A History of Zero
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Sripati: an Eleventh-Century Indian Mathematician
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector HlSTORlA MATHEMATICA 12 (1985). 2.544 Sripati: An Eleventh-Century Indian Mathematician KRIPA NATH SINHA University of Kalyani, P.O. Kalyani, District Nadia, West Bengal, India Srlpati (fl. A.D. 1039-1056) is best known for his writings on astronomy, arithmetic, mensuration, and algebra. This article discusses Sripati’s arithmetic, the Ganitatilaka, as well as the arithmetical and algebraic chapters of the SiddhdntaSekhara. In addition to discussing the kinds of problems considered by Srlpati and the techniques he used to solve them, the article considers the sources upon which Sripati drew. A glossary of Indian treatises and technical terms is provided. o 1985 Academic PKSS. IOC. Srlpati (actif vers 1039-1056 ap. J.C.) est surtout connu pour ses Ccrits sur I’astronomie, I’arithmetique, le toise, et I’algebre. Dans cet article, nous abordons I’arithmetique de Sripati, le Ganitatilaka, de m&me que les chapitres arithmetiques et algebriques de son SiddhrintaSekh’ara. En plus d’aborder les types de problemes Ctudies par Sripati et les techniques qu’il a employees pour les resoudre, nous exminons aussi les sources auxquelles Srlpati fait appel. Un glossaire des trait& et des termes techniques indiens complete cet article. 0 1985 Academic Press, Inc. Sripati, der zwischen 1039 und 1056 wirkte, ist vor allem durch seine Schriften tiber Astronomie, Arithmetik, Mensuration und Algebra bekannt. In diesem Beitrag werden seine Arithmetik, die Gavitatilaka, und die arithmetischen und algebraischen Kapitel aus SiddhhtaSekhara behandelt. Neben der Art der Probleme, die Srlpati studierte, und den von ihm verwendeten Losungsmethoden werden such die von ihm benutzten Quellen be- trachtet. -
KHA in ANCIENT INDIAN MATHEMATICS It Was in Ancient
KHA IN ANCIENT INDIAN MATHEMATICS AMARTYA KUMAR DUTTA It was in ancient India that zero received its first clear acceptance as an integer in its own right. In 628 CE, Brahmagupta describes in detail rules of operations with integers | positive, negative and zero | and thus, in effect, imparts a ring structure on integers with zero as the additive identity. The various Sanskrit names for zero include kha, ´s¯unya, p¯urn. a. There was an awareness about the perils of zero and yet ancient Indian mathematicians not only embraced zero as an integer but allowed it to participate in all four arithmetic operations, including as a divisor in a division. But division by zero is strictly forbidden in the present edifice of mathematics. Consequently, verses from ancient stalwarts like Brahmagupta and Bh¯askar¯ac¯arya referring to numbers with \zero in the denominator" shock the modern reader. Certain examples in the B¯ıjagan. ita (1150 CE) of Bh¯askar¯ac¯arya appear as absurd nonsense. But then there was a time when square roots of negative numbers were considered non- existent and forbidden; even the validity of subtracting a bigger number from a smaller number (i.e., the existence of negative numbers) took a long time to gain universal acceptance. Is it not possible that we too have confined ourselves to a certain safe convention regarding the zero and that there could be other approaches where the ideas of Brahmagupta and Bh¯askar¯ac¯arya, and even the examples of Bh¯askar¯ac¯arya, will appear not only valid but even natural? Enterprising modern mathematicians have created elaborate legal (or technical) machinery to overcome the limitations imposed by the prohibition against use of zero in the denominator. -
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. -
Ancient Indian Mathematical Evolution Since Counting
Journal of Statistics and Mathematical Engineering e-ISSN: 2581-7647 Volume 5 Issue 3 Ancient Indian Mathematical Evolution since Counting 1 2 Sankar Prasad Mukherjee , Sandip Ghanta* 1Research Guide, 2Research Scholar 1,2Department of Mathematics, Seacom Skills University, Kolkata, West Bengal, India Email: *[email protected] DOI: Abstract This paper is an endeavor how chronologically since inception and into growth of mathematics occurred in Ancient India with an effort of counting to establish the numeral system through different ages, i.e., Rigveda, Yajurvada, Buddhist, Indo-Bactrian, Bramhi, Gupta and Devanagari Periods. Ancient India’s such contribution was of immense value helped to accelerate the progress of Mathematical development up to modern age as we see today. Keywords: Brahmi numerals, centesimal scale, devanagari, rigveda, kharosthi numerals, yajurveda INTRODUCTION main striking feature being counting and This research paper is an endeavor to evolution of numeral system thereby. synchronize all the historical research with essence of pre-historic and post-historic Mathematical Evolution in Vedic Period respectively interwoven into a texture of Decimal Number System in the Rigveda evolution process of Mathematics. The first Numbers are represented in decimal system form of writing human race was not (i.e., base 10) in the Rigveda, in all other literature but Mathematics. Arithmetic Vedic treatises, and in all subsequent Indian what is today was felt as an essential need texts. No other base occurs in ancient for day to day necessity of human race. In Indian texts, except a few instances of base various countries at various point of time, 100 (or higher powers of 10). -
In Praise of Her Supreme Holiness Shri Mataji Nirmala Devi
In praise of Her Supreme Holiness Shri Mataji Nirmala Devi 2016 Edition The original Sahaja Yoga Mantrabook was compiled by Sahaja Yoga Austria and gibven as a Guru Puja gift in 1989 0 'Now the name of your Mother is very powerful. You know that is the most powerful name, than all the other names, the most powerful mantra. But you must know how to take it. With that complete dedication you have to take that name. Not like any other.' Her Supreme Holiness Shri Mataji Nirmala Devi ‘Aum Twameva sakshat, Shri Nirmala Devyai namo namaḥ. That’s the biggest mantra, I tell you. That’s the biggest mantra. Try it’ Birthday Puja, Melbourne, 17-03-85. 1 This book is dedicated to Our Beloved Divine M other Her Supreme Holiness Shri MMMatajiM ataji Nirmal aaa DevDeviiii,,,, the Source of This Knowledge and All Knowledge . May this humble offering be pleasing in Her Sight. May Her Joy always be known and Her P raises always sung on this speck of rock in the Solar System. Feb 2016 No copyright is held on this material which is for the emancipation of humanity. But we respectfully request people not to publish any of the contents in a substantially changed or modified manner which may be misleading. 2 Contents Sanskrit Pronunciation .................................... 8 Mantras in Sahaja Yoga ................................... 10 Correspondence with the Chakras ....................... 14 The Three Levels of Sahasrara .......................... 16 Om ................................................. 17 Mantra Forms ........................................ 19 Mantras for the Chakras .................................. 20 Mantras for Special Purposes ............................. 28 The Affirmations ......................................... 30 Short Prayers for the Chakras ............................. 33 Gāyatrī Mantra ...................................... -
A Historical Survey of Methods of Solving Cubic Equations Minna Burgess Connor
University of Richmond UR Scholarship Repository Master's Theses Student Research 7-1-1956 A historical survey of methods of solving cubic equations Minna Burgess Connor Follow this and additional works at: http://scholarship.richmond.edu/masters-theses Recommended Citation Connor, Minna Burgess, "A historical survey of methods of solving cubic equations" (1956). Master's Theses. Paper 114. This Thesis is brought to you for free and open access by the Student Research at UR Scholarship Repository. It has been accepted for inclusion in Master's Theses by an authorized administrator of UR Scholarship Repository. For more information, please contact [email protected]. A HISTORICAL SURVEY OF METHODS OF SOLVING CUBIC E<~UATIONS A Thesis Presented' to the Faculty or the Department of Mathematics University of Richmond In Partial Fulfillment ot the Requirements tor the Degree Master of Science by Minna Burgess Connor August 1956 LIBRARY UNIVERStTY OF RICHMOND VIRGlNIA 23173 - . TABLE Olf CONTENTS CHAPTER PAGE OUTLINE OF HISTORY INTRODUCTION' I. THE BABYLONIANS l) II. THE GREEKS 16 III. THE HINDUS 32 IV. THE CHINESE, lAPANESE AND 31 ARABS v. THE RENAISSANCE 47 VI. THE SEVEW.l'EEl'iTH AND S6 EIGHTEENTH CENTURIES VII. THE NINETEENTH AND 70 TWENTIETH C:BNTURIES VIII• CONCLUSION, BIBLIOGRAPHY 76 AND NOTES OUTLINE OF HISTORY OF SOLUTIONS I. The Babylonians (1800 B. c.) Solutions by use ot. :tables II. The Greeks·. cs·oo ·B.c,. - )00 A~D.) Hippocrates of Chios (~440) Hippias ot Elis (•420) (the quadratrix) Archytas (~400) _ .M~naeobmus J ""375) ,{,conic section~) Archimedes (-240) {conioisections) Nicomedea (-180) (the conchoid) Diophantus ot Alexander (75) (right-angled tr~angle) Pappus (300) · III. -
The Unicode Standard, Version 4.0--Online Edition
This PDF file is an excerpt from The Unicode Standard, Version 4.0, issued by the Unicode Consor- tium and published by Addison-Wesley. The material has been modified slightly for this online edi- tion, however the PDF files have not been modified to reflect the corrections found on the Updates and Errata page (http://www.unicode.org/errata/). For information on more recent versions of the standard, see http://www.unicode.org/standard/versions/enumeratedversions.html. Many of the designations used by manufacturers and sellers to distinguish their products are claimed as trademarks. Where those designations appear in this book, and Addison-Wesley was aware of a trademark claim, the designations have been printed in initial capital letters. However, not all words in initial capital letters are trademark designations. The Unicode® Consortium is a registered trademark, and Unicode™ is a trademark of Unicode, Inc. The Unicode logo is a trademark of Unicode, Inc., and may be registered in some jurisdictions. The authors and publisher have taken care in preparation of this book, but make no expressed or implied warranty of any kind and assume no responsibility for errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of the use of the information or programs contained herein. The Unicode Character Database and other files are provided as-is by Unicode®, Inc. No claims are made as to fitness for any particular purpose. No warranties of any kind are expressed or implied. The recipient agrees to determine applicability of information provided. Dai Kan-Wa Jiten used as the source of reference Kanji codes was written by Tetsuji Morohashi and published by Taishukan Shoten. -
Nepali Letters Ka Kha Pair
Nepali Letters Ka Kha Noam addles tropically. Which Roscoe biases so bilaterally that Malcolm sub her Cheshire? Ventilated Hussein shotguns lachrymosely. Materials on nepali, ka is considered to nepali tool is the more consonants Kannada alphabets and pronounce the initial consonant letters, a unique in syllabic writing the schwa is not a question. Tone letter and nepali ka is an otherwise standard ligature may need to collect and common conjunct consonant letters, or on our prior permission. Write in devanagari is written in preetin font widely used in your can find nepali. Bilingual for the letters ka kha syllables are ready to fit in internet about nepali font, where you can be very popular nepali writing system a script. Stop solution for making this page, malayalam joins letters also used to copy the leading to nepali. Passwords can now here once you should be able to the support materials on daraz nepal you are looking for? Leading to read below or as being common conjunct forms, some writers and easily from left to nepali. It suitable unicode and kha can now here once ruled the previously transliterated text area will be a written. Syllabic writing system; in english to the real sound in the sidebar. Fit in indic scripts, with a search videos in the adjacent characters or tirhuta, you to nepali. Copyright the universal history of what you know how those words into preeti nepali is for? Fuzzy phonetic alphabet, it is through pure ligatures are other? Appended to convert the letters kha is not an historic period or tirhuta, then please check the better you will allow you can find the sounds. -
An Introduction to Indic Scripts
An Introduction to Indic Scripts Richard Ishida W3C [email protected] HTML version: http://www.w3.org/2002/Talks/09-ri-indic/indic-paper.html PDF version: http://www.w3.org/2002/Talks/09-ri-indic/indic-paper.pdf Introduction This paper provides an introduction to the major Indic scripts used on the Indian mainland. Those addressed in this paper include specifically Bengali, Devanagari, Gujarati, Gurmukhi, Kannada, Malayalam, Oriya, Tamil, and Telugu. I have used XHTML encoded in UTF-8 for the base version of this paper. Most of the XHTML file can be viewed if you are running Windows XP with all associated Indic font and rendering support, and the Arial Unicode MS font. For examples that require complex rendering in scripts not yet supported by this configuration, such as Bengali, Oriya, and Malayalam, I have used non- Unicode fonts supplied with Gamma's Unitype. To view all fonts as intended without the above you can view the PDF file whose URL is given above. Although the Indic scripts are often described as similar, there is a large amount of variation at the detailed implementation level. To provide a detailed account of how each Indic script implements particular features on a letter by letter basis would require too much time and space for the task at hand. Nevertheless, despite the detail variations, the basic mechanisms are to a large extent the same, and at the general level there is a great deal of similarity between these scripts. It is certainly possible to structure a discussion of the relevant features along the same lines for each of the scripts in the set. -
Brahminet-ITRANS Transliteration Scheme
BrahmiNet-ITRANS Transliteration Scheme Anoop Kunchukuttan August 2020 The BrahmiNet-ITRANS notation (Kunchukuttan et al., 2015) provides a scheme for transcription of major Indian scripts in Roman, using ASCII characters. It extends the ITRANS1 transliteration scheme to cover characters not covered in the original scheme. Tables 1a shows the ITRANS mappings for vowels and diacritics (matras). Table 1b shows the ITRANS mappings for consonants. These tables also show the Unicode offset for each character. By Unicode offset, we mean the offset of the character in the Unicode range assigned to the script. For Indic scripts, logically equivalent characters are assigned the same offset in their respective Unicode codepoint ranges. For illustration, we also show the Devanagari characters corresponding to the transliteration. ka kha ga gha ∼Na, N^a ITRANS Unicode Offset Devanagari 15 16 17 18 19 a 05 अ क ख ग घ ङ aa, A 06, 3E आ, ◌ा cha Cha ja jha ∼na, JNa i 07, 3F इ, ि◌ 1A 1B 1C 1D 1E ii, I 08, 40 ई, ◌ी च छ ज झ ञ u 09, 41 उ, ◌ु Ta Tha Da Dha Na uu, U 0A, 42 ऊ, ◌ू 1F 20 21 22 23 RRi, R^i 0B, 43 ऋ, ◌ृ ट ठ ड ढ ण RRI, R^I 60, 44 ॠ, ◌ॄ ta tha da dha na LLi, L^i 0C, 62 ऌ, ◌ॢ 24 25 26 27 28 LLI, L^I 61, 63 ॡ, ◌ॣ त थ द ध न pa pha ba bha ma .e 0E, 46 ऎ, ◌ॆ 2A 2B 2C 2D 2E e 0F, 47 ए, ◌े प फ ब भ म ai 10,48 ऐ, ◌ै ya ra la va, wa .o 12, 4A ऒ, ◌ॊ 2F 30 32 35 o 13, 4B ओ, ◌ो य र ल व au 14, 4C औ, ◌ौ sha Sha sa ha aM 05 02, 02 अं 36 37 38 39 aH 05 03, 03 अः श ष स ह .m 02 ◌ं Ra lda, La zha .h 03 ◌ः 31 33 34 (a) Vowels ऱ ळ ऴ (b) Consonants Version: v1.0 (9 August 2020) NOTES: 1. -
Equation Solving in Indian Mathematics
U.U.D.M. Project Report 2018:27 Equation Solving in Indian Mathematics Rania Al Homsi Examensarbete i matematik, 15 hp Handledare: Veronica Crispin Quinonez Examinator: Martin Herschend Juni 2018 Department of Mathematics Uppsala University Equation Solving in Indian Mathematics Rania Al Homsi “We owe a lot to the ancient Indians teaching us how to count. Without which most modern scientific discoveries would have been impossible” Albert Einstein Sammanfattning Matematik i antika och medeltida Indien har påverkat utvecklingen av modern matematik signifi- kant. Vissa människor vet de matematiska prestationer som har sitt urspring i Indien och har haft djupgående inverkan på matematiska världen, medan andra gör det inte. Ekvationer var ett av de områden som indiska lärda var mycket intresserade av. Vad är de viktigaste indiska bidrag i mate- matik? Hur kunde de indiska matematikerna lösa matematiska problem samt ekvationer? Indiska matematiker uppfann geniala metoder för att hitta lösningar för ekvationer av första graden med en eller flera okända. De studerade också ekvationer av andra graden och hittade heltalslösningar för dem. Denna uppsats presenterar en litteraturstudie om indisk matematik. Den ger en kort översyn om ma- tematikens historia i Indien under många hundra år och handlar om de olika indiska metoderna för att lösa olika typer av ekvationer. Uppsatsen kommer att delas in i fyra avsnitt: 1) Kvadratisk och kubisk extraktion av Aryabhata 2) Kuttaka av Aryabhata för att lösa den linjära ekvationen på formen 푐 = 푎푥 + 푏푦 3) Bhavana-metoden av Brahmagupta för att lösa kvadratisk ekvation på formen 퐷푥2 + 1 = 푦2 4) Chakravala-metoden som är en annan metod av Bhaskara och Jayadeva för att lösa kvadratisk ekvation 퐷푥2 + 1 = 푦2. -
On the Shoulders of Hipparchus a Reappraisal of Ancient Greek Combinatorics
Arch. Hist. Exact Sci. 57 (2003) 465–502 Digital Object Identifier (DOI) 10.1007/s00407-003-0067-0 On the Shoulders of Hipparchus A Reappraisal of Ancient Greek Combinatorics F. Acerbi Communicated by A. Jones 1. Introduction To write about combinatorics in ancient Greek mathematics is to write about an empty subject. The surviving evidence is so scanty that even the possibility that “the Greeks took [any] interest in these matters” has been denied by some historians of mathe- matics.1 Tranchant judgments of this sort reveal, if not a cursory scrutiny of the extant sources, at least a defective acquaintance with the strong selectivity of the process of textual transmission the ancient mathematical corpus has been exposed to–afactthat should induce, about a sparingly attested field of research, a cautious attitude rather than a priori negative assessments.2 (Should not the onus probandi be required also when the existence in the past of a certain domain of research is being denied?) I suspect that, behind such a strongly negative historiographic position, two different motives could have conspired: the prejudice of “ancient Greek mathematicians” as geometri- cally-minded and the attempt to revalue certain aspects of non-western mathematics by tendentiously maintaining that such aspects could not have been even conceived by “the Greeks”. Combinatorics is the ideal field in this respect, since many interesting instances of combinatorial calculations come from other cultures,3 whereas combina- torial examples in the ancient Greek mathematical corpus have been considered, as we have seen, worse than disappointing, and in any event such as to justify a very negative attitude.