KENDRIYA VIDYALAYA BIRPUR DEHRADUN

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INDIAN MATHEMATICIANS1) Aryabhat Aryabhat was an acclaimed mathematician-astronomer. He was born in Kusumapura (present day Patna) in Bihar, India.

His contribution to mathematics, science and astronomy is immense, and yet he has not been accorded the recognition in the world history of science. At the age of 24, he wrote his famed “Aryabhatiya”. He was aware of the concept of zero, as well as the use of large numbers up to 1018. He was the first to calculate the value for ‘pi’ accurately to the fourth decimal point. He devised the formula for calculating areas of triangles and circles.

He calculated the circumference of the earth as 62,832 miles, which is

an excellent approximation, and suggested that the apparent rotation of the heavens was due to the axial rotation of the earth on its axis. He was the first known astronomer to devise a continuous Counting of solar days,designating each day with a number. He asserted that the planets shine due to the reflection of sunlight, and that and that the eclipses occur due to the shadows of moon and earth. moon and moon and earth. His observations discount the “flat earth” concept,

and lay the foundation for the belief that earth and other planets orbit

the sun.Childhood & Early Life Aryabhata’s birthplace is uncertain, but it may have been in the area known in ancient texts as Ashmaka, which may have been Maharashtra or Dhaka or in Kusumapura in present day Patna.

Some archaeological evidence suggests that he came from the present day

Kodungallur, the historical capital city of Thiruvanchikkulam of ancient Kerala

this theory is strengthened by the several commentaries on him having come

from Kerala. He went to Kusumapura for advanced studies and lived there for

some time. Both Hindu and Buddhist traditions, as well as Bhāskara I, the 7th

Century mathematician, identify Kusumapura as modern Patna.

Career & Later Life

A verse mentions that Aryabhata was the head of an institution (kulapa) at Kusumapura. Since, the University of Nalanda was in Pataliputra, and had an astronomical observatory; it is probable that he was its head too.Direct details of his work are known only from the Aryabhatiya. His disciple Bhaskara I calls it Ashmakatantra (or the treatise from the Ashmaka).

The Aryabhatiya is also occasionally referred to as Arya-shatas-aShTa (literally,

Aryabhata’s 108), because there are 108 verses in the text. It also has 13

introductory verses, and is divided into four pādas or chapters. Aryabhatiya’s

first chapter, Gitikapada, with its large units of time — kalpa, manvantra,

manvantra, and Yuga — introduces a different cosmology. The duration of the

planetary revolutions during a mahayuga is given as 4.32 million years.

Ganitapada, the second chapter of Aryabhatiya has 33 verses covering

mensuration (kṣetra vyāvahāra), arithmetic and geometric progressions,

gnomon or shadows (shanku-chhAyA), simple, quadratic, simultaneous, and

indeterminate equations.Aryabhatiya’s third chapter Kalakriyapada explains different units of

time, amethod for determining the positions of planets for a given day, and

a sevenday week with names for the days of week.

MAJOR WORKS

Aryabhata’s 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 Aryabhatiya covers arithmetic,

algebra, and trigonometry.

BRAHMAGUPTA

Brahmagupta was born in 598 CE according to his own statement. He lived

in Bhillamala (modern Bhinmal) during the reign of the Chapa dynasty ruler

Vyagrahamukha.

He was the son of Jishnugupta. He was a Shaivite byreligion.Even though most scholars assume that Brahmagupta was bornin Bhillamala, there is no conclusive evidence for it.

Life and career

However, he lived andworked there for a good part of his life. PrithudakaSvamin, a latercommentator, called him Bhillamalacharya, the teacher fromBhillamala.Sociologist G. S. Ghurye believed that he might have beenfrom the Multan region or the Abu region. Bhillamala, called pi-lo-molo by Xuanzang, was the apparent capital of the Gurjaradesa, the secondlargest kingdom of Western India, comprising the southern Rajasthan andnorthern Gujarat in modern-day India. It was also a center of learning formathematics and astronomy. Brahmagupta became an astronomer of  Brahmapaksha school (one of the four major schools of Indian astronomy this period). He studied the five traditional siddhanthas on Indian as well as the work of other astronomers including Aryabhata I, Latadeva,Pradyumna, Varahamihira, Simha, Srisena, Vijayanandin and Vishnuchandra.

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In the year 628, at an age of 30, he composed Brāhmasphuṭasiddhānta (the improved treatise of Brahma) which is believed to be a revised version of the received siddhanta of the Brahmapaksha school. Scholars state that he has incorported a great deal of originality to his revision, adding a considerable amount of new material. The book consists of 24 chapters with 1008 verses in the ārya meter. A good deal of it is astronomy, but it also contains key chapters on mathematics, including algebra, geometry, trigonometry and algorithmics, which are believed to contain new insights due to Brahmagupta himself.[4]

BRAHMAGUPTA'S FORMULA

Brahmagupta's most famous result in geometry is his formula for cyclic quadrilaterals. Given the lengths of the sides of any cyclic quadrilateral, Brahmagupta gave an approximate and an exact formula for the figure's area,

12.21. The approximate area is the product of the halves of the sums of the sides and opposite sides of a triangle and a quadrilateral. The accurate [area] is the square root from the product of the halves of the sums of the sides diminished by [each] side of the quadrilateral.

So given the lengths p, q, r and s of a cyclic quadrilateral, the approximate area is p + r/2 · q + s/2 while, letting t = p + q + r + s/2, the exact area is

√(t − p)(t − q)(t − r)(t − s). Although Brahmagupta does not explicitly state that these quadrilaterals

are cyclic, it is apparent from his rules that this is the case. Heron's formula is a special case of this formula and it can be derived by setting one of the sides equal to zero.

ACHARYA HEMACHANDRA Acharya Hemachandra was a Jain scholar, poet,

and polymath whowrote on grammar, philosophy, prosody, and contemporary history. Noted as a prodigy by his contemporaries, he gained the title kalikālasarvajña, "the all-knowing of the Kali Yuga".

EARLY LIFE

Hemachandra was born in Dhandhuka, in present-day Gujarat, on Kartika Sud Purnima (the full moon day of Kartika month). His date of birth differs according to sources but 1088 is generally accepted. His father, Chachiga-deva was a Modh Bania Vaishnava. His mother, Pahini, was a Jain. Hemchandra's original given name was Changadeva. In his childhood, the Jain monk Devachandra Suri visited Dhandhuka and was impressed by the young Hemachandra's intellect. His mother and maternal uncle concurred with Devachandra, in opposition to his father, that Hemachandra be a disciple of his. Devachandra took Hemachandra to Khambhat, where Hemachandra was placed under the care of the local governor Udayana. Chachiga came to Udayana's place to take his son back, but was so overwhelmed by the kind treatment he received, that he decided to willingly leave his son with Devachandra.

Some years later, Hemachandra was initiated a Jain monk on Magha Sud Chauth (4th day of the bright half of Magha month) and was given a new name, Somchandra. Udayana helped Devchandra Suri in the ceremony. He was trained in religious discourse, philosophy, logic and grammar and became well versed in Jain and non–Jain scriptures. At the age of 21, he was ordained an acharya of the Śvētāmbara school of Jainism at Nagaur in present-day Rajasthan. At this time, he was named Hemachandra Suri.WorksInstruction by Monks, Folio from the SiddhahemashabdanushasanaWorship of Parshvanatha, Folio from the SiddhahemashabdanushasanaA prodigious writer, Hemachandra wrote grammars of Sanskrit and Prakrit, poetry, prosody, lexicons, texts on science and logic and many branches of Indian philosophy. It is said that Hemachandra composed 3.5 crore verses in total, many of which are now lost.[

BHASKARA (C. 600 – C. 680) Bhaskara  (Bengali: ভাস্কর; Marathi:   भास्कर commonly

called Bhaskara I to avoid confusion with the 12th century mathematician Bhāskara II) was a 7th-century mathematician, who was the first to write numbers in the Hindu 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 Aryabhatta's work.

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.

This commentary, Āryabhaṭīyabhāṣya,

BiographyLittle is known about Bhāskara's life. He was probably a Marathi astronomer. He was born at Bori, in Parbhani district of Maharashtra state in India in 7th century.His astronomical education was given by his father. Bhaskara is considered the most important scholar of Aryabhata's astronomical school. He and Brahmagupta are two of 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 years ago. 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.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.

SRINIVASA RAMANUJAN

Srinivasa Ramanujan Iyengar  22 December 1887 – 26 April 1920) was an Indian mathematician and autodidact who lived during the British Raj. Though he had almost no formal training in pure mathematics, he made substantial contributions to mathematical analysis, number theory, infinite series, and continued fractions. Ramanujan initially developed his own mathematical research in isolation; it was quickly recognized by Indian mathematicians. When his skills became obvious and known to the wider mathematical community, centred in Europe at the time, he began a partnership with the English mathematician G. H. Hardy. The Cambridge professor realized that Ramanujan had produced new theorems in addition to rediscovering previously known ones.

During his short life, Ramanujan independently compiled nearly 3,900 results (mostly identities and equations).Nearly all his claims have now been proven correct.His original and highly unconventional results, such as the Ramanujan prime and the Ramanujan theta function, have inspired a vast amount of further research.The Ramanujan Journal, a peer-reviewed scientific journal, was established to publish work in all areas of mathematics influenced by Ramanujan.

Deeply religious,Ramanujan credited his substantial mathematical capacities to divinity: '"An equation for me has no meaning," he once said, "unless it expresses a thought of God."

EARLY LIFE

Ramanujan was born on 22 December 1887 into a Tamil Brahmin Iyengar family in Erode, Madras Presidency (now Tamil Nadu), at the residence of his maternal grandparents.His father, K. Srinivasa Iyengar, worked as a clerk in a sari shop and hailed from Thanjavur district.His mother, Komalatammal, was a housewife and also sang at a local temple.They lived in a small traditional home on Sarangapani Sannidhi Street in the town of Kumbakonam. The family home is now a museum. When Ramanujan was a year and a half old, his mother gave birth to a son, Sadagopan, who died less than three months later. In December 1889, Ramanujan contracted smallpox, but unlike the thousands in the Thanjavur district who died of the disease that year, he recovered. He moved with his mother to her parents' house in Kanchipuram, near Madras (now Chennai). His mother gave birth to two more children, in 1891 and 1894, but both died in infancy.

On 1 October 1892, Ramanujan was enrolled at the local school. After his maternal grandfather lost his job as a court official in Kanchipuram, Ramanujan and his mother moved back to Kumbakonam and he was enrolled in the Kangayan Primary School.

When his paternal grandfather died, he was sent back to his maternal grandparents, then living in Madras. He did not like school in Madras, and tried to avoid attending. His family enlisted a local constable to make sure the boy attended school. Within six months, Ramanujan was back in Kumbakonam.

Since Ramanujan's father was at work most of the day, his mother took care of the boy as a child. He had a close relationship with her. From her, he learned about tradition and puranas. He learned to sing religious songs, to attend pujas at the temple, and to maintain particular eating habits – all of which are part of Brahmin culture. At the Kangayan Primary School, Ramanujan performed well. Just before turning 10, in November 1897, he passed his primary examinations in English, Tamil, geography and arithmetic with the best scores in the district. That year, Ramanujan entered Town Higher Secondary School, where he encountered formal mathematics for the first time.

By age 11, he had exhausted the mathematical knowledge of two college students who were lodgers at his home. He was later lent a book by S. L. Loney on advanced trigonometryetry. He mastered this by the age of 13 while discovering sophisticated theorems on his own. By 14, he was receiving merit certificates and academic awards that continued throughout his school career, and he assisted the school in the logistics of assigning its 1200 students (each with differing needs) to its 35-odd teachers. He completed mathematical exams in half the allotted time, and showed a familiarity with geometry and infinite series. Ramanujan was shown how to solve cubic equations in 1902; he developed his own method to solve the quartic. The following year, not knowing that the quintic could not be solved by radicals, he tried to do so. In 1903, when he was 16, Ramanujan obtained from a friend a library copy of a A Synopsis of Elementary Results in Pure and Applied Mathematics, G. S. Carr's collection of 5,000 theorems.

 Ramanujan reportedly studied the contents of the book in detail. The book is generally acknowledged as a key element in awakening his genius. The next year, Ramanujan independently developed and investigated the Bernoulli numbers and calculated the Euler–Mascheroni constant Euler–Mascheroni constant 

up to 15 decimal places. His peers at the time commented that they "rarely understood him" and "stood in respectful awe" of him.Mathematical achievements

In mathematics, there is a distinction between having an insight and having a proof. Ramanujan proposed an abundance of formulae that could be investigated later in depth. G. H. Hardy said that Ramanujan's discoveries are unusually rich and that there is often more to them than initially meets

the eye. As a byproduct of his work, new directions of research were opened up. Examples of the most interesting of these formulae include the intriguing infinite series for π, one of which is given below:

{\displaystyle {\frac {1}{\pi }}={\frac {2{\sqrt {2}}}{9801}}\sum _{k=0}^{\infty

}{\frac {(4k)!(1103+26390k)}{(k!)^{4}396^{4k}}}.} This result is based on the negative fundamental discriminant d = −4 × 58 =

−232 with class number h(d) = 2. 26390 = 5 × 7 × 13 × 58 and 16 × 9801 = 3962 and is related to the fact that

{\textstyle e^{\pi {\sqrt {58}}}=396^{4}-104.000000177\dots .} This might be compared to Heegner numbers, which have class number 1 and

yield similar formulae. Ramanujan's series for π converges extraordinarily rapidly (exponentially) and

forms the basis of some of the fastest algorithms currently used to calculate π. Truncating the sum to the first term also gives the approximation 9801√2/4412 for π, which is correct to six decimal places. See also the more general Ramanujan–Sato series.

OTHER MATHEMATICIANS' VIEWS OF RAMANUJAN

Hardy said: "He combined a power of generalization, a feeling for form, and a capacity for rapid modification of his hypotheses, that were often really startling, and made him, in his own peculiar field, without a rival in his day.Thelimitations of his knowledge were as startling as its profundity. Here was a man who could work out modular equations and theorems... to orders unheard of, whose mastery of continued fractions was... beyond that of any mathematician in the world, who had found for himself the functional equation of the zeta function and the dominant terms of many of the most famous problems in the analytic theory of numbers; and yet he had never heard of a doubly periodic function or of Cauchy's theorem, and had indeed but the vaguest idea of what a function of a complex variable was...". When asked about the methods Ramanujan employed to arrive at his solutions, Hardy said that they were "arrived at by a process of mingled argument, intuition, and induction, of which he was entirely unable to give any coherent account." He also stated that he had "never met his equal, and can compare him only with Euler or Jacobi."

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