Archive for the ‘Indian Mathematicians’ Category

Srinivasa Ramanujan Indian Mathematician Biography

October 26, 2008

Srinivasa Ramanujan (Dec. 22, 1887 — April 26, 1920)

K. Srinivasa Rao


Srinivasa Ramanujan was one of India’s greatest mathematical geniuses. He made substantial contributions to the analytical theory of numbers and worked on elliptic functions, continued fractions, and infinite series.

Ramanujan was born in his grandmother’s house in Erode, a small village about 400 km southwest of Madras. When Ramanujan was a year old his mother took him to the town of Kumbakonam, about 160 km nearer Madras. His father worked in Kumbakonam as a clerk in a cloth merchant’s shop. In December 1889 he contracted smallpox.

When he was nearly five years old, Ramanujan entered the primary school in Kumbakonam although he would attend several different primary schools before entering the Town High School in Kumbakonam in January 1898. At the Town High School, Ramanujan was to do well in all his school subjects and showed himself an able all round scholar. In 1900 he began to work on his own on mathematics summing geometric and arithmetic series.

Ramanujan was shown how to solve cubic equations in 1902 and he went on to find his own method to solve the quartic. The following year, not knowing that the quintic could not be solved by radicals, he tried (and of course failed) to solve the quintic.

It was in the Town High School that Ramanujan came across a mathematics book by G S Carr called Synopsis of elementary results in pure mathematics. This book, with its very concise style, allowed Ramanujan to teach himself mathematics, but the style of the book was to have a rather unfortunate effect on the way Ramanujan was later to write down mathematics since it provided the only model that he had of written mathematical arguments. The book contained theorems, formulae and short proofs. It also contained an index to papers on pure mathematics which had been published in the European Journals of Learned Societies during the first half of the 19th century. The book, published in 1856, was of course well out of date by the time Ramanujan used it.

By 1904 Ramanujan had begun to undertake deep research. He investigated the series ∑(1/n) and calculated Euler’s constant to 15 decimal places. He began to study the Bernoulli numbers, although this was entirely his own independent discovery.

Ramanujan, on the strength of his good school work, was given a scholarship to the Government College in Kumbakonam which he entered in 1904. However the following year his scholarship was not renewed because Ramanujan devoted more and more of his time to mathematics and neglected his other subjects. Without money he was soon in difficulties and, without telling his parents, he ran away to the town of Vizagapatnam about 650 km north of Madras. He continued his mathematical work, however, and at this time he worked on hypergeometric series and investigated relations between integrals and series. He was to discover later that he had been studying elliptic functions.

In 1906 Ramanujan went to Madras where he entered Pachaiyappa’s College. His aim was to pass the First Arts examination which would allow him to be admitted to the University of Madras. He attended lectures at Pachaiyappa’s College but became ill after three months study. He took the First Arts examination after having left the course. He passed in mathematics but failed all his other subjects and therefore failed the examination. This meant that he could not enter the University of Madras. In the following years he worked on mathematics developing his own ideas without any help and without any real idea of the then current research topics other than that provided by Carr’s book.

Continuing his mathematical work Ramanujan studied continued fractions and divergent series in 1908. At this stage he became seriously ill again and underwent an operation in April 1909 after which he took him some considerable time to recover. He married on 14 July 1909 when his mother arranged for him to marry a ten year old girl S Janaki Ammal. Ramanujan did not live with his wife, however, until she was twelve years old.

Ramanujan continued to develop his mathematical ideas and began to pose problems and solve problems in the Journal of the Indian Mathematical Society. He devoloped relations between elliptic modular equations in 1910. After publication of a brilliant research paper on Bernoulli numbers in 1911 in the Journal of the Indian Mathematical Society he gained recognition for his work. Despite his lack of a university education, he was becoming well known in the Madras area as a mathematical genius.

In 1911 Ramanujan approached the founder of the Indian Mathematical Society for advice on a job. After this he was appointed to his first job, a temporary post in the Accountant General’s Office in Madras. It was then suggested that he approach Ramachandra Rao who was a Collector at Nellore. Ramachandra Rao was a founder member of the Indian Mathematical Society who had helped start the mathematics library. He writes in [30]:-

A short uncouth figure, stout, unshaven, not over clean, with one conspicuous feature-shining eyes- walked in with a frayed notebook under his arm. He was miserably poor. … He opened his book and began to explain some of his discoveries. I saw quite at once that there was something out of the way; but my knowledge did not permit me to judge whether he talked sense or nonsense. … I asked him what he wanted. He said he wanted a pittance to live on so that he might pursue his researches.

Ramachandra Rao told him to return to Madras and he tried, unsuccessfully, to arrange a scholarship for Ramanujan. In 1912 Ramanujan applied for the post of clerk in the accounts section of the Madras Port Trust. In his letter of application he wrote [3]:-

I have passed the Matriculation Examination and studied up to the First Arts but was prevented from pursuing my studies further owing to several untoward circumstances. I have, however, been devoting all my time to Mathematics and developing the subject.

Despite the fact that he had no university education, Ramanujan was clearly well known to the university mathematicians in Madras for, with his letter of application, Ramanujan included a reference from E W Middlemast who was the Professor of Mathematics at The Presidency College in Madras. Middlemast, a graduate of St John’s College, Cambridge, wrote [3]:-

I can strongly recommend the applicant. He is a young man of quite exceptional capacity in mathematics and especially in work relating to numbers. He has a natural aptitude for computation and is very quick at figure work.

On the strength of the recommendation Ramanujan was appointed to the post of clerk and began his duties on 1 March 1912. Ramanujan was quite lucky to have a number of people working round him with a training in mathematics. In fact the Chief Accountant for the Madras Port Trust, S N Aiyar, was trained as a mathematician and published a paper On the distribution of primes in 1913 on Ramanujan’s work. The professor of civil engineering at the Madras Engineering College C L T Griffith was also interested in Ramanujan’s abilities and, having been educated at University College London, knew the professor of mathematics there, namely M J M Hill. He wrote to Hill on 12 November 1912 sending some of Ramanujan’s work and a copy of his 1911 paper on Bernoulli numbers.

Hill replied in a fairly encouraging way but showed that he had failed to understand Ramanujan’s results on divergent series. The recommendation to Ramanujan that he read Bromwich’s Theory of infinite series did not please Ramanujan much. Ramanujan wrote to E W Hobson and H F Baker trying to interest them in his results but neither replied. In January 1913 Ramanujan wrote to G H Hardy having seen a copy of his 1910 book Orders of infinity. In Ramanujan’s letter to Hardy he introduced himself and his work [10]:-

I have had no university education but I have undergone the ordinary school course. After leaving school I have been employing the spare time at my disposal to work at mathematics. I have not trodden through the conventional regular course which is followed in a university course, but I am striking out a new path for myself. I have made a special investigation of divergent series in general and the results I get are termed by the local mathematicians as ‘startling’.

Hardy, together with Littlewood, studied the long list of unproved theorems which Ramanujan enclosed with his letter. On 8 February he replied to Ramanujan [3], the letter beginning:-

I was exceedingly interested by your letter and by the theorems which you state. You will however understand that, before I can judge properly of the value of what you have done, it is essential that I should see proofs of some of your assertions. Your results seem to me to fall into roughly three classes:
(1) there are a number of results that are already known, or easily deducible from known theorems;
(2) there are results which, so far as I know, are new and interesting, but interesting rather from their curiosity and apparent difficulty than their importance;
(3) there are results which appear to be new and important…

Ramanujan was delighted with Hardy’s reply and when he wrote again he said [8]:-

I have found a friend in you who views my labours sympathetically. … I am already a half starving man. To preserve my brains I want food and this is my first consideration. Any sympathetic letter from you will be helpful to me here to get a scholarship either from the university of from the government.

Indeed the University of Madras did give Ramanujan a scholarship in May 1913 for two years and, in 1914, Hardy brought Ramanujan to Trinity College, Cambridge, to begin an extraordinary collaboration. Setting this up was not an easy matter. Ramanujan was an orthodox Brahmin and so was a strict vegetarian. His religion should have prevented him from travelling but this difficulty was overcome, partly by the work of E H Neville who was a colleague of Hardy’s at Trinity College and who met with Ramanujan while lecturing in India.

Ramanujan sailed from India on 17 March 1914. It was a calm voyage except for three days on which Ramanujan was seasick. He arrived in London on 14 April 1914 and was met by Neville. After four days in London they went to Cambridge and Ramanujan spent a couple of weeks in Neville’s home before moving into rooms in Trinity College on 30th April. Right from the beginning, however, he had problems with his diet. The outbreak of World War I made obtaining special items of food harder and it was not long before Ramanujan had health problems.

Right from the start Ramanujan’s collaboration with Hardy led to important results. Hardy was, however, unsure how to approach the problem of Ramanujan’s lack of formal education. He wrote [1]:-

What was to be done in the way of teaching him modern mathematics? The limitations of his knowledge were as startling as its profundity.

Littlewood was asked to help teach Ramanujan rigorous mathematical methods. However he said ([31]):-

… that it was extremely difficult because every time some matter, which it was thought that Ramanujan needed to know, was mentioned, Ramanujan’s response was an avalanche of original ideas which made it almost impossible for Littlewood to persist in his original intention.

The war soon took Littlewood away on war duty but Hardy remained in Cambridge to work with Ramanujan. Even in his first winter in England, Ramanujan was ill and he wrote in March 1915 that he had been ill due to the winter weather and had not been able to publish anything for five months. What he did publish was the work he did in England, the decision having been made that the results he had obtained while in India, many of which he had communicated to Hardy in his letters, would not be published until the war had ended.

On 16 March 1916 Ramanujan graduated from Cambridge with a Bachelor of Science by Research (the degree was called a Ph.D. from 1920). He had been allowed to enrol in June 1914 despite not having the proper qualifications. Ramanujan’s dissertation was on Highly composite numbers and consisted of seven of his papers published in England.

Ramanujan fell seriously ill in 1917 and his doctors feared that he would die. He did improve a little by September but spent most of his time in various nursing homes. In February 1918 Hardy wrote (see [3]):-

Batty Shaw found out, what other doctors did not know, that he had undergone an operation about four years ago. His worst theory was that this had really been for the removal of a malignant growth, wrongly diagnosed. In view of the fact that Ramanujan is no worse than six months ago, he has now abandoned this theory – the other doctors never gave it any support. Tubercle has been the provisionally accepted theory, apart from this, since the original idea of gastric ulcer was given up. … Like all Indians he is fatalistic, and it is terribly hard to get him to take care of himself.

On 18 February 1918 Ramanujan was elected a fellow of the Cambridge Philosophical Society and then three days later, the greatest honour that he would receive, his name appeared on the list for election as a fellow of the Royal Society of London. He had been proposed by an impressive list of mathematicians, namely Hardy, MacMahon, Grace, Larmor, Bromwich, Hobson, Baker, Littlewood, Nicholson, Young, Whittaker, Forsyth and Whitehead. His election as a fellow of the Royal Society was confirmed on 2 May 1918, then on 10 October 1918 he was elected a Fellow of Trinity College Cambridge, the fellowship to run for six years.

The honours which were bestowed on Ramanujan seemed to help his health improve a little and he renewed his effors at producing mathematics. By the end of November 1918 Ramanujan’s health had greatly improved. Hardy wrote in a letter [3]:-

I think we may now hope that he has turned to corner, and is on the road to a real recovery. His temperature has ceased to be irregular, and he has gained nearly a stone in weight. … There has never been any sign of any diminuation in his extraordinary mathematical talents. He has produced less, naturally, during his illness but the quality has been the same. ….

He will return to India with a scientific standing and reputation such as no Indian has enjoyed before, and I am confident that India will regard him as the treasure he is. His natural simplicity and modesty has never been affected in the least by success – indeed all that is wanted is to get him to realise that he really is a success.

Ramanujan sailed to India on 27 February 1919 arriving on 13 March. However his health was very poor and, despite medical treatment, he died there the following year.

The letters Ramanujan wrote to Hardy in 1913 had contained many fascinating results. Ramanujan worked out the Riemann series, the elliptic integrals, hypergeometric series and functional equations of the zeta function. On the other hand he had only a vague idea of what constitutes a mathematical proof. Despite many brilliant results, some of his theorems on prime numbers were completely wrong.

Ramanujan independently discovered results of Gauss, Kummer and others on hypergeometric series. Ramanujan’s own work on partial sums and products of hypergeometric series have led to major development in the topic. Perhaps his most famous work was on the number p(n) of partitions of an integer n into summands. MacMahon had produced tables of the value of p(n) for small numbers n, and Ramanujan used this numerical data to conjecture some remarkable properties some of which he proved using elliptic functions. Other were only proved after Ramanujan’s death.

In a joint paper with Hardy, Ramanujan gave an asymptotic formula for p(n). It had the remarkable property that it appeared to give the correct value of p(n), and this was later proved by Rademacher.

Ramanujan left a number of unpublished notebooks filled with theorems that mathematicians have continued to study. G N Watson, Mason Professor of Pure Mathematics at Birmingham from 1918 to 1951 published 14 papers under the general title Theorems stated by Ramanujan and in all he published nearly 30 papers which were inspired by Ramanujan’s work. Hardy passed on to Watson the large number of manuscripts of Ramanujan that he had, both written before 1914 and some written in Ramanujan’s last year in India before his death.

The picture above is taken from a stamp issued by the Indian Post Office to celebrate the 75th anniversary of his birth.

Aryabhatta – Facts about the Great Indian Astronomer & Mathematician

October 25, 2008

Āryabhatta (b. 476 AD – 550) is the first in the line of great mathematician-astronomers from the classical age of Indian mathematics and Indian astronomy. His most famous works are the Aryabhatiya (499) and Arya-Siddhanta.

Biography

Aryabhata was born in the region lying between Narmada and Godavari, which was known as Ashmaka,and is now identified with Maharashtra, though early Buddhist texts describe Ashmaka as being further south, dakShiNApath or the Deccan, while other texts describe the Ashmakas as having fought Alexander, which would put them further north. Other traditions in India claim that he was from Kerala and that he travelled to the North, or that he was a Maga Brahmin from Gujarat.

However, it is fairly certain that at some point, he went to Kusumapura for higher studies, and that he lived here for some time. Bhāskara I (AD 629) identifies Kusumapura as Pataliputra (modern Patna). He lived there in the dying years of the Gupta empire, the time which is known as the golden age of India, when it was already under Hun attack in the Northeast, during the reign of Buddhagupta and some of the smaller kings before Vishnugupta.

His first name “Arya” is a term used for respect, such as “Sri”, whereas Bhatta is a typical north Indian name — found today usually among the “Bania” (or trader) community in Bihar.

Works

Aryabhata is the author of several treatises on mathematics and astronomy, some of which are lost. His major work, Aryabhatiya, a compendium of mathematics and astronomy, was extensively referred to in the Indian mathematical literature, and has survived to modern times.

The Arya-siddhanta, a lost work on astronomical computations, is known through the writings of Aryabhata’s contemporary Varahamihira, as well as through later mathematicians and commentators including Brahmagupta and Bhaskara I. This work appears to be based on the older Surya Siddhanta, and uses the midnight-day-reckoning, as opposed to sunrise in Aryabhatiya. This also contained a description of several astronomical instruments, the gnomon (shanku-yantra), a shadow instrument (chhAyA-yantra), possibly angle-measuring devices, semi-circle and circle shaped (dhanur-yantra / chakra-yantra), a cylindrical stick yasti-yantra, an umbrella-shaped device called chhatra-yantra, and water clocks of at least two types, bow-shaped and cylindrical.

A third text that may have survived in Arabic translation is the Al ntf or Al-nanf, which claims to be a translation of Aryabhata, but the Sanskrit name of this work is not known. Probably dating from the ninth c., it is mentioned by the Persian scholar and chronicler of India, Abū Rayhān al-Bīrūnī.

Aryabhatiya

Direct details of Aryabhata’s work are therefore known only from the Aryabhatiya. The name Aryabhatiya is due to later commentators, Aryabhata himself may not have given it a name; it is referred by his disciple Bhaskara I as Ashmakatantra or the treatise from the Ashmaka. It is also occasionally referred to as Arya-shatas-aShTa, lit., Aryabhata’s 108, which is the number of verses in the text. It is written in the very terse style typical of the sutra literature, where each line is an aid to memory for a complex system. Thus, the explication of meaning is due to commentators. The entire text consists of 108 verses, plus an introductory 13, the whole being divided into four pAdas or chapters:

gitikApAda: (13 verses) large units of time – kalpa, manvantra, yuga, which present a cosmology that differs from earlier texts such as Lagadha’s Vedanga Jyotisha(ca. 1st c. BC). Also includes the table of sines (jya), given in a single verse. For the planetary revolutions during a mahayuga, the number of 4.32mn years is given.

gaNitapAda (33 verses), covering mensuration (kShetra vyAvahAra), arithmetic and geometric progressions, gnomon / shadows (shanku-chhAyA), simple, quadratic, simultaneous, and indeterminate equations (kuTTaka)

kAlakriyApAda (25 verses) : different units of time and method of determination of positions of planets for a given day. Calculations concerning the intercalary month (adhikamAsa), kShaya-tithis. Presents a seven-day week, with names for days of week.

golapAda (50 verses): Geometric/trigonometric aspects of the celestial sphere, features of the ecliptic, celestial equator, node, shape of the earth, cause of day and night, rising of zodiacal signs on horizon etc.

In addition, some versions cite a few colophons added at the end, extolling the virtues of the work, etc.

The Aryabhatiya presented a number of innovations in mathematics and astronomy in verse form, which were influential for many centuries. The extreme brevity of the text was elaborated in commentaries by his disciple Bhaskara I (Bhashya, ca. 600) and by Nilakantha Somayaji in his Aryabhatiya Bhasya, (1465).

Statue of Aryabhata on the grounds of IUCAA, Pune.
Mathematics

Place Value system and zero

The number place-value system, first seen in the 3rd century Bakhshali Manuscript was clearly in place in his work; he certainly did not use the symbol, but the French mathematician Georges Ifrah argues that knowledge of zero was implicit in Aryabhata’s place-value system as a place holder for the powers of ten with null coefficients.

However, Aryabhata did not use the brahmi numerals; continuing the Sanskritic tradition from Vedic times, he used letters of the alphabet to denote numbers, expressing quantities (such as the table of sines) in a mnemonic form.

Pi as Irrational
Aryabhata worked on the approximation for Pi (π), and may have realized that π is irrational. In the second part of the Aryabhatiyam , he writes

chaturadhikam śatamaśaguam dvāśaśistathā sahasrāām
Ayutadvayaviśkambhasyāsanno vrîttapariaha.

“Add four to 100, multiply by eight and then add 62,000. By this rule the circumference of a circle of diameter 20,000 can be approached.”

In other words, π= ~ 62832/20000 = 3.1416, correct to five digits. The commentator Nilakantha Somayaji, (Kerala School, 15th c.) interprets the word āsanna (approaching), appearing just before the last word, as saying that not only that is this an approximation, but that the value is incommensurable (or irrational). If this is correct, it is quite a sophisticated insight, for the irrationality of pi was proved in Europe only in 1761 by Lambert).

After Aryabhatiya was translated into Arabic (ca. 820 AD) this approximation was mentioned in Al-Khwarizmi’s book on algebra.

Mensuration and trigonometry
In Ganitapada 6, Aryabhata gives the area of triangle as

tribhujasya phalashariram samadalakoti bhujardhasamvargah

that translates to: for a triangle, the result of a perpendicular with the half-side is the area.

Indeterminate Equations
A problem of great interest to Indian mathematicians since ancient times has been to find integer solutions to equations that have the form ax + b = cy, a topic that has come to be known as diophantine equations. Here is an example from Bhaskara’s commentary on Aryabhatiya

Find the number which gives 5 as the remainder when divided by 8; 4 as the remainder when divided by 9; and 1 as the remainder when divided by 7.

i.e. find N = 8x+5 = 9y+4 = 7z+1. It turns out that the smallest value for N is 85. In general, diophantine equations can be notoriously difficult. Such equations were considered extensively in the ancient Vedic text Sulba Sutras, the more ancient parts of which may date back to 800 BCE. Aryabhata’s method of solving such problems, called the kuttaka method. Kuttaka means pulverizing, that is breaking into small pieces, and the method involved a recursive algorithm for writing the original factors in terms of smaller numbers. Today this algorithm, as elaborated by Bhaskara in AD 621, is the standard method for solving first order Diophantine equations, and it is often referred to as the Aryabhata algorithm.

The diophantine equations are of interest in cryptology, and the RSA Conference, 2006, focused on the kuttaka method and earlier work in the Sulvasutras.

Astronomy

Aryabhata’s system of astronomy was called the audAyaka system (days are reckoned from uday, dawn at lanka, equator). Some of his later writings on astronomy, which apparently proposed a second model (ardha-rAtrikA, midnight), are lost, but can be partly reconstructed from the discussion in Brahmagupta’s khanDakhAdyaka. In some texts he seems to ascribe the apparent motions of the heavens to the earth’s rotation.

Motions of the Solar System
Aryabhata appears to have believed that the earth rotates about its axis. This is made clear in the statement, referring to Lanka , which describes the movement of the stars as a relative motion caused by the rotation of the earth:

Like a man in a boat moving forward sees the stationary objects as moving backward, just so are the stationary stars seen by the people in lankA (i.e. on the equator) as moving exactly towards the West. [achalAni bhAni samapashchimagAni – golapAda.]

But the next verse describes the motion of the stars and planets as real movements: “The cause of their rising and setting is due to the fact the circle of the asterisms together with the planets driven by the provector wind, constantly moves westwards at Lanka”.

Lanka (Sri Lanka) is here a reference point on the equator, which was taken as the equivalent to the reference meridian for astronomical calculations.

Aryabhata described a geocentric model of the solar system, in which the Sun and Moon are each carried by epicycles which in turn revolve around the Earth. In this model, which is also found in the Paitāmahasiddhānta (ca. AD 425), the motions of the planets are each governed by two epicycles, a smaller manda (slow) epicycle and a larger śīghra (fast) epicycle. The order of the planets in terms of distance from earth are taken as: the Moon, Mercury, Venus, the Sun, Mars, Jupiter, Saturn, and the asterisms.

The positions and periods of the planets were calculated relative to uniformly moving points, which in the case of Mercury and Venus, move around the Earth at the same speed as the mean Sun and in the case of Mars, Jupiter, and Saturn move around the Earth at specific speeds representing each planet’s motion through the zodiac. Most historians of astronomy consider that this two epicycle model reflects elements of pre-Ptolemaic Greek astronomy. Another element in Aryabhata’s model, the śīghrocca, the basic planetary period in relation to the Sun, is seen by some historians as a sign of an underlying heliocentric model.

Eclipses

He states that the Moon and planets shine by reflected sunlight. Instead of the prevailing cosmogyny where eclipses were caused by pseudo-planetary nodes Rahu and Ketu, he explains eclipses in terms of shadows cast by and falling on earth. Thus the lunar eclipse occurs when the moon enters into the earth-shadow (verse gola.37), and discusses at length the size and extent of this earth-shadow (verses gola.38-48), and then the computation, and the size of the eclipsed part during eclipses. Subsequent Indian astronomers improved on these calculations, but his methods provided the core. This computational paradigm was so accurate that the 18th century scientist Guillaume le Gentil, during a visit to Pondicherry, found the Indian computations of the duration of the lunar eclipse of 1765-08-30 to be short by 41 seconds, whereas his charts (by Tobias Mayer, 1752) were long by 68 seconds.

Aryabhata’s computation of Earth’s circumference as 24,835 miles, which was only 0.2% smaller than the actual value of 24,902 miles. This approximation might have improved on the computation by the Greek mathematician Eratosthenes (c.200 BC), whose exact computation is not known in modern units.

Sidereal periods
Considered in modern English units of time, Aryabhata calculated the sidereal rotation (the rotation of the earth referenced the fixed stars) as 23 hours 56 minutes and 4.1 seconds; the modern value is 23:56:4.091. Similarly, his value for the length of the sidereal year at 365 days 6 hours 12 minutes 30 seconds is an error of 3 minutes 20 seconds over the length of a year. The notion of sidereal time was known in most other astronomical systems of the time, but this computation was likely the most accurate in the period.

Heliocentrism

Āryabhata claims that the Earth turns on its own axis and some elements of his planetary epicyclic models rotate at the same speed as the motion of the planet around the Sun. This has suggested to some interpreters that Āryabhata’s calculations were based on an underlying heliocentric model in which the planets orbit the Sun.[12][13] A detailed rebuttal to this heliocentric interpretation is in a review which describes B. L. van der Waerden’s book as “show[ing] a complete misunderstanding of Indian planetary theory [that] is flatly contradicted by every word of Āryabhata’s description,” although some concede that Āryabhata’s system stems from an earlier heliocentric model of which he was unaware. It has even been claimed that he considered the planet’s paths to be elliptical, although no primary evidence for this has been cited. Though Aristarchus of Samos (3rd century BC) and sometimes Heraclides of Pontus (4th century BC) are usually credited with knowing the heliocentric theory, the version of Greek astronomy known in ancient India, Paulisa Siddhanta (possibly by a Paul of Alexandria) makes no reference to a Heliocentric theory.

Legacy

Aryabhata’s work was of great influence in the Indian astronomical tradition, and influenced several neighbouring cultures through translations. The Arabic translation during the Islamic Golden Age (ca. 820), was particularly influential. Some of his results are cited by Al-Khwarizmi, and he is referred to by the 10th century Arabic scholar Al-Biruni, who states that Āryabhata’s followers believed the Earth to rotate on its axis.

His definitions of sine, as well as cosine (kojya), versine (ukramajya), and inverse sine (otkram jya), influenced the birth of trigonometry. He was also the first to specify sine and versine (1 – cosx) tables, in 3.75° intervals from 0° to 90°, to an accuracy of 4 decimal places.

In fact, the modern names “sine” and “cosine”, are a mis-transcription of the words jya and kojya as introduced by Aryabhata. They were transcribed as jiba and kojiba in Arabic. They were then misinterpreted by Gerard of Cremona while translating an Arabic geometry text to Latin; he took jiba to be the Arabic word jaib, which means “fold in a garment”, L. sinus (c.1150).

Aryabhata’s astronomical calculation methods were also very influential. Along with the trigonometric tables, they came to be widely used in the Islamic world, and were used to compute many Arabic astronomical tables (zijes). In particular, the astronomical tables in the work of the Arabic Spain scientist Al-Zarqali (11th c.), were translated into Latin as the Tables of Toledo (12th c.), and remained the most accurate Ephemeris used in Europe for centuries.

Calendric calculations worked out by Aryabhata and followers have been in continuous use in India for the practical purposes of fixing the Panchanga, or Hindu calendar, These were also transmitted to the Islamic world, and formed the basis for the Jalali calendar introduced 1073 by a group of astronomers including Omar Khayyam, versions of which (modified in 1925) are the national calendars in use in Iran and Afghanistan today. The Jalali calendar determines its dates based on actual solar transit, as in Aryabhata (and earlier Siddhanta calendars). This type of calendar requires an Ephemeris for calculating dates. Although dates were difficult to compute, seasonal errors were lower in the Jalali calendar than in the Gregorian calendar.

India’s first satellite Aryabhata, was named after him.

The lunar crater Aryabhata is named in his honour.

Famous Indian Mathematicians Biography

September 3, 2008

The most fundamental contribution of ancient India in mathematics is the invention of decimal system of enumeration, including the invention of zero. The decimal system uses nine digits (1 to 9) and the symbol zero (for nothing) to denote all natural numbers by assigning a place value to the digits. The Arabs carried this system to Africa and Europe.

The Vedas and Valmiki Ramayana used this system, though the exact dates of these works are not known. MohanjoDaro and Harappa excavations (which may be around 3000 B.C. old) also give specimens of writing in India. Aryans came 1000 years later, around 2000 B.C. Being very religious people, they were deeply interested in planetary positions to calculate auspicious times, and they developed astronomy and mathematics towards this end. They identified various nakshatras (constellations) and named the months after them. They could count up to 1012, while the Greeks could count up to 104 and Romans up to 108. Values of irrational numbers such as and were also known to them to a high degree of approximation. Pythagoras Theorem can be also traced to the Aryan’s Sulbasutras. These Sutras, estimated to be between 800 B.C. and 500 B.C., cover a large number of geometric principles. Jaina religious works (dating from 500 B.C. to 100 B.C.) show they knew how to solve quadratic equations (though ancient Chinese and Babylonians also knew this prior to 2000 B.C.). Jainas used as the value of (circumference = x Diameter). They were very fond of large numbers, and they classified numbers as enumerable, unenumerable and infinite. The Jainas also worked out formulae for permutations and combinations though this knowledge may have existed in Vedic times. Sushruta Samhita (famous medicinal work, around 6th century B.C.) mentions that 63 combinations can be made out of 6 different rasas (tastes -bitter, sour, sweet, salty, astringent and hot).

In the year 1881 A.D., at a village named Bakhshali near Peshawar, a farmer found a manuscript during excavation. About 70 leaves were found, and are now famous as the Bakhshali Manuscript. Western scholars estimate its date as about third or fourth century A.D. It is devoted mostly to arithmetic and algebra, with a few problems on geometry and mensuration.

With this historical background, we come to the famous Indian mathematicians.

Aryabhata (475 A.D. -550 A.D.) is the first well known Indian mathematician. Born in Kerala, he completed his studies at the university of Nalanda. In the section Ganita (calculations) of his astronomical treatise Aryabhatiya (499 A.D.), he made the fundamental advance in finding the lengths of chords of circles, by using the half chord rather than the full chord method used by Greeks. He gave the value of as 3.1416, claiming, for the first time, that it was an approximation. (He gave it in the form that the approximate circumference of a circle of diameter 20000 is 62832.) He also gave methods for extracting square roots, summing arithmetic series, solving indeterminate equations of the type ax -by = c, and also gave what later came to be known as the table of Sines. He also wrote a text book for astronomical calculations, Aryabhatasiddhanta. Even today, this data is used in preparing Hindu calendars (Panchangs). In recognition to his contributions to astronomy and mathematics, India’s first satellite was named Aryabhata.

Brahmagupta (598 A.D. -665 A.D.) is renowned for introduction of negative numbers and operations on zero into arithmetic. His main work was Brahmasphutasiddhanta, which was a corrected version of old astronomical treatise Brahmasiddhanta. This work was later translated into Arabic as Sind Hind. He formulated the rule of three and proposed rules for the solution of quadratic and simultaneous equations. He gave the formula for the area of a cyclic quadrilateral as where s is the semi perimeter. He was the first mathematician to treat algebra and arithmetic as two different branches of mathematics. He gave the solution of the indeterminate equation Nx²+1 = y². He is also the founder of the branch of higher mathematics known as “Numerical Analysis”.

After Brahmagupta, the mathematician of some consequence was Sridhara, who wrote Patiganita Sara, a book on algebra, in 750 A.D. Even Bhaskara refers to his works. After Sridhara, the most celebrated mathematician was Mahaviracharaya or Mahavira. He wrote Ganita Sara Sangraha in 850 A.D., which is the first text book on arithmetic in present day form. He is the only Indian mathematician who has briefly referred to the ellipse (which he called Ayatvrit). The Greeks, by contrast, had studied conic sections in great detail.

Bhaskara (1114 A.D. -1185 A.D.) or Bhaskaracharaya is the most well known ancient Indian mathematician. He was born in 1114 A.D. at Bijjada Bida (Bijapur, Karnataka) in the Sahyadari Hills. He was the first to declare that any number divided by zero is infinity and that the sum of any number and infinity is also infinity. He is famous for his book Siddhanta Siromani (1150 A.D.). It is divided into four sections -Leelavati (a book on arithmetic), Bijaganita (algebra), Goladhayaya (chapter on sphere -celestial globe), and Grahaganita (mathematics of the planets). Leelavati contains many interesting problems and was a very popular text book. Bhaskara introduced chakrawal, or the cyclic method, to solve algebraic equations. Six centuries later, European mathematicians like Galois, Euler and Lagrange rediscovered this method and called it “inverse cyclic”. Bhaskara can also be called the founder of differential calculus. He gave an example of what is now called “differential coefficient” and the basic idea of what is now called “Rolle’s theorem”. Unfortunately, later Indian mathematicians did not take any notice of this. Five centuries later, Newton and Leibniz developed this subject. As an astronomer, Bhaskara is renowned for his concept of Tatkalikagati (instantaneous motion).

After this period, India was repeatedly raided by muslims and other rulers and there was a lull in scientific research. Industrial revolution and Renaissance passed India by. Before Ramanujan, the only noteworthy mathematician was Sawai Jai Singh II, who founded the present city of Jaipur in 1727 A.D. This Hindu king was a great patron of mathematicians and astronomers. He is known for building observatories (Jantar Mantar) at Delhi, Jaipur, Ujjain, Varanasi and Mathura. Among the instruments he designed himself are Samrat Yantra, Ram Yantra and Jai Parkash.

Well known Indian mathematicians of 20th century are:

Srinivasa Aaiyangar Ramanujan is undoubtedly the most celebrated Indian Mathematical genius. He was born in a poor family at Erode in Tamil Nadu on December 22, 1887. Largely self taught, he feasted on Loney’s Trigonometry at the age of 13, and at the age of 15, his senior friends gave him Synopsis of Elementary Results in Pure and Applied Mathematics by George Carr. He used to write his ideas and results on loose sheets. His three filled notebooks are now famous as Ramanujan’s Frayed Notebooks. Though he had no qualifying degree, the University of Madras granted him a monthly scholarship of Rs. 75 in 1913. A few months earlier, he had sent a letter to great mathematician G.H. Hardy, in which he mentioned 120 theorems and formulae. Hardy and his colleague at Cambridge University, J.E. Littlewood immediately recognised his genius. Ramanujan sailed for Britain on March 17, 1914. Between 1914 and 1917, Ramanujan published 21 papers, some in collaboration with Hardy. His achievements include Hardy-Ramanujan-Littlewood circle method in number theory, Roger-Ramanujan’s identities in partition of numbers, work on algebra of inequalities, elliptic functions, continued fractions, partial sums and products of hypergeometric series, etc. He was the second Indian to be elected Fellow of the Royal Society in February, 1918. Later that year, he became the first Indian to be elected Fellow of Trinity College, Cambridge. Ramanujan had an intimate familiarity with numbers. During an illness in England, Hardy visited Ramanujan in the hospital. When Hardy remarked that he had taken taxi number 1729, a singularly unexceptional number, Ramanujan immediately responded that this number was actually quite remarkable: it is the smallest integer that can be represented in two ways by the sum of two cubes: 1729=1³+12³=9³+10³.
Unfortunately, Ramanujan’s health deteriorated due to tuberculosis, and he returnted to India in 1919. He died in Madras on April 26, 1920.

P.C. Mahalanobis : He founded the Indian Statistical Research Institute in Calcutta. In 1958, he started the National Sample Surveys which gained international fame. He died in 1972 at the age of 79.

C.R. Rao : A well known statistician, famous for his “theory of estimation”(1945). His formulae and theory include “Cramer -Rao inequality”, “Fischer -Rao theorem” and “Rao – Blackwellisation”.

D.R. Kaprekar (1905-1988) : Fond of numbers. Well known for “Kaprekar Constant” 6174. Take any four digit number in which all digits are not alike. Arrange its digits in descending order and subtract from it the number formed by arranging the digits in ascending order. If this process is repeated with reminders, ultimately number 6174 is obtained, which then generates itself.

Harish Chandra (1923-1983) : Greatly developed the branch of higher mathematics known as the infinite dimensional group representation theory.

Narendra Karmarkar : India born Narendra Karmarkar, working at Bell Labs USA, stunned the world in 1984 with his new algorithm to solve linear programming problems. This made the complex calculations much faster, and had immediate applications in airports, warehouses, communication networks etc.



Famous Indian Mathematicians.. – not talking about ancient Indian math stalwarts from centuries ago like Brahamagupta, Bhaskara or Aryabhatta nor THE famous Indian math wizard of the last couple centuries – Srinivasa Ramanujan ..
But want to bring to your attention two mathematicians making the headlines/news in the last 5-6 years..
1. Chandrakant Khare – Read my blog post about ‘Chandrakant Khare and Fermat’s Last Theorem’ for further details..
2. Also another mathematician making the news recently.. – Divakar Viswanath of Univ. of Michigan-Ann Arbor (PhD at Cornell). In 1999, he wrote a paper about “Divakar’s constant” which is the ‘limit of the ratio of consecutive random fibonacci numbers‘. Lot of discussion around this since with some people calling this new constant almost as important as the “golden ratio” (I need to find a reference/citation for this bold statement – for someone calling the constant as important as the much vaunted golden ratio – it was quoted to me by a friend).

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