The Algorithmic Approach of Indian Mathematics
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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 "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. -
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. -
Rationale of the Chakravala Process of Jayadeva and Bhaskara Ii
HISTORIA MATHEMATICA 2 (1975) , 167-184 RATIONALE OF THE CHAKRAVALA PROCESS OF JAYADEVA AND BHASKARA II BY CLAS-OLOF SELENIUS UNIVERSITY OF UPPSALA SUMMARIES The old Indian chakravala method for solving the Bhaskara-Pell equation or varga-prakrti x 2- Dy 2 = 1 is investigated and explained in detail. Previous mis- conceptions are corrected, for example that chakravgla, the short cut method bhavana included, corresponds to the quick-method of Fermat. The chakravala process corresponds to a half-regular best approximating algorithm of minimal length which has several deep minimization properties. The periodically appearing quantities (jyestha-mfila, kanistha-mfila, ksepaka, kuttak~ra, etc.) are correctly understood only with the new theory. Den fornindiska metoden cakravala att l~sa Bhaskara- Pell-ekvationen eller varga-prakrti x 2 - Dy 2 = 1 detaljunders~ks och f~rklaras h~r. Tidigare missuppfatt- 0 ningar r~ttas, sasom att cakravala, genv~gsmetoden bhavana inbegripen, motsvarade Fermats snabbmetod. Cakravalaprocessen motsvarar en halvregelbunden b~st- approximerande algoritm av minimal l~ngd med flera djupt liggande minimeringsegenskaper. De periodvis upptr~dande storheterna (jyestha-m~la, kanistha-mula, ksepaka, kuttakara, os~) blir forstaellga0. 0 . f~rst genom den nya teorin. Die alte indische Methode cakrav~la zur Lbsung der Bhaskara-Pell-Gleichung oder varga-prakrti x 2 - Dy 2 = 1 wird hier im einzelnen untersucht und erkl~rt. Fr~here Missverst~ndnisse werden aufgekl~rt, z.B. dass cakrav~la, einschliesslich der Richtwegmethode bhavana, der Fermat- schen Schnellmethode entspreche. Der cakravala-Prozess entspricht einem halbregelm~ssigen bestapproximierenden Algorithmus von minimaler L~nge und mit mehreren tief- liegenden Minimierungseigenschaften. Die periodisch auftretenden Quantit~ten (jyestha-mfila, kanistha-mfila, ksepaka, kuttak~ra, usw.) werden erst durch die neue Theorie verst~ndlich. -
IJR-1, Mathematics for All ... Syed Samsul Alam
January 31, 2015 [IISRR-International Journal of Research ] MATHEMATICS FOR ALL AND FOREVER Prof. Syed Samsul Alam Former Vice-Chancellor Alaih University, Kolkata, India; Former Professor & Head, Department of Mathematics, IIT Kharagpur; Ch. Md Koya chair Professor, Mahatma Gandhi University, Kottayam, Kerala , Dr. S. N. Alam Assistant Professor, Department of Metallurgical and Materials Engineering, National Institute of Technology Rourkela, Rourkela, India This article briefly summarizes the journey of mathematics. The subject is expanding at a fast rate Abstract and it sometimes makes it essential to look back into the history of this marvelous subject. The pillars of this subject and their contributions have been briefly studied here. Since early civilization, mathematics has helped mankind solve very complicated problems. Mathematics has been a common language which has united mankind. Mathematics has been the heart of our education system right from the school level. Creating interest in this subject and making it friendlier to students’ right from early ages is essential. Understanding the subject as well as its history are both equally important. This article briefly discusses the ancient, the medieval, and the present age of mathematics and some notable mathematicians who belonged to these periods. Mathematics is the abstract study of different areas that include, but not limited to, numbers, 1.Introduction quantity, space, structure, and change. In other words, it is the science of structure, order, and relation that has evolved from elemental practices of counting, measuring, and describing the shapes of objects. Mathematicians seek out patterns and formulate new conjectures. They resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity. -
Quantum Algorithm for Solving Pell's Equation
CS682 Project Report Hallgren's Efficient Quantum Algorithm for Solving Pell's Equation by Ashish Dwivedi (17111261) under the guidance of Prof Rajat Mittal November 15, 2017 Abstract In this project we aim to study an excellent result on the quantum computation model. Hallgren in 2002 [Hal07] showed that a seemingly difficult problem in classical computa- tion model, solving Pell's equation, is efficiently solvable in quantum computation model. This project explains the idea of Hallgren's quantum algorithm to solve Pell's equation. Contents 0.1 Introduction..................................1 0.2 Background..................................1 0.3 Hallgren's periodic function.........................3 0.4 Quantum Algorithm to find irrational period on R .............4 0.4.1 Discretization of the periodic function...............4 0.4.2 Quantum Algorithm.........................5 0.4.3 Classical Post Processing.......................7 0.5 Summary...................................8 0.1 Introduction In this project we study one of those problems which have no known efficient solution on the classical computation model and are assumed hard to be solved efficiently, but in recent years efficient quantum algorithms have been found for them. One of these problems is the problem of finding solution to Pell's equation. Pell's equation are equations of the form x2 − dy2 = 1, where d is some given non- square positive integer and x and y are indeterminate. Clearly, (1; 0) is a solution of this equation which are called trivial solutions. We want some non-trivial integer solutions satisfying these equations. There is no efficient algorithm known for solving such equations on classical compu- tation model. But in 2002, Sean Hallgren [Hal07] gave an algorithm on quantum compu- tation model which takes time only polynomial in input size (log d). -
Aryabhatiya with English Commentary
ARYABHATIYA OF ARYABHATA Critically edited with Introduction, English Translation. Notes, Comments and Indexes By KRIPA SHANKAR SHUKLA Deptt. of Mathematics and Astronomy University of Lucknow in collaboration with K. V. SARMA Studies V. V. B. Institute of Sanskrit and Indological Panjab University INDIAN NATIONAL SCIENCE ACADEMY NEW DELHI 1 Published for THE NATIONAL COMMISSION FOR THE COMPILATION OF HISTORY OF SCIENCES IN INDIA by The Indian National Science Academy Bahadur Shah Zafar Marg, New Delhi— © Indian National Science Academy 1976 Rs. 21.50 (in India) $ 7.00 ; £ 2.75 (outside India) EDITORIAL COMMITTEE Chairman : F. C. Auluck Secretary : B. V. Subbarayappa Member : R. S. Sharma Editors : K. S. Shukla and K. V. Sarma Printed in India At the Vishveshvaranand Vedic Research Institute Press Sadhu Ashram, Hosbiarpur (Pb.) CONTENTS Page FOREWORD iii INTRODUCTION xvii 1. Aryabhata— The author xvii 2. His place xvii 1. Kusumapura xvii 2. Asmaka xix 3. His time xix 4. His pupils xxii 5. Aryabhata's works xxiii 6. The Aryabhatiya xxiii 1. Its contents xxiii 2. A collection of two compositions xxv 3. A work of the Brahma school xxvi 4. Its notable features xxvii 1. The alphabetical system of numeral notation xxvii 2. Circumference-diameter ratio, viz., tz xxviii table of sine-differences xxviii . 3. The 4. Formula for sin 0, when 6>rc/2 xxviii 5. Solution of indeterminate equations xxviii 6. Theory of the Earth's rotation xxix 7. The astronomical parameters xxix 8. Time and divisions of time xxix 9. Theory of planetary motion xxxi - 10. Innovations in planetary computation xxxiii 11. -
Vedic Math Seminar
Seminar on Vedic Mathematics Dr. Chandrasekharan Raman December 5, 2015 Bridgewater Temple Hall, NJ Ancient Indian Mathematics • Sulba Sutras (700 BC) – rational approximation to √2, proof to Pythagoras theorem etc. • Pingala’s Chandas (300 BC) – combinatorics • Jain Mathematicians (300 BC) – concept of infinity and zero (shunya) • Classical period (400 AD – 1600 AD) ▫ Aryabhata – sine table, trigonometry, π ▫ Brahmagupta – cyclic quadrilateral, indeterm Equ. ▫ Bhaskara II – Lilavati, Bijaganita ▫ Madhava – infinite series for π • Excellent source : Wikipedia (Indian Mathematics) Vedic Mathematics What is Vedic Mathematics? “Vedic Mathematics” is the name given to a work in Indian Mathematics by Sri Bharati Krsna Tirthaji (1884-1960). Vedic Math is based on sixteen Sutras or principles What it is not? It is not from the Vedas It is not ancient Why Vedic Mathematics? Gives an insight into the structure of numbers Very much amenable to mental calculations Decimal Number System in Ancient India • The decimal number system – representing numbers in base 10, was a contribution to the world by Indians • The Place Value System was also a contribution of India Name Value Name Value Eka 100 Arbudam 107 Dasa 101 Nyarbudam 108 Shatam 102 Samudra 109 Sahasram 103 Madhyam 1010 Ayutam 104 Anta 1011 Niyutam 105 Parardha 1012 Prayutam 106 Maths in day-to-day life of a vendor in India 1 11 21 31 41 • You buy some stuff from 2 12 22 32 42 a vendor for Rs 23 3 13 23 33 43 • You pay a 50-rupee note 4 14 24 34 44 5 15 25 35 45 • He pays you back 6 -
Mineral Development and Ecological Sustainability Dr
Science Horizon Volume 5 Issue 11 November, 2020 President, Odisha Bigyan Academy Editorial Board Prof. Pramod Chandra Mishra Prof. Niranjan Barik Editor Prof. Ramesh Chandra Parida Er. Mayadhar Swain Dr. Choudhury Satyabrata Nanda Managing Editor Dr. Rajballav Mohanty Er. Bhagat Charan Mohanty Secretary, Odisha Bigyan Academy Dr. Nilambar Biswal CONTENTS Subject Author Page 1. Editorial : Climate Change and Carbon Neutrality Er. Mayadhar Swain 2 2. The Mysterious Concept of Mass - III Dr. Bijay Kumar Parida 4 3. Mineral Development and Ecological Sustainability Dr. Sundara Narayana Patro 7 4. Behaviour Development of The Child Dr. (Mrs.) Pritishri Parhi 12 5. Let Us Find L.C.M. Dr. Archana Panigrahi 15 6. A Brief History of Indian Math-Magicians Purusottam Sahoo 17 7. O Mighty Planet Venus Vaibhavi Karthik 24 8. One Interesting Series of Nilakantha Er. Mayadhar Swain 25 9. Dragonfly: The Hawk of the Insect World Dr. Raj Ballav Mohanty 26 Dr. Taranisen Panda 10. Stories of Genes: II The Genetic Code G.B.N. Chainy 27 11. Mask: A Need For Survival Dr. Pranab Kumar Ghosh 32 12. Humus - The Skeleton of Soil Meenakhi Prusty 37 Monika Ray 13. Quiz: Covid-19 Pandemic Dr. Bibhuti Narayan Biswal 43 14. Recent News on Science & Technology 45 Cover Design : Kalakar Sahoo NOVEMBER, 2020 // EDITORIAL // CLIMATE CHANGE AND CARBON NEUTRALITY At present Global Warming and Climate the atmosphere from burning of fossil fuels Change are the main environmental problems like coal, petroleum oil and natural gas in faced by Earth. The problem is created by factories and thermal power plants. Also lot of man. -
A Combined Decimal and Binary Floating-Point Divider
A Combined Decimal and Binary Floating-point Divider Sonia Gonz´alez-Navarro Alberto Nannarelli Michael Schulte Charles Tsen University of M´alaga Technical University of Denmark University of Wisconsin-Madison Nvidia Corporation [email protected] [email protected] [email protected] [email protected] Abstract—In this paper, we present the hardware design of proposed for multiplication [6]. Combined binary and BCD a combined decimal and binary floating-point divider, based units have been proposed for addition [7], multiplication [8], on specifications in the IEEE 754-2008 Standard for Floating- [9] and division [10]. point Arithmetic. In contrast to most recent decimal divider designs, which are based on the Binary Coded Decimal (BCD) The combined unit we propose operates on either 64-bit encoding, our divider operates on either 64-bit binary encoded BID-encoded DFP numbers or 64-bit BFP numbers and it is decimal floating-point (DFP) numbers or 64-bit binary floating- based on the radix-10 division unit presented in [11]. This unit point (BFP) numbers. The division approach implemented in is implemented using the digit-recurrence approach [12] and our design is based on a digit-recurrence algorithm. We describe it has been modified to also support 64-bit BFP numbers. The the hardware resources shared between the two floating-point datatypes and demonstrate that hardware sharing is advanta- BFP division is implemented with a retimed radix-16 (two geous. Compared to a standalone DFP divider, the combined overlapped radix-4 stages) digit-recurrence unit with selection divider has the same worst case delay and 17% more area. -
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. -
Mathematics Newsletter Volume 21. No4, March 2012
MATHEMATICS NEWSLETTER EDITORIAL BOARD S. Ponnusamy (Chief Editor) Department of Mathematics Indian Institute of Technology Madras Chennai - 600 036, Tamilnadu, India Phone : +91-44-2257 4615 (office) +91-44-2257 6615, 2257 0298 (home) [email protected] http://mat.iitm.ac.in/home/samy/public_html/index.html S. D. Adhikari G. K. Srinivasan Harish-Chandra Research Institute Department of Mathematics, (Former Mehta Research Institute ) Indian Institute of Technology Chhatnag Road, Jhusi Bombay Allahabad 211 019, India Powai, Mumbai 400076, India [email protected] [email protected] C. S. Aravinda B. Sury, TIFR Centre for Applicable Mathematics Stat-Math Unit, Sharadanagar, Indian Statistical Institute, Chikkabommasandra 8th Mile Mysore Road, Post Bag No. 6503 Bangalore 560059, India. Bangalore - 560 065 [email protected], [email protected] [email protected] M. Krishna G. P. Youvaraj The Institute of Mathematical Sciences Ramanujan Institute CIT Campus, Taramani for Advanced Study in Mathematics Chennai-600 113, India University of Madras, Chepauk, [email protected] Chennai-600 005, India [email protected] Stefan Banach (1892–1945) R. Anantharaman SUNY/College, Old Westbury, NY 11568 E-mail: rajan−[email protected] To the memory of Jong P. Lee Abstract. Stefan Banach ranks quite high among the founders and developers of Functional Analysis. We give a brief summary of his life, work and methods. Introduction (equivalent of middle/high school) there. Even as a student Stefan revealed his talent in mathematics. He passed the high Stefan Banach and his school in Poland were (among) the school in 1910 but not with high honors [M].