgriechische Mathematik. Volume 1: Vorgeschichte und Ägypten (Mathematische Studienhefte, 1; Hanover: Hermann Schroedel / Paderborn: Ferdinand Schöningh, 1958); Richard J. Gillings, Mathematics in the Time of the Pharaohs (Cambridge, MA: MIT Press, 1972). After the manuscript for the present article was finished (essentially in 1997), Marshall Clagett has published Ancient Egyptian Science. A Source Book. Volume 3: Ancient Egyptian Mathematics (Memoirs of the American Philosophical Society, 232; Philadelphia, PA: American Philosophical Society, 1999). This volume reproduces the facsimile edition of Chace’svolume2and includes a translation kept close to that of the same volume as well as an extensive analysis. Extensive analysis of RMP as well as other Middle Kingdom sources is also found in Annette Imhausen, Ägyptische Algorithmen. Eine Untersuchung zu den mittelägyptischen mathematischen Aufgabentexten (Ägyptologische Abhandlungen, 65; Wiesbaden: Harrassowitz, 2003). Egyptian mathematics in its social setting is dealt 144 Egyptian Mathematics 145 Papyrus (MMP).2 To these can be added a few shorter papyri containing mathematical problems; a couple of listings of equivalent fractions; and a larger number of accounting papyri that apply the metrology and show how the basic arithmetical techniques were used. The RMP is a copy from a Middle Kingdom original (Amenemhet III, c. 1800), made during the Hyksos period; it is a teacher’sorcalculator’s manual, containing several tables (to which we shall return) and some eighty problems with solution. The MMP seems to be a late Middle Kingdom copy from an earlier Middle Kingdom original; it is a collection of student’s answers to problems, provided with the teacher’s approval (at times justly refused). The other properly mathematical sources are from the Middle through New Kingdom.3 A source of particular character is the New Kingdom fictional “satirical letter” or Papyrus Anastasi I,4 in which a scribe chides a colleague for his professional ignorance; it shows that a military scribe was supposed to be familiar with Palestinian geography and with the determination of the manpower, rations, and other requirements of construction work. Accounting papyri show that other categories of scribes had analogous tasks. Some administrative records go back to the Old Kingdom (and a few docu- ments containing numbers to the early dynastic period); even in this respect, however, the Middle through New Kingdom is much richer. The mathematical sources of the Pharaonic period are written in hieratic script (evidently, some hieroglyphic documents contain numbers); part of the metrological terminology seems to have been created in hieratic and to have acquired hieroglyphic equivalents only at a later moment (when at all). with in Annette Imhausen, Mathematics in Ancient Egypt: A Contextual History (Princeton & Oxford: Princeton University Press, 2016). 2 Edition with translation and extensive commentary: W. W. Struve, Mathematischer Papyrus des Staatlichen Museums der Schönen Künste in Moskau (Quellen und Studien zur Geschichte der Mathematik. Abteilung A: Quellen, 1. Band; Berlin: Julius Springer, 1930). The edition is reproduced in Marshall Clagett, Ancient Egyptian Science, vol. 3, which also contains an English translation and a commentary. 3 Marshall Clagett, following recent research, gives the following approximate dates (Ancient Egyptian Science. A Source Book. Volume I: Knowledge and Order (Memoirs of the American Philosophical Society, 184 A+B; Philadelphia, PA: American Philosophical Society, 1989), pp. 629–35): Early dynastic period (dynasties 1–2): 3110–2665. Old Kingdom (dynasties 3–8): 2664–2155. First intermediate period (dynasties 9–10): 2154–2052. Middle Kingdom (dynasties 11–13): 2040–1640. Second intermediate period (Hyksos dynasties 15–16, Theban dynasty 17): 1640–1532. New Kingdom (dynasties 18–20): 1550–1070. Third intermediate period (dynasties 21–(initial) 25): 1070–712. Late period (dynasties (final) 25–31, including the Assyrian hegemony during dynasty 26 and the Persian dynasties 27 and 31): 712–332. Greco-Roman period: 332 bce to 395 ce. 4 Full edition and translation in Alan H. Gardiner, Egyptian Hieratic Texts. Series I: Literary Texts from the New Kingdom. Part I: The Papyrus Anastasi I and the Papyrus Koller, together with Parallel Texts (Leipzig: J. C. Hinrichs’sche Buchhandlung, 1911). 146 Jens Høyrup From the demotic phase (late and Greco-Roman periods), several papyri containing mathematical problems and tables survive5 – with the possible exception of one uncertain fragment all of Greco-Roman date. NUMBERS AND METROLOGY The Egyptian number system was decadic. Already in early dynastic times, individual signs for 1, 10, 100, 1,000, 10,000, 100,000, and 1,000,000 existed.6 In hieroglyphic, other numbers were constructed additively, by mere repetition of these signs; in hieratic, thesewerecontractedintoindividualsigns for 1, 2, ...,9,10,20,..., 90, 100, 200, etc. From the Middle Kingdom onward, fractional numbers were expressed as sums of aliquot parts (including two-thirds); in order to keep close to theEgyptiannotationwemaytranscribe ″ 2 ′ 1 ′ 1 1 ″ ′ ′ them as follows: 3 (= /3), 2 (= /2), 3 (= /3), 4(= /4), 5,etc.(3 ,2,and3 had special signs); the others were denoted by the sign ro (“mouth,” here “part”)or by a dot above the number (in hieroglyphic respectively hieratic writing), according to a canon that did not allow repetition of the same aliquot part but expressed for instance 5 5as3′ 15 (juxtaposition means addition). Essential metrologies, however, would operate with subunits instead of these fractions. Many of the problems in the mathematical texts deal with the difficulties to which the non-decadic metrologies would give rise. Closest to decadic principles is the length system. The basic length unit was the “royal cubit” “ ” “fi ” “ ” (mh˙ , c. 52cm), subdivided into 7 palms of 4 ngers each (a short cubit of 6 palms was also in use). 100 royal cubits was a “rope” (khet). Land might be measured in setat (that is, square khet), divided by successive halvings into 1 subunits with special names (down to /32 setat); in surveying practice, the “cubit of land” (1 cubit versus 1 khet) and the “thousand of land” (1,000 cubit versus 1 khet) were mostly preferred. The central capacity unit was the h̩ ek̩ at, divided according to one system into 10 henu or 320 ro (the “part” again, but in a different use), according to another 1 7 by successive halvings (down to /64). Multiples of the h̩ ek̩ at were expressed by special signs or by non-standard use of the standard numerals. A special unit for 2 bulkysubstancesisthekhar,equalto20h̩ ek̩ at and to /3 of a cubic royal cubit (probably a secondary normalization of originally distinct units). 5 Edition with translation and extensive commentary in Richard A. Parker (ed.), Demotic Mathematical Papyri (Providence, RI and London: Brown University Press, 1972). 6 The standard reference for Egyptian numerals and number words remains Kurt Sethe, Von Zahlen und Zahlworten bei den Alten Ägyptern, und was für andere Völker und Sprachen daraus zu lernen ist (Schriften der Wissenschaftlichen Gesellschaft in Straßburg, 25; Straßburg: Karl J. Trübner, 1916). In the second millennium, the sign for 1,000,000, and afterwards that for 100,000, went out of use; instead, multiplicative notations were used. 7 The hieroglyphic signs for the successive halves of the h̩ ek̩ at can be put together to the standard representation of the healing sacred eye of Horus, as pointed out by Peet, The Rhind Mathematical Papyrus,
Load more

Recommended publications
  • Numbers - the Essence
    Numbers - the essence A computer consists in essence of a sequence of electrical on/off switches. Big question: How do switches become a computer? ITEC 1000 Introduction to Information Technologies 1 Friday, September 25, 2009 • A single switch is called a bit • A sequence of 8 bits is called a byte • Two or more bytes are often grouped to form word • Modern laptop/desktop standardly will have memory of 1 gigabyte = 1,000,000,000 bytes ITEC 1000 Introduction to Information Technologies 2 Friday, September 25, 2009 Question: How do encode information with switches ? Answer: With numbers - binary numbers using only 0’s and 1’s where for each switch on position = 1 off position = 0 ITEC 1000 Introduction to Information Technologies 3 Friday, September 25, 2009 • For a byte consisting of 8 bits there are a total of 28 = 256 ways to place 0’s and 1’s. Example 0 1 0 1 1 0 1 1 • For each grouping of 4 bits there are 24 = 16 ways to place 0’s and 1’s. • For a byte the first and second groupings of four bits can be represented by two hexadecimal digits • this will be made clear in following slides. ITEC 1000 Introduction to Information Technologies 4 Friday, September 25, 2009 Bits & Bytes imply creation of data and computer analysis of data • storing data (reading) • processing data - arithmetic operations, evaluations & comparisons • output of results ITEC 1000 Introduction to Information Technologies 5 Friday, September 25, 2009 Numeral systems A numeral system is any method of symbolically representing the counting numbers 1, 2, 3, 4, 5, .
    [Show full text]
  • On the Origin of the Indian Brahma Alphabet
    - ON THE <)|{I<; IN <>F TIIK INDIAN BRAHMA ALPHABET GEORG BtfHLKi; SECOND REVISED EDITION OF INDIAN STUDIES, NO III. TOGETHER WITH TWO APPENDICES ON THE OKU; IN OF THE KHAROSTHI ALPHABET AND OF THK SO-CALLED LETTER-NUMERALS OF THE BRAHMI. WITH TIIKKK PLATES. STRASSBUKi-. K A K 1. I. 1 1M I: \ I I; 1898. I'lintccl liy Adolf Ilcil/.haiisi'ii, Vicniiii. Preface to the Second Edition. .As the few separate copies of the Indian Studies No. Ill, struck off in 1895, were sold very soon and rather numerous requests for additional ones were addressed both to me and to the bookseller of the Imperial Academy, Messrs. Carl Gerold's Sohn, I asked the Academy for permission to issue a second edition, which Mr. Karl J. Trlibner had consented to publish. My petition was readily granted. In addition Messrs, von Holder, the publishers of the Wiener Zeitschrift fur die Kunde des Morgenlandes, kindly allowed me to reprint my article on the origin of the Kharosthi, which had appeared in vol. IX of that Journal and is now given in Appendix I. To these two sections I have added, in Appendix II, a brief review of the arguments for Dr. Burnell's hypothesis, which derives the so-called letter- numerals or numerical symbols of the Brahma alphabet from the ancient Egyptian numeral signs, together with a third com- parative table, in order to include in this volume all those points, which require fuller discussion, and in order to make it a serviceable companion to the palaeography of the Grund- riss.
    [Show full text]
  • 1 Oriental Metrology and the Politics of Antiquity
    1 Oriental Metrology and the Politics of Antiquity in Nineteenth-century Survey Sciences Simon Schaffer University of Cambridge E-mail: [email protected] Argument Metrological techniques to establish shared quantitative measures have often been seen as signs of rational modernisation. The cases considered here show instead the close relation of such techniques with antiquarian and revivalist programmes under imperial regimes. Enterprises in survey sciences in Egypt in the wake of the French invasion of 1798 and in India during the East India Company’s revenue surveys involved the promotion of a new kind of oriental metrology designed to represent colonisers’ measures as restorations of ancient values to be applied to current systems of survey and measurement. Surveyors’ practice and hardware help clarify the significance of the complex historical and political functions of scientific standards. The balance of the paper discusses the survey work of later nineteenth century indigenous Egyptian astronomers at a conjuncture of major economic and political dislocation to explore the various versions of antiquity at stake in these metrological programmes. 2 Introduction: survey sciences and metrology’s invention “Egyptian genius always seems to take pleasure in veiling from the world the principle of its lovely creations, concealing it from profane eyes, perhaps so as better to give them a divine origin, keep them pure and guard them from time’s injuries. Thus one sees in use in Egypt, but without being able to understand the principle, a measurement system apparently crude but in fact the most exact of all known systems”: Mahmud al-Falaki, “The current Egyptian measurement system” (Mahmud 1873, 67) Metrological equipment relies on material measures that somehow embody agreed standards used by a specific community to help make its world knowable in quantitative form.
    [Show full text]
  • Lecture 2: Arithmetic
    E-320: Teaching Math with a Historical Perspective Oliver Knill, 2010-2017 Lecture 2: Arithmetic The oldest mathematical discipline is arithmetic. It is the theory of the construction and manip- ulation of numbers. The earliest steps were done by Babylonian, Egyptian, Chinese, Indian and Greek thinkers. Building up the number system starts with the natural numbers 1; 2; 3; 4::: which can be added and multiplied. Addition is natural: join 3 sticks to 5 sticks to get 8 sticks. Multiplication ∗ is more subtle: 3 ∗ 4 means to take 3 copies of 4 and get 4 + 4 + 4 = 12 while 4 ∗ 3 means to take 4 copies of 3 to get 3 + 3 + 3 + 3 = 12. The first factor counts the number of operations while the second factor counts the objects. To motivate 3 ∗ 4 = 4 ∗ 3, spacial insight motivates to arrange the 12 objects in a rectangle. This commutativity axiom will be carried over to larger number systems. Realizing an addition and multiplicative structure on the natural numbers requires to define 0 and 1. It leads naturally to more general numbers. There are two major motivations to to build new numbers: we want to 1. invert operations and still get results. 2. solve equations. To find an additive inverse of 3 means solving x + 3 = 0. The answer is a negative number. To solve x ∗ 3 = 1, we get to a rational number x = 1=3. To solve x2 = 2 one need to escape to real numbers. To solve x2 = −2 requires complex numbers. Numbers Operation to complete Examples of equations to solve Natural numbers addition and multiplication 5 + x = 9 Positive fractions addition and
    [Show full text]
  • Arabic Numeral
    CHAPTER 4 Number Representation and Calculation Copyright © 2015, 2011, 2007 Pearson Education, Inc. Section 4.4, Slide 1 4.4 Looking Back at Early Numeration Systems Copyright © 2015, 2011, 2007 Pearson Education, Inc. Section 4.4, Slide 2 Objectives 1. Understand and use the Egyptian system. 2. Understand and use the Roman system. 3. Understand and use the traditional Chinese system. 4. Understand and use the Ionic Greek system. Copyright © 2015, 2011, 2007 Pearson Education, Inc. Section 4.4, Slide 3 The Egyptian Numeration System The Egyptians used the oldest numeration system called hieroglyphic notation. Copyright © 2015, 2011, 2007 Pearson Education, Inc. Section 4.4, Slide 4 Example: Using the Egyptian Numeration System Write the following numeral as a Hindu-Arabic numeral: Solution: Using the table, find the value of each of the Egyptian numerals. Then add them. 1,000,000 + 10,000 + 10,000 + 10 + 10 + 10 + 1 + 1 + 1 + 1 = 1,020,034 Copyright © 2015, 2011, 2007 Pearson Education, Inc. Section 4.4, Slide 5 Example: Using the Egyptian Numeration System Write 1752 as an Egyptian numeral. Solution: First break down the Hindu-Arabic numeral into quantities that match the Egyptian numerals: 1752 = 1000 + 700 + 50 + 2 = 1000 + 100 + 100 + 100 + 100 + 100 + 100 + 100 + 10 + 10 + 10 + 10 + 10 + 1 + 1 Now use the table to find the Egyptian symbol that matches each quantity. Thus, 1752 can be expressed as Copyright © 2015, 2011, 2007 Pearson Education, Inc. Section 4.4, Slide 6 The Roman Numeration System Roman I V X L C D M Numeral Hindu- 1 5 10 50 100 500 1000 Arabic Numeral The Roman numerals were used until the eighteenth century and are still commonly used today for outlining, on clocks, and in numbering some pages in books.
    [Show full text]
  • 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).
    [Show full text]
  • Different and Dissonant Values in Measuring Dimensions in Ancient Egypt “A Comparative Study with Contemporary Measurements”
    Annals of Archaeology Volume 3, Issue 1, 2020, PP 12-29 ISSN 2639-3662 Different and Dissonant Values in Measuring Dimensions in Ancient Egypt “A Comparative Study with Contemporary Measurements” Dr. Ayman Waziry* Professor of Archaeology/Egyptology, Faculty of Archaeology, Fayoum University, Main building of Faculty of Archeology, Postcode: 63514, Al-Fayoum City, Fayoum Governorate, Egypt *Corresponding Author: Dr. Ayman Waziry, Professor of Archaeology/Egyptology, Faculty of Archaeology, Fayoum University, Main building of Faculty of Archeology, Postcode: 63514, Al- Fayoum City, Fayoum Governorate, Egypt ABSTRACT This paper focuses on How there are different and dissonant values in measuring dimensions in ancient Egypt? The ancient Egyptians relied on a natural method to measure dimensions like the arm that was used as a measure of length, approximately equal to the length of a forearm. Traditionally, it was the length from the bent elbow to the tips of the fingers. Typically, almost 18 inches or 44 cm, however there was a long cubit of about 21 inches or 52 cm. The second natural method was the width of the palm of the hand. As well as, the human fingers used as digits for measuring width, where the four digits equal the sign of one palm and seven palms equals one cubit. In present-day trigonometry, cotangent requires same units for both horizontal run and vertical rise, however ancient sources like Rhind Papyrus uses palms for the run and cubits for the rise, resulting in these different, yet characteristic mathematics. In ancient Egypt there were seven palms in a cubit, in addition to the seqed that was seven times the cotangent.
    [Show full text]
  • Mathematics in Ancient Egypt: a Contextual History (Book Review)
    Digital Collections @ Dordt Faculty Work Comprehensive List 6-8-2016 Mathematics in Ancient Egypt: A Contextual History (Book Review) Calvin Jongsma Dordt College, [email protected] Follow this and additional works at: https://digitalcollections.dordt.edu/faculty_work Part of the Christianity Commons Recommended Citation Jongsma, C. (2016). Mathematics in Ancient Egypt: A Contextual History (Book Review). MAA Reviews Retrieved from https://digitalcollections.dordt.edu/faculty_work/515 This Book Review is brought to you for free and open access by Digital Collections @ Dordt. It has been accepted for inclusion in Faculty Work Comprehensive List by an authorized administrator of Digital Collections @ Dordt. For more information, please contact [email protected]. Mathematics in Ancient Egypt: A Contextual History (Book Review) Abstract Reviewed Title: Mathematics in Ancient Egypt: A Contextual History by Annette Imhausen. Princeton, NJ: Princeton University Press, 2016. 234 pp. ISBN: 9780691117133. Keywords book review, Annette Imhausen, Mathematics in Ancient Egypt, history Disciplines Christianity Comments Access book review from publisher's site: http://www.maa.org/press/maa-reviews/mathematics-in-ancient-egypt-a-contextual-history This book review is available at Digital Collections @ Dordt: https://digitalcollections.dordt.edu/faculty_work/515 Mathematics in Ancient Egypt: A Contextual History Mathematics in Ancient Egypt: A Contextual History is one of the latest outstanding monographs in history of mathematics published by Princeton University Press. Over the last decade and a half Princeton has produced around three dozen new books in this area, several of them definitive contributions to our understanding of ancient and non-Western mathematics. Annette Imhausen’s book does for ancient Egyptian mathematics what Eleanor Robson’s did for Mesopotamian mathematics and Kim Plofker’s did for Indian mathematics: it delivers a deeply informed up-to-date contextual history and overview of a culture’s practice and development of mathematics.
    [Show full text]
  • Ancient Egyptian Cubits – Origin and Evolution
    Ancient Egyptian Cubits – Origin and Evolution by Antoine Pierre Hirsch A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Near and Middle Eastern Civilizations University of Toronto © Copyright by Antoine Pierre Hirsch 2013 i Ancient Egyptian Cubits – Origin and Evolution Antoine Pierre Hirsch Doctor of Philosophy Near and Middle Eastern Civilizations University of Toronto 2013 Abstract This thesis suggests that prior to Ptolemaic and Roman times, ancient Egypt had two distinct and parallel linear systems: the royal system limited to official architectural projects and land measurements, and a great (aA) system used for everyday measurements. A key 1/3 ratio explains ancient Egyptian linear measurements and their agricultural origin. Emmer is 1/3 lighter than barley, consequently, for an equal weight, a container filled with emmer will be 1/3 greater than a container filled with barley. The lengths derived from both containers share the same 1/3 ratio. The second chapter, Previous Studies, lists the work of scholars involved directly or indirectly with ancient Egyptian metrology. The third chapter, The Royal Cubit as a Converter and the Scribe’s Palette as a Measuring Device, capitalizes on the colour scheme (black and white on the reproduction of Appendix A) appearing on the Amenemope cubit artifact to show the presence of two cubits and two systems: the black (royal system) and the white (great [aA] system) materialized by the scribe's palette of 30, 40, and 50 cm. The royal cubit artifacts provide a conversion bridge between the royal and the great systems. The information derived from the visual clues on the Amenemope cubit artifact are tested against a database of artifacts scattered in museums around the world.
    [Show full text]
  • Count Like an Egyptian
    The School and University Partnership for Educational Renewal in Mathematics An NSF-funded Graduate STEM Fellows in K–12 Education Project University of Hawai‘i at Manoa,¯ Department of Mathematics Count Like an Egyptian Grade Levels This lesson is intended for grades 2-4. Common Core Standard(s) • 2nd Grade: – 2.NBT.A.1.B - The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones). – 2.NBT.A.4 - Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the results of comparisons. • 4th Grade: – 4.NBT.A.2 - Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form. Compare two multi-digit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons. Objectives and Topics Students use a place-value strategy to compare and order multi-digit whole numbers. Materials and Resources • access to reference books, articles, etc. that describe how different cultures have used symbols to represent numbers (optional) • Internet access (optional) • world map or globe • base-10 blocks Introduction and Outline This lesson can be used as the mathematical element of a cross-curricular unit. UHM Department of Mathematics 1 Dayna Kitsuwa superm.math.hawaii.edu Grade Level: 2-4 The School and University Partnership for Educational Renewal in Mathematics An NSF-funded Graduate STEM Fellows in K–12 Education Project University of Hawai‘i at Manoa,¯ Department of Mathematics 0.1 Student Investigation (optional) How do other cultures from the past and present use symbols to represent the number 4,125? Project and read the investigation question.
    [Show full text]
  • The Egyptian Numeration System
    www.wrightmath.info/media/Math 230/The Egyptian Number System.pdf The Egyptian Numeration System Egyptian numerals have been found on the writings on the stones of monument walls of ancient time. Numbers have also been found on pottery, limestone plaques, and on the fragile fibers of the papyrus. The language is composed of heiroglyphs, pictorial signs that represent people, animals, plants, and numbers. The Egyptians used a written numeration that was changed into hieroglyphic writing, which enabled them to note whole numbers to 1,000,000. The system had a decimal base and allowed for the additive principle. In this notation there was a special sign for every power of ten. For I, a vertical line; for 10, a sign with the shape of an upside down U; for 100, a spiral rope; for 1000, a lotus blossom; for 10,000 , a raised finger, slightly bent; for 100,000 , a tadpole; and for 1,000,000, an “astonished man” with upraised arms. Decimal Egyptian Number Symbol 1 = staff 10 = heel bone 100 = coil of rope 1000 = lotus flower 10,000 = pointing finger 100,000 = tadpole 1,000,000 = astonished man This hieroglyphic numeration was a written version of a concrete counting system using material objects, i.e., a “grouping tally system”. To represent a number, the sign for each decimal order was repeated as many times as necessary. To make it easier to read the repeated signs they were placed in groups of two, three, or four and arranged vertically. Project note: The Rhind papyrus and the Moscow papyrus are two ancient Egyptian documents that show the solutions to quite a number of mathematical problems, some practical in nature and some purely investigating the nature of numbers.
    [Show full text]
  • A Study of Our Present Numbering System: an Histo
    A STUDY OF OUR PRESENT NUMBERING SYSTEM: AN HISTO~t CAL A~RO&~H A THESIS SUBMITTED TO THE FACULTY OF ATLANTA UNIVERSITY IN PARTIAL FUIJFILLMBNT OF THE REQUIREMBNTS FOR THE DE(~?.EE OF MASTER OF SCIENCE BY MAE FRANCES WIlSON DEPA1~rMBNT OF MATHEMATICS ATLANTA, GEORGIA AUGUST 196t~ ii ~‘- TABLE OF CONTENTS Page LIST OF FIGURES. • . • • • • Iii Chapter I. INTRODUCTION. .• . 1 II. NU_~4ERATION . 3 III. SYSTE~ ND THEIR PROPERTIES. 10 B IBLIOGf?JkFHY. , • • . • • 22 ii EGYPTIAN NUMERALS I I A VtR.TICaL 10 (1 A HE~i.- BONE 10 A [email protected] I. 0~ A~~° A LÔTUS 1. ü~ I k POINTIkI6 IU A 6u9%8o1 FsU 1. 0 ~ A- (4111” 1W A-$TON~$JtM~VT Figure 1 iii AT ~ a.xn~ç~ ill III UU~Od,d, III UUU~Ø~ =(Ub4(QIJS-~o I) b4 (~c?I)/ ~b..cb1 l(1IVVU&~ =(V(oI)E09~h~Z III L)ULI = (i)S4-(o~)L 5L IIIL)L) (,)~-f-(oOZ = Ii V :~7:J ~ ~ (~~o” # ~ 0/)£ ~ (‘i, 01)1 ~1Q £ I S?~I~EJWnN NVIJ~J~-D~ A 6 —~(4oV4Q14.Q1 AAA >~~PI~P’>I~ L2 Q~. — Q?4O~ 1,1 A> II > 0l AAIAA~AA1L b AAAAAAA L 9 S ~AAA ~A4 Al Z A I ~~Wri/q fV~bNINQ7A~’d~ STht~3’~WnN ~io~ I~T~U1N ~TvINQ~UEya OOb, L~ Ob e b c9oo~’ LL8 ILOOL cOL X009 09 9 c?,QQS (LOS (LOOI7 71 Oh ~L- OQE D OO~ )I OZ 0~ 00/ 7 0/ ~IY~{~W11t~ )I~O S a.zn2T~ COb 00~ CCL 009 00.9 OOtT 001 002 00/ L~ L UQLL~ S~T3~QNnH Oh Q~OL 09 OS 0*701 OZ 0? SN3± b~~9S-&~ZI qLL[LLLLCt~~ S~LINfl $IV~WflN ~ 9 e.in2t~ ~,1- co’ 4- 01 itt b 37d WYX~ 9 -ç 4.7 z I 5 7j1≥13(AJPAJ 3~3/IVd4’I— 3S3IVIfI) 741>13W fiN ~‘flQ ~I~HflN ~S~vdvf-~9~NIHD L ‘~T~!i •~7’~74,4J Qb’ FQ~/ ~v ~, Ob~ ~J9h-6 ~ZL ‘q~’ wi ~44OA/ -% A’~VW..t f7~fs~3,4/g4/ ~ b VA L C’? ~ ~4 I Q’b’ 914 ~9A 000/ P~ ~ $~TVV~flN ~IEWV-flG$EIR CHI&PrER I INTRODUCTION The meaningful approach to the teaching of arithmetic is widely accepted.
    [Show full text]