Mathematicians

AncientApastamba · Baudhayana · Katyayana · Manava · Pāṇini · Pingala · Yajnavalkya

Classical

Aryabhata I · Aryabhata II · Bhāskara I · Bhāskara II · Melpathur Narayana Bhattathiri · Brahmadeva · Brahmagupta · Brihaddeshi · Halayudha · Jyesthadeva · Madhava of Sangamagrama · Mahavira · Mahendra Suri · Munishvara · Narayana Pandit · Parameshvara · Achyuta Pisharati · Jagannatha Samrat · Nilakantha Somayaji · Sripati · Sridhara · Gangesha Upadhyaya · Varahamihira · Sankara Variar · Virasena

Modern

Shreeram Shankar Abhyankar · A. A. Krishnaswami Ayyangar · Raj Chandra Bose · Satyendra Nath Bose · Harish-Chandra · Subrahmanyan Chandrasekhar · D. K. Ray-Chaudhuri · Sarvadaman Chowla · Narendra Karmarkar · Prasanta Chandra Mahalanobis · Jayant Narlikar · Vijay Kumar Patodi · Srinivasa Ramanujan · Calyampudi Radhakrishna Rao · S. N. Roy · S. S. Shrikhande · Navin M. Singhi · Mathukumalli V. Subbarao · S. R. Srinivasa Varadhan

Treatises

Aryabhatiya · Bakhshali manuscript · Brahmasphutasiddhanta · Karanapaddhati · Maha-Siddhanta · Paulisa Siddhanta · Paitamaha Siddhanta · Romaka Siddhanta · Sadratnamala  · Śulba Sūtras · Surya Siddhanta · Tantrasamgraha · Vasishtha Siddhanta · Ve ṇ vāroha  · Yuktibasha · Yavanajataka

CentersJantar Mantar (Jaipur) · Kerala school of astronomy and mathematics · Ujjain · Yantra Mandir · Yantra Mandir (Delhi)

Influences Babylonian mathematics · Greek mathematics · Islamic mathematics

Influenced Chinese mathematics · Islamic mathematics · European mathematics

Indian mathematics also known as Hindu mathematics[1] refers to the mathematics that emerged in the Indian subcontinent,[2] from ancient times until the end of the 18th century. In the classical period of Indian mathematics (400 AD to 1200 AD), important contributions were made by scholars like Aryabhata, Brahmagupta, and Bhaskara II. The decimal number system in use today[3] and the binary number system [4] were first recorded in Indian mathematics.[5] Indian mathematicians made early contributions to the study of the concept of zero as a number,[6] negative numbers,[7] arithmetic, and algebra.[8] In addition, trigonometry [9] was further advanced in India, and, in particular, the modern definitions of sine and cosine were developed there.[10] These mathematical concepts were transmitted to the Middle East, China, and Europe [8] and led to further developments that now form the foundations of many areas of mathematics.

Ancient and medieval Indian mathematical works, all composed in Sanskrit, usually consisted of a section of sutras in which a set of rules or problems were stated with great economy in verse in order to aid memorization by a student. This was followed by a second section consisting of a prose commentary (sometimes multiple commentaries by different scholars) that explained the problem in more detail and provided justification for the solution. In the prose section, the form (and therefore its memorization) was not considered as important as the ideas involved.[2][11] All mathematical works were orally transmitted until approximately 500 BCE; thereafter, they were transmitted both orally and in manuscript form. The oldest extant mathematical document

produced on the Indian subcontinent is the birch bark Bakhshali Manuscript, discovered in 1881 in the village of Bakhshali, near Peshawar (modern day Pakistan) and is likely from the 7th century CE.[12][13]

A later landmark in Indian mathematics was the development of the series expansions for trigonometric functions (sine, cosine, and arc tangent) by mathematicians of the Kerala school in the 15th century CE. Their remarkable work, completed two centuries before the invention of calculus in Europe, provided what is now considered the first example of a power series (apart from geometric series).[14] However, they did not formulate a systematic theory of differentiation and integration, nor is there any direct evidence of their results being transmitted outside Kerala.[15][16][17][18]

Aryabhata (IAST: Āryabhaṭa; Sanskrit: आर्य�भटः�) (476–550 CE) was 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 CE, when he was 23 years old) and the Arya-siddhanta.

Name

While there is a tendency to misspell his name as "Aryabhatta" by analogy with other names having the "bhatta" suffix, his name is properly spelled Aryabhata: every astronomical text spells his name thus,[1] including Brahmagupta's references to him "in more than a hundred places by name".[2] Furthermore, in most instances "Aryabhatta" does not fit the metre either.[1]

Birth

Aryabhata mentions in the Aryabhatiya that it was composed 3,600 years into the Kali Yuga, when he was 23 years old. This corresponds to 499 CE, and implies that he was born in 476 CE.[1]

Aryabhata provides no information about his place of birth. The only information comes from Bhāskara I, who describes Aryabhata as āśmakīya, "one belonging to the aśmaka country." While aśmaka was originally situated in the northwest of India, it is widely attested that, during the Buddha's time, a branch of the Aśmaka people settled in the region between the Narmada and Godavari rivers, in the South Gujarat–North Maharashtra region of central India. Aryabhata is believed to have been born there.[1][3] However, early Buddhist texts describe Ashmaka as being further south, in dakshinapath or the Deccan, while other texts describe the Ashmakas as having fought Alexander, which would put them further north.[3]

work

It is fairly certain that, at some point, he went to Kusumapura for advanced studies and that he lived there for some time.[4] Both Hindu and Buddhist tradition, as well as Bhāskara I (CE 629), identify Kusumapura as Pā ṭ aliputra , modern Patna.[1] A verse mentions that Aryabhata was the head of an institution (kulapa) at Kusumapura, and, because the university of Nalanda was in Pataliputra at the time and had an astronomical observatory, it is speculated that Aryabhata might have been the head of the Nalanda university as well.[1] Aryabhata is also reputed to have set up an observatory at the Sun temple in Taregana, Bihar.[5]

Other hypotheses

It was suggested that Aryabhata may have been from Kerala, but K. V. Sarma, an authority on Kerala's astronomical tradition, disagreed[1] and pointed out several errors in this hypothesis.[6]

Aryabhata mentions "Lanka" on several occasions in the Aryabhatiya, but his "Lanka" is an abstraction, standing for a point on the equator at the same longitude as his Ujjayini.[7]

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 mathematical part of the Aryabhatiya covers arithmetic, algebra, plane trigonometry, and spherical trigonometry. It also contains continued fractions, quadratic equations, sums-of-power series, and a table of sines.

The Arya-siddhanta, a lost work on astronomical computations, is known through the writings of Aryabhata's contemporary, Varahamihira, and 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. It also contained a description of several astronomical instruments: the gnomon (shanku-yantra), a shadow instrument (chhAyA-yantra), possibly angle-measuring devices, semicircular and circular (dhanur-yantra / chakra-yantra), a cylindrical stick yasti-yantra, an umbrella-shaped device called the chhatra-yantra, and water clocks of at least two types, bow-shaped and cylindrical.[3]

A third text, which may have survived in the Arabic translation, is Al ntf or Al-nanf. It claims that it is a translation by Aryabhata, but the Sanskrit name of this work is not known. Probably dating from the 9th century, it is mentioned by the Persian scholar and chronicler of India, Abū Rayhān al-Bīrūnī.

[3]Aryabhatiya

Direct details of Aryabhata's work are known only from the Aryabhatiya. The name "Aryabhatiya" is due to later commentators. Aryabhata himself may not have given it a name. His disciple Bhaskara I calls it Ashmakatantra (or the treatise from the Ashmaka). It is also occasionally referred to as Arya-shatas-aShTa (literally, Aryabhata's 108), because there are 108 verses in the text. It is written in the very terse style typical of sutra literature, in which each line is an aid to memory for a complex system. Thus, the explication of meaning is due to commentators. The text consists of the 108 verses and 13 introductory verses, and is divided into four pādas or chapters:

1. Gitikapada: (13 verses): large units of time—kalpa, manvantra, and yuga—which present a cosmology different from earlier texts such as Lagadha's Vedanga Jyotisha(ca. 1st century BCE). There is also a table of sines (jya), given in a single verse. The duration of the planetary revolutions during a mahayuga is given as 4.32 million years.

2. Ganitapada (33 verses): covering mensuration (kṣetra vyāvahāra), arithmetic and geometric progressions, gnomon / shadows (shanku-chhAyA), simple, quadratic, simultaneous, and indeterminate equations (kuTTaka)

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

4. 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 CE) and by Nilakantha Somayaji in his Aryabhatiya Bhasya, (1465 CE).

Bhāskara (c. 600 – c. 680) (Marathi: भ�स्कर commonly called Bhaskara I to avoid confusion with the 12th century mathematician Bhāskara II) was a 7th century Indian mathematician, who was apparently the first to write numbers in the Hindu-Arabic decimal system with a circle for the zero, and who gave a unique and remarkable rational approximation of the sine function in his commentary on Aryabhata's work.[1] This commentary, Āryabhaṭīyabhāṣya, written in 629 CE, is the oldest known prose work in Sanskrit on mathematics and astronomy. He also wrote two astronomical works in the line of Aryabhata's school, the Mahābhāskarīya and the Laghubhāskarīya.[2]

We know little about Bhāskara's life. He was "probably a Marathi astronomer".[3]

His astronomical education was given by his father. Bhaskara is considered the most important scholar of Aryabhata's astronomical school. He and Brahmagupta are the most renowned Indian mathematicians who made considerable contributions to the study of fractions.

Representation of numbers

Bhaskara's probably most important mathematical contribution concerns the representation of numbers in a positional system. The first positional representations were known to Indian astronomers about 500. However, the numbers were not written in figures, but in words or allegories, and were organized in verses. For instance, the number 1 was given as moon, since it exists only once; the number 2 was represented by wings, twins, or eyes, since they always occur in pairs; the number 5 was given by the (5) senses. Similar to our current decimal system, these words were aligned such that each number assigns the factor of the power of ten corresponding to its position, only in reverse order: the higher powers were right from the lower ones.

His system is truly positional, since the same words representing, can also be used to represent the values 40 or 400.[4] Quite remarkably, he often explains a number given in this system, using the formula ankair api ("in figures this reads"), by repeating it written with the first nine Brahmi numerals, using a small circle for the zero . Contrary to his word number system, however, the figures are written in descending valuedness from left to right, exactly as we do it today. Therefore, at least since 629 the decimal system is definitely known to the Indian scientists. Presumably, Bhaskara did not invent it, but he was the first having no compunctions to use the Brahmi numerals in a scientific contribution in Sanskrit.

Brahmagupta (Sanskrit: ब्रह्मगु�प्त; ( listen (help·info)) (598–668) was an Indian mathematician and astronomer who had written numerous important works on mathematics and astronomy. His most well known being the Brahmasphutasiddhanta (Correctly Established

Doctrine of Brahma). Written in 628 in Bhinmal, it composed of 25 chapters detailing several

unprecedented mathematical conjectures.

Life and work

Brahmagupta was born in 598 CE(it is believed) in Bhinmal city in the state of Rajasthan of northwest India. He likely lived most of his life in Bhillamala (modern Bhinmal in Rajasthan) in the empire of Harsha during the reign (and possibly under the patronage) of King Vyaghramukha.[1] As a result, Brahmagupta is often referred to as Bhillamalacarya, that is, the teacher from Bhillamala Bhinmal. He was the head of the astronomical observatory at Ujjain, and during his tenure there wrote four texts on mathematics and astronomy: the Cadamekela in 624, the Brahmasphutasiddhanta in 628, the Khandakhadyaka in 665, and the Durkeamynarda in 672. The Brahmasphutasiddhanta (Corrected Treatise of Brahma) is arguably his most famous work. The historian al-Biruni (c. 1050) in his book Tariq al-Hind states that the Abbasid caliph al-Ma'mun had an embassy in India and from India a book was brought to Baghdad which was translated into Arabic as Sindhind. It is generally presumed that Sindhind is none other than Brahmagupta's Brahmasphuta-siddhanta.[2]

Although Brahmagupta was familiar with the works of astronomers following the tradition of Aryabhatiya, it is not known if he was familiar with the work of Bhaskara I, a contemporary.[1] Brahmagupta had a plethora of criticism directed towards the work of rival astronomers, and in his Brahmasphutasiddhanta is found one of the earliest attested schisms among Indian mathematicians. The division was primarily about the application of mathematics to the physical world, rather than about the mathematics itself. In Brahmagupta's case, the disagreements stemmed largely from the choice of astronomical parameters and theories.[1] Critiques of rival theories appear throughout the first ten astronomical chapters and the eleventh chapter is entirely

devoted to criticism of these theories, although no criticisms appear in the twelfth and eighteenth chapters.[1]

Brahmagupta's most famous work is his Brahmasphutasiddhanta. It is composed in elliptic verse, as was common practice in Indian mathematics, and consequently has a poetic ring to it. As no proofs are given, it is not known how Brahmagupta's mathematics was derived.[3]

Mahavira was a 9th-century Indian mathematician from Gulbarga who asserted that the square root of a negative number did not exist. He gave the sum of a series whose terms are squares of an arithmetical progression and empirical rules for area and perimeter of an ellipse. He was patronised by the great Rashtrakuta king Amoghavarsha.[1]

Mahavira was the author of Ganit Saar Sangraha. He separated Astrology from Mathematics. He expounded on the same subjects on which Aryabhata and Brahmagupta contended, but he expressed them more clearly. He is highly respected among Indian Mathematicians, because of his establishment of terminology for concepts such as equilateral, and isosceles triangle; rhombus; circle and semicircle. Mahavira's eminence spread in all South India and his books proved inspirational to other Mathematicians in Southern India.[2] It was translated into Telugu language by Pavuluri Mallana as Saar Sangraha Ganitam.

Pavuluri Mallana, who followed Adikavi Nannaya is a mathematician of 11th century. He composed 'Ganitam' is the first Telugu rendering of an original Sanskrit work on mathematics.[1]

He was contemporary of king Rajaraja Narendra (1022–1063 AD). He has translated Ganitasara, a mathematical treatise of Mahivaracharya into Telugu language as Sara Sangraha Ganitamu.[2] He also wrote Bhadradri Rama Satakamu published by Vavilla Ramaswamy Sastrulu and Sons in 1916.[3]

Rajaraja Narendra has donated Navakhandavada agraharam near Pithapuram to a Brahmin named Mallana. His grandson also named Mallana is the famous writer. This Pavuluru village is presently in Parchuru Mandal of Prakasam district.

Daivajna Varāhamihira (Devanagari: वर�हमि�हिहर; 505 – 587), also called Varaha, or Mihira was an Indian astronomer, mathematician, and astrologer who lived in Ujjain. He is considered to be one of the nine jewels (Navaratnas) of the court of legendary king Vikramaditya (thought to be the Gupta emperor Chandragupta II Vikramaditya).

Works

Pancha-Siddhantika

Varahamihira's main work is the book Pañcasiddhāntikā (or Pancha-Siddhantika, "[Treatise] on the Five [Astronomical] Canons) dated ca. 575 CE gives us information about older Indian texts which are now lost. The work is a treatise on mathematical astronomy and it summarises five earlier astronomical treatises, namely the Surya Siddhanta, Romaka Siddhanta, Paulisa Siddhanta, Vasishtha Siddhanta and Paitamaha Siddhantas. It is a compendium of native Indian as well as Hellenistic astronomy (including Greek, Egyptian and Roman elements).[1]

The 11th century Arabian scholar Alberuni also described the details of "The Five Astronomical Canons":

"They [the Indians] have 5 Siddhāntas:

Sūrya-Siddhānta, ie. the Siddhānta of the Sun, composed by Lāṭadeva, Vasishtha-siddhānta, so called from one of the stars of the Great Bear, composed

by Vishnucandra, Pulisa-siddhānta, so called from Paulisa, the Greek, from the city of Saintra,

which is supposed to be Alexandria, composed by Pulisa. Romaka-siddhānta, so called from the Rūm, ie. the subjects of the Roman Empire,

composed by Śrīsheṇa. Brahma-siddhānta, so called from Brahman, composed by Brahmagupta, the son

of Jishṇu, from the town of Bhillamāla between Multān and Anhilwāra, 16 yojanas from the latter place."[2]

===Brihat-Samhita=== saurabh

Main article: Brihat-Samhita

Varahamihira's other most important contribution is the encyclopedic Brihat-Samhita.

Varahamihira also made important contributions to mathematics. He was also an astrologer. He wrote on all the three main branches of Jyotisha astrology:

Brihat Jataka - is considered as one the five main treatises on Hindu astrology on horoscopy.

Daivaigya Vallabha Laghu Jataka Yoga Yatra Vivaha Patal His son Prithuyasas also contributed in the Hindu astrology; his book "Hora Saara" is a

famous book on horoscopy.

Narayana Pandita (Sanskrit: ना�र�र्यण पण्डि��त) (1340–1400) was a major mathematician of the Kerala school. He wrote the Ganita Kaumudi in 1356 about mathematical operations. The work anticipated many developments in combinatorics.

Narayana Pandit had written two works, an arithmetical treatise called Ganita Kaumudi and an algebraic treatise called Bijganita Vatamsa. Narayanan is also thought to be the author of an elaborate commentary of Bhaskara II's Lilavathi, titled Karmapradipika (or Karma-Paddhati).[1] Although the Karmapradipika contains little original work, it contains seven different methods for squaring numbers, a contribution that is wholly original to the author, as well as contributions to algebra and magic squares.[1]

Narayanan's other major works contain a variety of mathematical developments, including a rule to calculate approximate values of square roots, investigations into the second order indeterminate equation nq2 + 1 = p2 (Pell's equation), solutions of indeterminate higher-order

equations, mathematical operations with zero, several geometrical rules, and a discussion of magic squares and similar figures.[1] Evidence also exists that Narayana made minor contributions to the ideas of differential calculus found in Bhaskara II's work. Narayana has also made contributions to the topic of cyclic quadrilaterals.[2] Narayana is also credited with developing a method for systematic generation of all permutations of a given sequence.

Vatasseri Paramesvara (Malayalam: വടശ്ശേ�രി� പരിശ്ശേ�ശ്വരിന്�‍)' (ca.1380–1460)[1] was a major Indian mathematician and astronomer of the Kerala school of astronomy and mathematics founded by Madhava of Sangamagrama. He was also an astrologer. Paramesvara was a proponent of observational astronomy in medieval India and he himself had made a series of eclipse observations to verify the accuracy of the computational methods then in use. Based on his eclipse observations, Paramesvara proposed several corrections to the astronomical parameters which had been in use since the times of Aryabhata. The computational scheme based on the revised set of parameters has come to be known as the Drgganita system. Paramesvara was also a prolific writer on matters relating to astronomy. At least 25 manuscripts have been identified as being authored by Paramesvara.[1]

Srīnivāsa Aiyangār Rāmānujan FRS, better known as Srinivasa Iyengar Ramanujan (Tamil: சீனி�வா�ச இரா�மா�னுஜன் or ஸ்ரீனி�வா�ஸ ஐயங்கா�ர் ரா�மா�னுஜன்) (22 December 1887 – 26 April 1920) was an Indian mathematician and autodidact who, with almost no formal training in pure mathematics, made substantial contributions to mathematical analysis, number theory, infinite series and continued fractions. Rāmānujan's talent was said, by the prominent English mathematician G.H. Hardy, to be in the same league as legendary mathematicians such as Euler, Gauss, Newton and Archimedes [1].

Born and raised in Erode, Tamil Nadu, India, Ramanujan first encountered formal mathematics at age 10. He demonstrated a natural ability, and was given books on advanced trigonometry written by S. L. Loney.[2] He had mastered them by age 12, and even discovered theorems of his own. He demonstrated unusual mathematical skills at school, winning accolades and awards. By 17, Ramanujan conducted his own mathematical research on Bernoulli numbers and the Euler–Mascheroni constant. He received a scholarship to study at Government College in Kumbakonam, but lost it when he failed his non-mathematical coursework. He joined another college to pursue independent mathematical research, working as a clerk in the Accountant-General's office at the Madras Port Trust Office to support himself.[3] In 1912–1913, he sent samples of his theorems to three academics at the University of Cambridge. Only G. H. Hardy recognized the brilliance of his work, subsequently inviting Ramanujan to visit and work with him at Cambridge. He became a Fellow of the Royal Society and a Fellow of Trinity College, Cambridge, dying of illness, malnutrition and possibly liver infection in 1920 at the age of 32.

Biographical details

Paramesvara was a Namputiri Brahmin of Bhrgugotra following the Ashvalayanasutra of the Rgveda. Paramesvara's family name (Illam) was Vatasseri (also called Vatasreni) and his family resided in in the village of Alattur (Sanskritised as Asvatthagrama) in Kerala. Alattur is situated on the northern bank of the river Nila (river Bharathappuzha) at its mouth in Kerala. He

was a grandson of a disciple of Govinda Bhattathiri (1237 – 1295 CE), a legendary figure in the astrological traditions of Kerala.

Paramesvara studied under teachers Rudra and Narayana, and also under Sangamagrama Madhava (c. 1350 – c. 1425) the founder of the Kerala school of astronomy and mathematics. Damodara, another prominent member of the Kerala school, was his son and also his pupil. Paramesvara was also a teacher of Nilakantha Somayaji (1444-1544) the author of the celebrated Tantrasamgraha.

Sridhara (c. 870 Bengal?, India – c. 930 India) was an Indian mathematician known for two treatises: Trisatika (sometimes called the Patiganitasara) and the Patiganita. He wrote on practical applications of algebra and was one of the first to give a formula for solving quadratic equations.

Nilakantha Somayaji (Malayalam: ന് ലകണ്�ഠ ശ്ശേ���യാ�ജി�, Hindi: ना�लक�ठ सो �र्य�जि") (1444–1544) (also referred to as Nilakantha Somayajin, Nilakantha Somasutvan and Kelallur Comatiri[1]) was a major mathematician and astronomer of theKerala school of astronomy and mathematics. One of his most influential works was the comprehensive astronomical treatise Tantrasamgraha completed in 1501. He had also composed an elaborate commentary on Aryabhatiya called the Aryabhatiya Bhasya. In this Bhasya, Nilakantha had discussed infinite series expansions of trigonometric functions and problems of algebra and spherical geometry. Grahapareeksakrama is a manual on making observations in astronomy based on instruments of the time.

Nilakantha Somayaji was one of the very few authors of the scholarly traditions of India who had cared to record details about his own life and times. So fortunately a few accurate particulars about Nilakantha Somayaji are known.[2][3]

In one of his works titled Siddhanta-darpana and also in his own commentary on Siddhanta-darpana, Nilakantha Somayaji has stated that he was born on Kali-day 1,660,181 which works out to 14 th June 1444 CE. A contemporary reference to Nilakantha Somayaji in a Malayalam work on astrology implies that Somayaji lived to a ripe old age even to become a centenarian. Sankara Variar, a pupil of Nilakantha Somayaji, in his commentary on Tantrasamgraha titled Tantrasamgraha-vyakhya, points out that the first and last verses of Tantrasamgraha contain chronograms specifying the Kali-days of the commencement (1,680,548) and of completion (1,680,553) of Somayaji's magnum opus Tantrasamgraha. Both these days occur in 1500 CE.

In Aryabhatiya-bhashya, Nilakantha Somayaji has stated that he was the son of Jatavedas and he had a brother named Sankara. Somayaji has further stated that he was a Bhatta belonging to the Gargya-gotra and was a follower of Asvalayana-sutra of Rgveda. References in his own Laghuramayana indicate that Nilakantha Somayaji was a member of the Kelallur family (Sanskritised as Kerala-sad-grama) residing at Kundagrama, now known as Trikkandiyur near

modern Tirur in Kerala. His wife was named Arya and he had two sons Rama and Dakshinamurti.

Nilakantha Somayaji studied vedanta and some aspects of astronomy under one Ravi. However, It was Damodara, son of Kerala-drgganita author Paramesvara, who initiated him into the science of astronomy and instructed him in the basic principles of mathematical computations. The great Malayalam poet Thunchaththu Ramanujan Ezhuthachan is said to have been a student of Nilakantha Somayaji.

The epithet Somayaji is a title assigned to or assumed by a Namputiri who has performed the vedic ritual of Somayajna.[4] So it could be surmised that Nilakantha Somayaji had also performed a Somayajna ritual and assumed the title of a Somayaji in later life. In colloquial Malayalam usage the word Somayaji has been corrupted to Comatiri.

Mādhavan of Sangamagrāmam (born as Irińńaŗappiļļy or Iriññinavaļļi Mādhavan Namboodiri. He had written that his house name was related to the Vihar where a plant called "bakuļam" was planted. According to Achyuta Pisharati , (who wrote a commentary on Veņwarõham written by Mādhavan) bakuļam was locally known as "iraňňi". Dr. K.V. Sarma, an authority on Mādhavan has the opinion that the house name is either Irińńāŗappiļļy or Iriññinavaļļy') (c. 1350 – c. 1425) was a prominent Kerala mathematician-astronomer from the town of Irińńālakkuţa, which was once known as 'Irińńāţikuţal' near Cochin, Kerala, India. Sangamagrāmam (lit. sangamam = union, grāmam = village) is a rough translation to Sanskrit from Dravidian word 'Irińńāţikuţal', which means 'iru (two) ańńāţi (market) kǖţal (union)' or the union of two markets. He is considered the founder of the Kerala School of Astronomy and Mathematics. He was the first to have developed infinite series approximations for a range of trigonometric functions, which has been called the "decisive step onward from the finite procedures of ancient mathematics to treat their limit-passage to infinity".[1] His discoveries opened the doors to what has today come to be known as Mathematical Analysis.[2] One of the greatest mathematician-astronomers of the Middle Ages, Mādhavan contributed to infinite series, calculus, trigonometry, geometry and algebra.

Some scholars have also suggested that Mādhava's work, through the writings of the Kerala school, may have been transmitted to Europe via Jesuit missionaries and traders who were active around the ancient port of Kochi at the time. As a result, it may have had an influence on later European developments in analysis and calculus.[3] This is due to wrong understanding of the authors concerned. It was almost impossible for the Jesuits in the sixteenth century, who are experts with the eminence of Mādhavan or his disciples, to study Sanskrit and Malayalam and to transmit them to European Mathematicians, instead of they themselves claiming the credit for the discovery.

Mahendra Suri is the 14th century Jain astronomer who wrote the Yantraraja, the first Indian treatise on the astrolabe.[1] He was a pupil of Madana Suri.

Sankara Variar (circa. 1500 - 1560 CE) was an astronomer-mathematician of the Kerala school of astronomy and mathematics who lived during the sixteenth century CE. His family were employed as temple-assistants in the Shiva-temple at Trkkutaveli near modern Ottapalam. [1]

Mathematical lineage

He was taught mainly by Nilakantha Somayaji (1444-1544), the author of the celebrated Tantrasamgraha and Jyesthadeva (1500 – 1575), the author of Yuktibhasa. Other teachers of Sankara Variar include Netranarayana, the patron of Nilakantha Somayaji and Chitrabhanu, the author of an astronomical treaties dated to 1530 and of a small work with solutions and proofs for algebraic equations

Vedic

Baudhayana Katyayana Panini , ca. 5th c. BC, Algebraic grammarian Yajnavalkya , credited with authorship of the Shatapatha Brahmana, which contains

calculations related to altar construction.

Classical

Post-Vedic Sanskrit to Pala period mathematicians (5th c. BC to 11th c. AD)

Aryabhata - Astronomer who gave accurate calculations for astronomical constants, 476-520

Aryabhata II Bhaskara I Bhaskara II Brahmagupta - Helped bring the concept of zero into arithmetic Mahavira Pavuluri Mallana - the first Telugu Mathematician Varahamihira Shridhara (between 650-850) - Gave a good rule for finding the volume of a sphere.

Medieval to Mughal period

13th century to 1800.13th century, Logician, mithila school

Ayush, son of Gangehsa, Logician, Mithila school Shankara Mishra, Logician, Mithila school Narayana Pandit Madhava of Sangamagrama some elements of Calculus hi Parameshvara (1360–1455), discovered drk-ganita, a mode of astronomy based on

observations, Madhava's Kerala school Nilakantha Somayaji ,1444-1545 - Mathematician and Astronomer, Madhava's Kerala

school Mahendra Suri (14th century) Shankara Variyar (c. 1530) Vasudeva Sarvabhauma, 1450–1525, Logician, Navadvipa school

Raghunatha Shiromani, (1475–1550), Logician, Navadvipa school Jyeshtadeva , 1500–1610, Author of Yuktibhasa, Madhava's Kerala school Achyuta Pisharati , 1550–1621, Astronomer/mathematician, Madhava's Kerala school Mathuranatha Tarkavagisha, c. 1575, Logician, Navadvipa school Jagadisha Tarkalankara, c. 1625, Logician, Navadvipa school Gadadhara Bhattacharya, c. 1650, Logician, Navadvipa school Munishvara (17th century) Kamalakara (1657) Jagannatha Samrat (1730)

Srinivasa Ramanujan

Born22 December 1887

Erode, Indian Empire

Died26 April 1920 (aged 32)

Chetput, (Madras), Indian Empire

Residence Tamil Nadu, India

Fields Mathematics

Alma mater

Government Arts College,

Pachaiyappa's College,

Trinity College, Cambridge

Academic advisors G. H. Hardy and J. E. Littlewood

Known for

Landau–Ramanujan constant

Mock theta functions

Ramanujan conjecture

Ramanujan prime

Ramanujan–Soldner constant

Ramanujan theta function

Ramanujan's sum

Rogers–Ramanujan identities

]