Number and Operation, Handbook for Elementary Mathematics Workshops
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
From Al-Khwarizmi to Algorithm (71-74)
Olympiads in Informatics, 2017, Vol. 11, Special Issue, 71–74 71 © 2017 IOI, Vilnius University DOI: 10.15388/ioi.2017.special.11 From Al-Khwarizmi to Algorithm Bahman MEHRI Sharif University of Technology, Iran e-mail: [email protected] Mohammad ibn Musa al-Khwarizmi (780–850), Latinized as Algoritmi, was a Persian mathematician, astronomer, and geographer during the Abbasid Caliphate, a scholar in the House of Wisdom in Baghdad. In the 12th century, Latin translations of his work on the Indian numerals introduced the decimal number system to the Western world. Al-Khwarizmi’s The Compendious Book on Calculation by Completion and Balancing resented the first systematic solu- tion of linear and quadratic equations in Arabic. He is often considered one of the fathers of algebra. Some words reflect the importance of al-Khwarizmi’s contributions to mathematics. “Algebra” (Fig. 1) is derived from al-jabr, one of the two operations he used to solve quadratic equations. Algorism and algorithm stem from Algoritmi, the Latin form of his name. Fig. 1. A page from al-Khwarizmi “Algebra”. 72 B. Mehri Few details of al-Khwarizmi’s life are known with certainty. He was born in a Per- sian family and Ibn al-Nadim gives his birthplace as Khwarazm in Greater Khorasan. Ibn al-Nadim’s Kitāb al-Fihrist includes a short biography on al-Khwarizmi together with a list of the books he wrote. Al-Khwārizmī accomplished most of his work in the period between 813 and 833 at the House of Wisdom in Baghdad. Al-Khwarizmi contributions to mathematics, geography, astronomy, and cartogra- phy established the basis for innovation in algebra and trigonometry. -
The Cambridge Mathematical Journal and Its Descendants: the Linchpin of a Research Community in the Early and Mid-Victorian Age ✩
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Historia Mathematica 31 (2004) 455–497 www.elsevier.com/locate/hm The Cambridge Mathematical Journal and its descendants: the linchpin of a research community in the early and mid-Victorian Age ✩ Tony Crilly ∗ Middlesex University Business School, Hendon, London NW4 4BT, UK Received 29 October 2002; revised 12 November 2003; accepted 8 March 2004 Abstract The Cambridge Mathematical Journal and its successors, the Cambridge and Dublin Mathematical Journal,and the Quarterly Journal of Pure and Applied Mathematics, were a vital link in the establishment of a research ethos in British mathematics in the period 1837–1870. From the beginning, the tension between academic objectives and economic viability shaped the often precarious existence of this line of communication between practitioners. Utilizing archival material, this paper presents episodes in the setting up and maintenance of these journals during their formative years. 2004 Elsevier Inc. All rights reserved. Résumé Dans la période 1837–1870, le Cambridge Mathematical Journal et les revues qui lui ont succédé, le Cambridge and Dublin Mathematical Journal et le Quarterly Journal of Pure and Applied Mathematics, ont joué un rôle essentiel pour promouvoir une culture de recherche dans les mathématiques britanniques. Dès le début, la tension entre les objectifs intellectuels et la rentabilité économique marqua l’existence, souvent précaire, de ce moyen de communication entre professionnels. Sur la base de documents d’archives, cet article présente les épisodes importants dans la création et l’existence de ces revues. 2004 Elsevier Inc. -
A Practical Sanskrit Introductory
A Practical Sanskrit Intro ductory This print le is available from ftpftpnacaczawiknersktintropsjan Preface This course of fteen lessons is intended to lift the Englishsp eaking studentwho knows nothing of Sanskrit to the level where he can intelligently apply Monier DhatuPat ha Williams dictionary and the to the study of the scriptures The rst ve lessons cover the pronunciation of the basic Sanskrit alphab et Devanagar together with its written form in b oth and transliterated Roman ash cards are included as an aid The notes on pronunciation are largely descriptive based on mouth p osition and eort with similar English Received Pronunciation sounds oered where p ossible The next four lessons describ e vowel emb ellishments to the consonants the principles of conjunct consonants Devanagar and additions to and variations in the alphab et Lessons ten and sandhi eleven present in grid form and explain their principles in sound The next three lessons p enetrate MonierWilliams dictionary through its four levels of alphab etical order and suggest strategies for nding dicult words The artha DhatuPat ha last lesson shows the extraction of the from the and the application of this and the dictionary to the study of the scriptures In addition to the primary course the rst eleven lessons include a B section whichintro duces the student to the principles of sentence structure in this fully inected language Six declension paradigms and class conjugation in the present tense are used with a minimal vo cabulary of nineteen words In the B part of -
The "Greatest European Mathematician of the Middle Ages"
Who was Fibonacci? The "greatest European mathematician of the middle ages", his full name was Leonardo of Pisa, or Leonardo Pisano in Italian since he was born in Pisa (Italy), the city with the famous Leaning Tower, about 1175 AD. Pisa was an important commercial town in its day and had links with many Mediterranean ports. Leonardo's father, Guglielmo Bonacci, was a kind of customs officer in the North African town of Bugia now called Bougie where wax candles were exported to France. They are still called "bougies" in French, but the town is a ruin today says D E Smith (see below). So Leonardo grew up with a North African education under the Moors and later travelled extensively around the Mediterranean coast. He would have met with many merchants and learned of their systems of doing arithmetic. He soon realised the many advantages of the "Hindu-Arabic" system over all the others. D E Smith points out that another famous Italian - St Francis of Assisi (a nearby Italian town) - was also alive at the same time as Fibonacci: St Francis was born about 1182 (after Fibonacci's around 1175) and died in 1226 (before Fibonacci's death commonly assumed to be around 1250). By the way, don't confuse Leonardo of Pisa with Leonardo da Vinci! Vinci was just a few miles from Pisa on the way to Florence, but Leonardo da Vinci was born in Vinci in 1452, about 200 years after the death of Leonardo of Pisa (Fibonacci). His names Fibonacci Leonardo of Pisa is now known as Fibonacci [pronounced fib-on-arch-ee] short for filius Bonacci. -
James Joseph Sylvester1
Chapter 8 JAMES JOSEPH SYLVESTER1 (1814-1897) James Joseph Sylvester was born in London, on the 3d of September, 1814. He was by descent a Jew. His father was Abraham Joseph Sylvester, and the future mathematician was the youngest but one of seven children. He received his elementary education at two private schools in London, and his secondary education at the Royal Institution in Liverpool. At the age of twenty he entered St. John’s College, Cambridge; and in the tripos examination he came out sec- ond wrangler. The senior wrangler of the year did not rise to any eminence; the fourth wrangler was George Green, celebrated for his contributions to mathe- matical physics; the fifth wrangler was Duncan F. Gregory, who subsequently wrote on the foundations of algebra. On account of his religion Sylvester could not sign the thirty-nine articles of the Church of England; and as a consequence he could neither receive the degree of Bachelor of Arts nor compete for the Smith’s prizes, and as a further consequence he was not eligible for a fellowship. To obtain a degree he turned to the University of Dublin. After the theological tests for degrees had been abolished at the Universities of Oxford and Cam- bridge in 1872, the University of Cambridge granted him his well-earned degree of Bachelor of Arts and also that of Master of Arts. On leaving Cambridge he at once commenced to write papers, and these were at first on applied mathematics. His first paper was entitled “An analytical development of Fresnel’s optical theory of crystals,” which was published in the Philosophical Magazine. -
Kharosthi Manuscripts: a Window on Gandharan Buddhism*
KHAROSTHI MANUSCRIPTS: A WINDOW ON GANDHARAN BUDDHISM* Andrew GLASS INTRODUCTION In the present article I offer a sketch of Gandharan Buddhism in the centuries around the turn of the common era by looking at various kinds of evidence which speak to us across the centuries. In doing so I hope to shed a little light on an important stage in the transmission of Buddhism as it spread from India, through Gandhara and Central Asia to China, Korea, and ultimately Japan. In particular, I will focus on the several collections of Kharo~thi manuscripts most of which are quite new to scholarship, the vast majority of these having been discovered only in the past ten years. I will also take a detailed look at the contents of one of these manuscripts in order to illustrate connections with other text collections in Pali and Chinese. Gandharan Buddhism is itself a large topic, which cannot be adequately described within the scope of the present article. I will therefore confine my observations to the period in which the Kharo~thi script was used as a literary medium, that is, from the time of Asoka in the middle of the third century B.C. until about the third century A.D., which I refer to as the Kharo~thi Period. In addition to looking at the new manuscript materials, other forms of evidence such as inscriptions, art and architecture will be touched upon, as they provide many complementary insights into the Buddhist culture of Gandhara. The travel accounts of the Chinese pilgrims * This article is based on a paper presented at Nagoya University on April 22nd 2004. -
OSU WPL # 27 (1983) 140- 164. the Elimination of Ergative Patterns Of
OSU WPL # 27 (1983) 140- 164. The Elimination of Ergative Patterns of Case-Marking and Verbal Agreement in Modern Indic Languages Gregory T. Stump Introduction. As is well known, many of the modern Indic languages are partially ergative, showing accusative patterns of case- marking and verbal agreement in nonpast tenses, but ergative patterns in some or all past tenses. This partial ergativity is not at all stable in these languages, however; what I wish to show in the present paper, in fact, is that a large array of factors is contributing to the elimination of partial ergativity in the modern Indic languages. The forces leading to the decay of ergativity are diverse in nature; and any one of these may exert a profound influence on the syntactic development of one language but remain ineffectual in another. Before discussing this erosion of partial ergativity in Modern lndic, 1 would like to review the history of what the I ndian grammar- ians call the prayogas ('constructions') of a past tense verb with its subject and direct object arguments; the decay of Indic ergativity is, I believe, best envisioned as the effect of analogical develop- ments on or within the system of prayogas. There are three prayogas in early Modern lndic. The first of these is the kartariprayoga, or ' active construction' of intransitive verbs. In the kartariprayoga, the verb agrees (in number and p,ender) with its subject, which is in the nominative case--thus, in Vernacular HindOstani: (1) kartariprayoga: 'aurat chali. mard chala. woman (nom.) went (fern. sg.) man (nom.) went (masc. -
Tai Lü / ᦺᦑᦟᦹᧉ Tai Lùe Romanization: KNAB 2012
Institute of the Estonian Language KNAB: Place Names Database 2012-10-11 Tai Lü / ᦺᦑᦟᦹᧉ Tai Lùe romanization: KNAB 2012 I. Consonant characters 1 ᦀ ’a 13 ᦌ sa 25 ᦘ pha 37 ᦤ da A 2 ᦁ a 14 ᦍ ya 26 ᦙ ma 38 ᦥ ba A 3 ᦂ k’a 15 ᦎ t’a 27 ᦚ f’a 39 ᦦ kw’a 4 ᦃ kh’a 16 ᦏ th’a 28 ᦛ v’a 40 ᦧ khw’a 5 ᦄ ng’a 17 ᦐ n’a 29 ᦜ l’a 41 ᦨ kwa 6 ᦅ ka 18 ᦑ ta 30 ᦝ fa 42 ᦩ khwa A 7 ᦆ kha 19 ᦒ tha 31 ᦞ va 43 ᦪ sw’a A A 8 ᦇ nga 20 ᦓ na 32 ᦟ la 44 ᦫ swa 9 ᦈ ts’a 21 ᦔ p’a 33 ᦠ h’a 45 ᧞ lae A 10 ᦉ s’a 22 ᦕ ph’a 34 ᦡ d’a 46 ᧟ laew A 11 ᦊ y’a 23 ᦖ m’a 35 ᦢ b’a 12 ᦋ tsa 24 ᦗ pa 36 ᦣ ha A Syllable-final forms of these characters: ᧅ -k, ᧂ -ng, ᧃ -n, ᧄ -m, ᧁ -u, ᧆ -d, ᧇ -b. See also Note D to Table II. II. Vowel characters (ᦀ stands for any consonant character) C 1 ᦀ a 6 ᦀᦴ u 11 ᦀᦹ ue 16 ᦀᦽ oi A 2 ᦰ ( ) 7 ᦵᦀ e 12 ᦵᦀᦲ oe 17 ᦀᦾ awy 3 ᦀᦱ aa 8 ᦶᦀ ae 13 ᦺᦀ ai 18 ᦀᦿ uei 4 ᦀᦲ i 9 ᦷᦀ o 14 ᦀᦻ aai 19 ᦀᧀ oei B D 5 ᦀᦳ ŭ,u 10 ᦀᦸ aw 15 ᦀᦼ ui A Indicates vowel shortness in the following cases: ᦀᦲᦰ ĭ [i], ᦵᦀᦰ ĕ [e], ᦶᦀᦰ ăe [ ∎ ], ᦷᦀᦰ ŏ [o], ᦀᦸᦰ ăw [ ], ᦀᦹᦰ ŭe [ ɯ ], ᦵᦀᦲᦰ ŏe [ ]. -
The Kharoṣṭhī Documents from Niya and Their Contribution to Gāndhārī Studies
The Kharoṣṭhī Documents from Niya and Their Contribution to Gāndhārī Studies Stefan Baums University of Munich [email protected] Niya Document 511 recto 1. viśu͚dha‐cakṣ̄u bhavati tathāgatānaṃ bhavatu prabhasvara hiterṣina viśu͚dha‐gātra sukhumāla jināna pūjā suchavi paramārtha‐darśana 4 ciraṃ ca āyu labhati anālpakaṃ 5. pratyeka‐budha ca karoṃti yo s̄ātravivegam āśṛta ganuktamasya 1 ekābhirāma giri‐kaṃtarālaya 2. na tasya gaṃḍa piṭakā svakartha‐yukta śamathe bhavaṃti gune rata śilipataṃ tatra vicārcikaṃ teṣaṃ pi pūjā bhavatu [v]ā svayaṃbhu[na] 4 1 suci sugaṃdha labhati sa āśraya 6. koḍinya‐gotra prathamana karoṃti yo s̄ātraśrāvaka {?} ganuktamasya 2 teṣaṃ ca yo āsi subha͚dra pac̄ima 3. viśāla‐netra bhavati etasmi abhyaṃdare ye prabhasvara atīta suvarna‐gātra abhirūpa jinorasa te pi bhavaṃtu darśani pujita 4 2 samaṃ ca pādo utarā7. imasmi dāna gana‐rāya prasaṃṭ́hita u͚tama karoṃti yo s̄ātra sthaira c̄a madhya navaka ganuktamasya 3 c̄a bhikṣ̄u m It might be going to far to say that Torwali is the direct lineal descendant of the Niya Prakrit, but there is no doubt that out of all the modern languages it shows the closest resemblance to it. [...] that area around Peshawar, where [...] there is most reason to believe was the original home of Niya Prakrit. That conclusion, which was reached for other reasons, is thus confirmed by the distribution of the modern dialects. (Burrow 1936) Under this name I propose to include those inscriptions of Aśoka which are recorded at Shahbazgaṛhi and Mansehra in the Kharoṣṭhī script, the vehicle for the remains of much of this dialect. To be included also are the following sources: the Buddhist literary text, the Dharmapada found in Khotan, written likewise in Kharoṣṭhī [...]; the Kharoṣṭhī documents on wood, leather, and silk from Caḍ́ota (the Niya site) on the border of the ancient kingdom of Khotan, which represented the official language of the capital Krorayina [...]. -
Sanskrit Alphabet
Sounds Sanskrit Alphabet with sounds with other letters: eg's: Vowels: a* aa kaa short and long ◌ к I ii ◌ ◌ к kii u uu ◌ ◌ к kuu r also shows as a small backwards hook ri* rri* on top when it preceeds a letter (rpa) and a ◌ ◌ down/left bar when comes after (kra) lri lree ◌ ◌ к klri e ai ◌ ◌ к ke o au* ◌ ◌ к kau am: ah ◌ं ◌ः कः kah Consonants: к ka х kha ga gha na Ê ca cha ja jha* na ta tha Ú da dha na* ta tha Ú da dha na pa pha º ba bha ma Semivowels: ya ra la* va Sibilants: sa ш sa sa ha ksa** (**Compound Consonant. See next page) *Modern/ Hindi Versions a Other ऋ r ॠ rr La, Laa (retro) औ au aum (stylized) ◌ silences the vowel, eg: к kam झ jha Numero: ण na (retro) १ ५ ॰ la 1 2 3 4 5 6 7 8 9 0 @ Davidya.ca Page 1 Sounds Numero: 0 1 2 3 4 5 6 7 8 910 १॰ ॰ १ २ ३ ४ ६ ७ varient: ५ ८ (shoonya eka- dva- tri- catúr- pancha- sás- saptán- astá- návan- dásan- = empty) works like our Arabic numbers @ Davidya.ca Compound Consanants: When 2 or more consonants are together, they blend into a compound letter. The 12 most common: jna/ tra ttagya dya ddhya ksa kta kra hma hna hva examples: for a whole chart, see: http://www.omniglot.com/writing/devanagari_conjuncts.php that page includes a download link but note the site uses the modern form Page 2 Alphabet Devanagari Alphabet : к х Ê Ú Ú º ш @ Davidya.ca Page 3 Pronounce Vowels T pronounce Consonants pronounce Semivowels pronounce 1 a g Another 17 к ka v Kit 42 ya p Yoga 2 aa g fAther 18 х kha v blocKHead -
Part 1: Introduction to The
PREVIEW OF THE IPA HANDBOOK Handbook of the International Phonetic Association: A guide to the use of the International Phonetic Alphabet PARTI Introduction to the IPA 1. What is the International Phonetic Alphabet? The aim of the International Phonetic Association is to promote the scientific study of phonetics and the various practical applications of that science. For both these it is necessary to have a consistent way of representing the sounds of language in written form. From its foundation in 1886 the Association has been concerned to develop a system of notation which would be convenient to use, but comprehensive enough to cope with the wide variety of sounds found in the languages of the world; and to encourage the use of thjs notation as widely as possible among those concerned with language. The system is generally known as the International Phonetic Alphabet. Both the Association and its Alphabet are widely referred to by the abbreviation IPA, but here 'IPA' will be used only for the Alphabet. The IPA is based on the Roman alphabet, which has the advantage of being widely familiar, but also includes letters and additional symbols from a variety of other sources. These additions are necessary because the variety of sounds in languages is much greater than the number of letters in the Roman alphabet. The use of sequences of phonetic symbols to represent speech is known as transcription. The IPA can be used for many different purposes. For instance, it can be used as a way to show pronunciation in a dictionary, to record a language in linguistic fieldwork, to form the basis of a writing system for a language, or to annotate acoustic and other displays in the analysis of speech. -
The Fascinating Story Behind Our Mathematics
The Fascinating Story Behind Our Mathematics Jimmie Lawson Louisiana State University Story of Mathematics – p. 1 Math was needed for everyday life: commercial transactions and accounting, government taxes and records, measurement, inheritance. Math was needed in developing branches of knowledge: astronomy, timekeeping, calendars, construction, surveying, navigation Math was interesting: In many ancient cultures (Egypt, Mesopotamia, India, China) mathematics became an independent subject, practiced by scribes and others. Introduction In every civilization that has developed writing we also find evidence for some level of mathematical knowledge. Story of Mathematics – p. 2 Math was needed in developing branches of knowledge: astronomy, timekeeping, calendars, construction, surveying, navigation Math was interesting: In many ancient cultures (Egypt, Mesopotamia, India, China) mathematics became an independent subject, practiced by scribes and others. Introduction In every civilization that has developed writing we also find evidence for some level of mathematical knowledge. Math was needed for everyday life: commercial transactions and accounting, government taxes and records, measurement, inheritance. Story of Mathematics – p. 2 Math was interesting: In many ancient cultures (Egypt, Mesopotamia, India, China) mathematics became an independent subject, practiced by scribes and others. Introduction In every civilization that has developed writing we also find evidence for some level of mathematical knowledge. Math was needed for everyday life: commercial transactions and accounting, government taxes and records, measurement, inheritance. Math was needed in developing branches of knowledge: astronomy, timekeeping, calendars, construction, surveying, navigation Story of Mathematics – p. 2 Introduction In every civilization that has developed writing we also find evidence for some level of mathematical knowledge. Math was needed for everyday life: commercial transactions and accounting, government taxes and records, measurement, inheritance.