MATHEMATICS PROJECT

MATHEMATICS PROJECTby:-Atishay JainClass:- x-cRoll no.:-10336

ACKNOWLEDGEMENT:-I would like to express my special thanks of gratitude to my teacher Mr.Vikram Yadav as well as our principal Mrs.R.R Dhaiya who gave me the golden opportunity to do this wonderful project on the topic Mathematicians Biography, which also helped me in doing a lot of Research and I came to know about so many new things I am really thankful to them.Secondly I would also like to thank my parents and friends who helped me a lot in finalizing this project within the limited time frame.

Topics:-1-Biography:-Aryabhata2-Biography:-Ramanujan

BIOGRAPHY:-

ARYABHATA

ARYABHATA

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, including Brahmagupta's references to him "in more than a hundred places by name". Furthermore, in most instances "Aryabhatta" would not fit the metre either.

TIME AND PLACE OF 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.

Aryabhata provides no information about his place of birth. The only information comes from Bhskara I, who describes Aryabhata as makya, "one belonging to the Amaka country." During the Buddha's time, a branch of the Amaka people settled in the region between the Narmada and Godavari rivers in central India; Aryabhata is believed to have been born there.

EDUCATION:-It is fairly certain that, at some point, he went to Kusumapura for advanced studies and lived there for some time. Both Hindu and Buddhist tradition, as well as Bhskara I (CE 629), identify Kusumapura as Paliputra, modern Patna. 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. Aryabhata is also reputed to have set up an observatory at the Sun temple in Taregana, Bihar.

MAJOR WORKS:-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.

Place value system and zero:-

The place-value system, first seen in the 3rd-century Bakhshali Manuscript, was clearly in place in his work. While he did not use a symbol for zero, 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.

Approximation of PI:-

Aryabhata worked on the approximation for pi (\pi), and may have come to the conclusion that \pi is irrational. In the second part of the Aryabhatiyam (gaitapda 10), he writes:

caturadhikam atamaaguam dvaistath sahasrm ayutadvayavikambhasysanno vttapariha. "Add four to 100, multiply by eight, and then add 62,000. By this rule the circumference of a circle with a diameter of 20,000 can be approached." This implies that the ratio of the circumference to the diameter is ((4 + 100) 8 + 62000)/20000 = 62832/20000 = 3.1416, which is accurate to five significant figures.

After Aryabhatiya was translated into Arabic (c. 820 CE) this approximation was mentioned in Al-Khwarizmi's book on algebra.

Trigonometry:-

Aryabhata discussed the concept of sine in his work by the name of ardha-jya, which literally means "half-chord". For simplicity, people started calling it jya. When Arabic writers translated his works from Sanskrit into Arabic, they referred it as jiba. However, in Arabic writings, vowels are omitted, and it was abbreviated as jb. Later writers substituted it with jaib, meaning "pocket" or "fold (in a garment)". (In Arabic, jiba is a meaningless word.) Later in the 12th century, when Gherardo of Cremona translated these writings from Arabic into Latin, he replaced the Arabic jaib with its Latin counterpart, sinus, which means "cove" or "bay"; thence comes the English word sine.

Algebra:-

In Aryabhatiya, Aryabhata provided elegant results for the summation of series of squares and cubes:-

1^2 + 2^2 + \cdots + n^2 = {n(n + 1)(2n + 1) \over 6}

and

1^3 + 2^3 + \cdots + n^3 = (1 + 2 + \cdots + n)^2 (see squared triangular number)

Biography:-Srinivasa ramanujan

Early Life:-Ramanujan was born on 22 December 1887 into a TamilBrahmin family in Erode, Madras Presidency (now TamilNadu), at the residence of his maternal grandparents.His father, K. Srinivasa Iyengar, worked as a clerk ina sari shop and hailed from Thanjavur district. Hismother, Komalatammal, was a housewife and also sang ata local temple. They lived in Sarangapani Street in a traditionalhome in the town of Kumbakonam. The familyhome is now a museum. When Ramanujan was a year anda half old, his mother gave birth to a son, Sadagopan, whodied less than three months later.

On 1 October 1892, Ramanujan was enrolled at the localschool. After his maternal grandfather lost his jobas a court official in Kanchipuram, Ramanujan andhis mother moved back to Kumbakonam and he was enrolledin the Kangayan Primary School. When his paternalgrandfather died, he was sent back to his maternalgrandparents, then living in Madras. He did not likeschool in Madras, and tried to avoid attending. His familyenlisted a local constable to make sure the boy attendedschool. Within six months, Ramanujan was backin Kumbakonam.

Attention towards mathematics:-Ramanujan met deputy collector V. Ramaswamy Aiyer,who had recently founded the Indian MathematicalSociety. Wishing for a job at the revenue departmentwhere Ayer worked, Ramanujan showed him his mathematicsnotebooks. As Ayer later recalled:I was struck by the extraordinary mathematicalresults contained in it [the notebooks].I had no mind to smother his genius by an appointmentin the lowest rungs of the revenuedepartment.

Aiyer sent Ramanujan, with letters of introduction, tohis mathematician friends in Madras. Some of themlooked at his work and gave him letters of introductionto R. Ramachandra Rao, the district collector forNellore and the secretary of the Indian MathematicalSociety. Rao was impressed by Ramanujans researchbut doubted that it was his own work. Ramanujanmentioned a correspondence he had with ProfessorSaldhana, a notable Bombay mathematician, in whichSaldhana expressed a lack of understanding of his workbut concluded that he was not a phony.

Works:-Ramanujan wrote his first formal paper forthe Journal on the properties of Bernoulli numbers. Oneproperty he discovered was that the denominators (sequenceA027642 in OEIS) of the fractions of Bernoullinumbers were always divisible by six. He also deviseda method of calculating Bn based on previous Bernoullinumbers. One of these methods follows:It will be observed that if n is even but not equal to zero,

(i) Bn is a fraction and the numerator of Bnn in its lowestterms is a prime number,(ii) the denominator of Bn contains each of the factors 2and 3 once and only once,(iii) 2n(2n - 1)Bnn is an integer and 2(2n - 1)Bn consequentlyis an odd integer.In his 17-page paper, Some Properties of BernoullisNumbers, Ramanujan gave three proofs, two corollariesand three conjectures. Ramanujans writing initiallyhad many flaws.

Life In England:-Ramanujan departed from Madras aboard the S.S.Nevasa on 17 March 1914. When he disembarked inLondon on 14 April, Neville was waiting for him witha car. Four days later, Neville took him to his houseon Chesterton Road in Cambridge. Ramanujan immediatelybegan his work with Littlewoods and Hardy. Aftersix weeks, Ramanujan moved out of Nevilles houseand took up residence on Whew ell's Court, a five-minutewalk from Hardys room. Hardy and Littlewoods beganto look at Ramanujans notebooks.

Illness and death:-Throughout his life, Ramanujan was plagued by healthproblems. His health worsened in England. He was diagnosedwith tuberculosis and a severe vitamin deficiency,and was confined to a sanatorium. In 1919 he returned toKumbakonam, Madras Presidency, and soon thereafter,in 1920, died at the age of 32. His widow, S. Janaki Ammal,moved to Bombay; in 1950 she returned to Chennai(formerly Madras), where she lived until her death in1994 at age 95.A 1994 analysis of Ramanujans medical records andsymptoms by Dr. D. A. B. Young concluded that it wasmuch more likely he had hepatic amoebiasis, an illnessthen widespread in Madras, rather than TB. He had twoepisodes of dysentery before he left India. When notproperly treated, dysentery can lie dormant for years andlead to hepatic amoebiasis.Although difficult to diagnose,it is a readily curable disease.

The ramanujan conjecture:-Although there are numerous statements that could haveborne the name Ramanujan conjecture, there is one thatwas very influential on later work. In particular, the connectionof this conjecture with conjectures of Andr Weilin algebraic geometry opened up new areas of research.That Ramanujan conjecture is an assertion on the size ofthe tau-function, which has as generating function the discriminantmodular form (q), a typical cusp form in thetheory of modular forms. It was finally proven in 1973,as a consequence of Pierre Deligne's proof of the Weilconjectures. The reduction step involved is complicated.Deligne won a Fields Medal in 1978 for that work.

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