Name - Meeran Ali Ahmad Class - X A Roll No. - 07
Indian mathematics emerged in the Indian subcontinentfrom 1200 BC until the end of the 18th century. In theclassical period of Indian mathematics (400 AD to 1200AD), important contributions were made by scholarslike Aryabhata, Brahmagupta, and Bhaskara II.The decimal number system in use today was firstrecorded in Indian mathematics. Indian mathematiciansmade early contributions to the study of the conceptof zero as a number, negative numbers, arithmetic,and algebra. In addition, trigonometry was furtheradvanced in India, and, in particular, the moderndefinitions of sine and cosine were developed there.These mathematical concepts were transmitted tothe Middle East, China, and Europe and led to furtherdevelopments that now form the foundations of manyareas of mathematics.
Vedic- Baudhayana Katyayana Panini, ca. 5th c. BC, Algebraic grammarian Yajnavalkya, credited with authorship ofthe Shatapatha Brahmana, which containscalculations related to altar construction.
Post-Vedic Sanskrit to Pala period mathematicians (5th c. BCto 11th c. AD) Aryabhata - Astronomer who gave accurate calculationsfor astronomical constants, 476AD-520AD Aryabhata II Bhaskara I Brahmagupta - Helped bring the concept of zero intoarithmetic (598 AD-670 AD) Bhāskara II Mahavira Pavuluri Mallana - the first Telugu Mathematician Varahamihira Shridhara (between 650-850) - Gave a good rule forfinding the volume of a sphere.
Narayana Pandit Madhava of Sangamagrama some elements ofCalculus hi Parameshvara (1360–1455), discovered drk-ganita,a mode of astronomy based onobservations, Madhavas Kerala school Nilakantha Somayaji,1444-1545 - Mathematicianand Astronomer, Madhavas Kerala school Mahendra Suri (14th century) Shankara Variyar (c. 1530) Raghunatha Siromani, (1475–1550), Logician,Navadvipa school
Aryabhata (475 A.D. -550 A.D.) is the first well knownIndian mathematician. Born in Kerala, he completed hisstudies at the university of Nalanda. In thesection Ganita (calculations) of his astronomical treatiseAryabhatiya (499 A.D.), he made the fundamental advancein finding the lengths of chords of circles, by using thehalf chord rather than the full chord method used byGreeks. He gave the value of as 3.1416, claiming, for thefirst time, that it was an approximation. (He gave it in theform that the approximate circumference of a circle ofdiameter 20000 is 62832.) He also gave methods forextracting square roots, summing arithmeticseries, solving indeterminate equations of the type ax -by= c, and also gave what later came to be known as the tableof Sines. He also wrote a text book for astronomicalcalculations, Aryabhatasiddhanta. Even today, this data isused in preparing Hindu calendars (Panchangs). Inrecognition to his contributions to astronomy andmathematics, Indias first satellite was named Aryabhatta.About-
Aryabhatta is the first writer on astronomy to whomthe Hindus do not allow the honour of a divineinspiration. Writers on mathematical sciencedistinctly state that he was the earliest uninspiredand a merely human writer on astronomy. This is anotice which sufficiently proves his being anhistorical character. He also ascribed to the epicycles, by which themotion of a planet is represented, a form varyingfrom the circle and nearly elliptic. His text specifies the earths diameter, 1050 yojanas;and the orbit or circumference of the earths wind[spiritus vector] 3393 yojanas; which, as the scholiastrightly argues, is no discrepancy.His contributions….
The great 7th Century Indian mathematician andastronomer Brahmagupta wrote some importantworks on both mathematics and astronomy. He wasfrom the state of Rajasthan of northwest India (he isoften referred to as Bhillamalacarya, the teacher fromBhillamala), and later became the head of theastronomical observatory at Ujjain in central India.Most of his works are composed in elliptic verse, acommon practice in Indian mathematics at the time,and consequently have something of a poetic ring tothem. It seems likely that Brahmaguptas works,especially his most famous text, the “Brahmasphut-asiddhanta”, were brought by the 8th CenturyAbbasid caliph Al-Mansur to his newly foundedcentre of learning at Baghdad on the banks of theTigris, providing an important link between Indianmathematics and astronomy and the nascent upsurgein science and mathematics in the Islamic world.About-
In his work on arithmetic, Brahmagupta explained how to findthe cube and cube-root of an integer and gave rules facilitatingthe computation of squares and square roots. He also gave rules for dealing with five types of combinationsof fractions. He gave the sum of the squares of the first n naturalnumbers as n(n + 1)(2n + 1)⁄ 6 and the sum of the cubes of thefirst nnatural numbers as (n(n + 1)⁄2)². Furthermore, he pointed out, quadratic equations (of thetype x2 + 2 = 11, for example) could in theory have two possiblesolutions, one of which could be negative, because 32 = 9 and -32 = 9. In addition to his work on solutions to general linear equationsand quadratic equations, Brahmagupta went yet further byconsidering systems of simultaneous equations (set of equationscontaining multiple variables), and solving quadratic equationswith two unknowns, something which was not even consideredin the West until a thousand years later, when Fermat wasconsidering similar problems in 1657.
Bhaskara (1114 A.D. -1185 A.D.) or Bhaskaracharaya is the mostwell known ancient Indian mathematician. He was born in 1114A.D. at Bijjada Bida (Bijapur, Karnataka) in the Sahyadari Hills.He was the first to declare that any number divided by zero isinfinity and that the sum of any number and infinity is alsoinfinity. He is famous for his book Siddhanta Siromani(1150A.D.). It is divided into four sections -Leelavati (a book onarithmetic),Bijaganita (algebra), Goladhayaya (chapter onsphere -celestial globe), andGrahaganita (mathematics of theplanets). Leelavati contains many interesting problems and wasa very popular text book. Bhaskara introducedchakrawal, or thecyclic method, to solve algebraic equations. Six centurieslater,European mathematicians like Galois, Euler and Lagrangerediscovered this method and called it "inverse cyclic". Bhaskaracan also be called the founder of differential calculus. He gavean example of what is now called "differential coefficient" andthe basic idea of what is now called "Rolles theorem".Unfortunately, later Indian mathematicians did not take anynotice of this. Five centuries later, Newton and Leibnizdeveloped this subject. As an astronomer, Bhaskara is renownedfor his concept of Tatkalikagati(instantaneous motion).About-
Terms for numbersIn English, the multiples of 1000 are termed as thousand, million, billion,trillion, quadrillion etc. These terms were named recently in English, butBhaskaracharya gave the terms for numbers in multiples of ten which are asfollows: eka(1), dasha(10), shata(100), sahastra(1000), ayuta(10,000),laksha(100,000), prayuta (1,000,000=million), koti(107), Kutarbuda(108),abja(109=billion), kharva (1010), nikharva (1011), mahapadma (1012=trillion),shanku(1013), jaladhi(1014), antya(1015=quadrillion), Madhya (1016) andparardha(1017).KuttakKuttak according to modern mathematics is indeterminate equation of firstorder. In the western world, the method of solving such equations was called aspulverizer. Bhaskara suggested a generalized solution to get multiple answersfor these equations.Simple mathematical methodsBhaskara II suggested simple methods to calculate the squares, square roots,cube, and cube roots of big numbers. The Pythagoras theorem was proved byhim in only two lines. Bhaskaras Khandameruis the famous Pascal Triangle.
Srinivasa Ramanujan (1887-1920) hailed as an all-time greatmathematician, like Euler, Gauss or Jacobi, for his natural genius, has leftbehind 4000 original theorems, despite his lack of formal education and ashort life-span. In his formative years, after having failed in his F.A. (Firstexamination in Arts) class at College, he ran from pillar to post in search ofa benefactor. It is during this period, 1903-1914, he kept a record of the finalresults of his original research work in the form of entries in two large-sizedNote Books. These were the ones which he showed to Dewan BahadurRamachandra Rao (Collector of Nellore), V. Ramaswamy Iyer (Founder ofIndian Mathematical Society), R. Narayana Iyer (Treasurer of IMS andManager, Madras Port Trust), and to several others to convince them of hisabilities as a Mathematician. The orchestrated efforts of hisadmirers, culminated in the encouragement he received from Prof. G.H.Hardy of Trinity College, Cambridge, whose warm response to the historicletter of Ramanujan which contained about 100 theorems, resulted ininducing the Madras University, to its lasting credit, to rise to the occasionthrice - in offering him the first research scholarship of the University inMay 1913 ; then in offering him a scholarship of 250 pounds a year for fiveyears with 100 pounds for passage by ship and for initial outfit to go toEngland in 1914 ; and finally, by granting Ramanujan 250 pounds a year asan allowance for 5 years commencing from April 1919 soon after histriumphant return from Cambridge ``with a scientific standing andreputation such as no Indian has enjoyed before.About-
Ramanujans arrival at Cambridge was the beginning of a verysuccessful five-year collaboration with Hardy. In some ways the twomade an odd pair: Hardy was a great exponent of rigor inanalysis, while Ramanujans results were (as Hardy put it) "arrived atby a process of mingled argument, intuition, and induction, of whichhe was entirely unable to give any coherent account". Hardy did hisbest to fill in the gaps in Ramanujans education without discouraginghim. He was amazed by Ramanujans uncanny formal intuition inmanipulating infinite series, continued fractions, and the like: "I havenever met his equal, and can compare him onlywith Euleror Jacobi."One remarkable result of the Hardy-Ramanujancollaboration was a formula for the number p(n) of partitions of anumber n. A partition of a positive integer n is just an expressionfor n as a sum of positive integers, regardless of order. Thus p(4) = 5because 4 can be written as 1+1+1+1, 1+1+2, 2+2, 1+3, or 4. Besides his published work, Ramanujan left behind severalnotebooks, which have been the object of much study. The Englishmathematician G. N. Watson wrote a long series of papers about them.More recently the American mathematician Bruce C. Berndt has writtena multi-volume study of the notebooks. In 1997 The RamanujanJournal was launched to publish work "in areas of mathematicsinfluenced by Ramanujan".
Calyampudi Radhakrishna Rao was born to C.D. Naidu and A.Laxmikantamma on 10 September 1920 in Huvvina Hadagalli inpresent day Karnataka. He was the eighth in a family of 10 children.After his father’sretirement, the family settled down inVishakapatnam in Andhra Pradesh. From his earliest years, Raohad an interest in mathematics. After completing high school hejoined the Mrs. A.V.N. College at Vishakapatnam for theIntermediate course. He received his M.A. in Mathematics with firstrank in 1940. Rao decided to pursue a research career inmathematics, but was denied a scholarship on the grounds of latesubmission of the application. He then went to Kolkata for an interview for a job. He did not getthe job, but by chance he visited the Indian Statistical Institute,then located in a couple of rooms in the Physics Department of thePresidency College, Kolkata. He applied for a one-year trainingcourse at the Institute and was admitted to the Training Section ofthe Institute from 1 January 1941. In July 1941 he joined the M.AStatistics program of the Calcutta University. By the time he passedthe M.A. exam in 1943, winning the gold medal of the University,he had already published some research papers!About-
The living legend and doyen of Indian Statistics, 91 year old Prof. CalyampudiRadhakrishna (C. R.) Rao was awarded the Guy Medal in Gold of the RoyalStatistical Society, UK on the 29th of June, 2011 "For his fundamentalcontributions to statistical theory and methodology, including unbiasedestimation, variance reduction by sufficiency, efficiency of estimation,information geometry, as well as the application of matrix theory in linearstatistical inference", the announcement stated. The Gold Medal is awarded by the Royal Statistical Society (triennially, exceptthe war period) and named after William Guy. There are Silver and BronzeMedals too, C. R. Rao already obtained the Silver Medal in 1965. Since 1892 he isthe 34th recipient of the Gold Medal. Previously, R. A. Fisher (1946), E. S.Pearson (1955), J. Neyman (1966), M. S. Bartlett (1969), H. Cramér (1972), and D.Cox (1973) received this prize, just to mention a few. Among the recipients onlyH. Cramér and J. Neyman were outside Great Britain. C. R. Rao is the first non-European and non-American to receive the award. I believe that he has longdeserved this prize. His formulae and theory include "Cramer -Rao inequality","Fischer -Rao theorem" and "Rao - Blackwellisation". . In 1980, 18th June shesolved the multiplication of 13 digit number 7,686,369,774,870 and2,465,099,745,779 picked up by the computer science department of imperialcollege, London. Shakuntala solved the question in a flash and took 28 secondsto solve the entire problem, and her answer was18,947,668,177,995,426,462,773,730. This amazing incident helped her get a placein the Guinness book of world record
Shakuntala Devi was born on 4th of November, 1939 inBengaluru in a well-known Brahmin priest family. Shedid card tricks with her father when she was only three.Shakuntala Devi received her early lessons inmathematics from her grandfather. By the age of 5,Shakuntala Devi became an expert in complex mentalarithmetic and was recognised as a child prodigy. Shedemonstrated her talents to a large assembly of studentsand professors at the University of Mysore a year later.And when she was 8 years old, she demonstrated hertalents at Annamalai University. Shakuntala Devi hasauthored a few books. She shares some of the methods ofmental calculations in her world famous book, Figuring:The Joy of Numbers. Puzzles to puzzle You, More Puzzlesto puzzle you, The Book of Numbers, Mathability:Awaken the Math Genius in Your Child, Astrology foryou, Perfect Murder, In the Wonderland of Numbers aresome of the popular books written by her. Her book, Inthe Wonderland of Numbers, talks about a girl Neha, andher fascination for numbers.About-
Shakuntala Devi was a genius and once in 1977 shementally solved the 23rd root of a 201 digit numberwithout any help from mechanical aid. She shares some of the methods of mental calculations inher world famous book, Figuring: The Joy of Numbers.Puzzles to puzzle You, More Puzzles to puzzle you, TheBook of Numbers, Mathability: Awaken the Math Geniusin Your Child, Astrology for you, Perfect Murder, In theWonderland of Numbers are some of the popular bookswritten by her. Her book, In the Wonderland of Numbers,talks about a girl Neha, and her fascination for numbers. She has been travelling around the globe performing forthe student community, Prime Ministers, Presidents,Politicians and Educationalists.
The most fundamental contribution of ancientIndia in mathematics is the invention of decimalsystem of enumeration, including the inventionof zero. The decimal system uses nine digits (1 to9) and the symbol zero (for nothing) to denote allnatural numbers by assigning a place value to thedigits. The Arabs carried this system to Africaand Europe. Indians have significantly contributed in thefield of mathematics and ,if God wills, they willdo the same in the near future.