Srinivasa Ramanujan

<p>" Srinivāsa Aiyangār Rāmānujan (Tamil: ஸ்ரீனிவாஸ ஐயங்கார் ராமானுஜன்) (December 22, 1887 – April 26, 1920) was an Indian mathematician and one of the greatest mathematical geniuses of the twentieth century. He is considered one of the greatest mathematical prodigy that the world has ever seen. He had uncanny mathematical manipulative abilities, as judged by experts in his field. He excelled in the heuristic aspects of number theory and insight into modular functions. He also made significant contributions to the development of partition functions and summation formulas involving constants such as π.</p>

<p>A child prodigy, he was largely self-taught in mathematics and had compiled over 3,000 theorems between 1914 and 1918 at the University of Cambridge. Often, his formulas were merely stated, without proof, and were only later proven to be true. His results were highly original and unconventional, and have inspired a large amount of research and many mathematical papers; however, some of his discoveries have been slow to enter the mathematical mainstream. Recently his formulae have started to be applied in the field of crystallography, and other applications in physics. The Ramanujan Journal was launched to publish work 'in areas of mathematics influenced by Ramanujan'. "</p>

<p>" Theorems and discoveries
It is said that Ramanujan's discoveries were unusually rich; that is, in many of them there was far more than initially met the eye. The following include both Ramanujan's own discoveries, and those developed or proven in collaboration with Hardy.</p>

<p>Properties of highly composite numbers
The partition function and its asymptotics
He also made major breakthroughs and discoveries in the areas of:</p>

<p>Gamma functions
Modular forms
Ramanujan's continued fractions
Divergent series
Hypergeometric series
Prime number theory. A type of prime numbers based on a 1919 publication by Ramanujan is named Ramanujan primes.
Mock theta functions "</p>

<p>" Childhood and early life
Ramanujan was born in 1887 in Erode, Tamil Nadu, India, the place of residence of his maternal grandparents. His father hailed from the fertile Thanjavur District (temple district), working, when Ramanujan was born in Kumbakonam, at a cloth merchant's shop. His mother is believed to have been well-educated in Indian mathematics and Ramanujan is conjectured by some to have been as well[1]. In 1898, at age 10, he entered the Town High School in Kumbakonam, where he may have encountered formal mathematics for the first time. At 11 he had mastered the mathematical knowledge of two lodgers at his home, both students at the Government College, and was lent books on advanced trigonometry written by S. L. Loney, which he mastered by age 13. His biographer reports that by 14 his true genius was beginning to become discernible. Not only did he achieve merit certificates and academic awards throughout his school years, he was also assisting the school in the logistics of assigning its 1200 students (each with their own needs) to its 35-odd teachers, completing mathematical exams in half the allotted time, and was showing familiarity with infinite series. His peers at the time commented later, "We, including teachers, rarely understood him" and "stood in respectful awe" of him. However, Ramanujan could not concentrate on other subjects and failed his high school exams. By age 17, he calculated Euler's constant to 15 decimal places. He began to study what he thought was a new class of numbers, but instead he had independently developed and investigated the Bernoulli numbers. At this time in his life, he was quite poor and was often near the point of starvation.</p>

<p>Adulthood in India
After marriage (on July 14, 1909) he began searching for work. With his packet of mathematical calculations, he travelled around the city of Madras (now Chennai) looking for a clerical position. He managed finally to get a job as an accountant in the General's Office at Madras. Ramanujan desired to focus completely on mathematics, and was advised by an Englishman to contact scholars in Cambridge. He doggedly solicited support from influential Indian individuals and published several papers in Indian mathematical journals, but was unsuccessful in his attempts to foster sponsorship. ( OR so the story goes - actually he was supported by R.Ramachandra Rao, then the Collector of the Nellore District and a distinguished civil servant. Ramachandra Rao, an amateur mathematician himself was the uncle of the well known mathematician, K. Ananda Rao, who went on to become the Principal of the Presidency college.) It was at this point that Sir Ashutosh Mukherjee tried to bolster his cause.</p>

<p>In late 1912 and early 1913 Ramanujan sent letters and examples of his theorems to three Cambridge academics: H. F. Baker, E. W. Hobson, and G. H. Hardy. Only Hardy, a Fellow of Trinity College to whom Ramanujan wrote in January 1913, recognized the genius demonstrated by the theorems.</p>

<p>Upon reading the initial unsolicited missive by an unknown and untrained Indian mathematician, Hardy and his colleague J.E. Littlewood commented that, “not one [theorem] could have been set in the most advanced mathematical examination in the world.” Although Hardy was one of the pre-eminent mathematicians of his day and an expert in several of the fields Ramanujan was writing about, he commented, 'many of them defeated me completely; I had never seen anything in the least like them before.' "</p>

<p>Cultural references
He was referred to in the film Good Will Hunting as an example of mathematical genius.
His biography was also highlighted in the Vernor Vinge book The Peace War.
The character 'Amita Ramanujan' in the CBS TV series Numb3rs (2005-) was named after him
The short story "Gomez", by Cyril Kornbluth, mentions Ramanujan by name as a comparison to its title character, another self-taught mathematical genius.</p>

<p>Where was this when I did a report on him three years ago. What he did is pretty amazing. I wish I was half that smart.</p>

<p>Nova did an episode on him years ago. It was fascinating:</p>

<p>Man Who Loved Numbers (The)
NOVA explores the life of Srinivasa Ramanujan, a poor clerk from India who astounded mathematicians in the 1910s with his brilliant insight into the world of numbers.
Original broadcast date: 03/22/88
Topic: biography</p>

<p>a movie based on him is coming up:
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<p>"When your automated teller machines divide and arrange your money before coughing it up, they are all using Ramanujan's partition theory"</p>

<p>Aryabhata (Hindi : आर्यभट, IAST: Āryabhaṭa) (476 – 550) is the first of the great mathematician-astronomers of the classical age of India. There exists no documentation to ascertain his exact birthplace. Available evidences suggest that he went to Kusumapura for higher studies. He lived in Kusumapura, which his commentator Bhāskara I (629 AD) identifies as Pataliputra (modern Patna).</p>

<p>" Aryabhata worked on the approximation for Pi, and may have realized that π is irrational. In the second part of the Aryabhatiya (gaṇitapāda 10), he writes:</p>

<p>chaturadhikam śatamaśṭaguṇam dvāśaśṭistathā sahasrāṇām
Ayutadvayaviśkambhasyāsanno vrîttapariṇahaḥ.</p>

<p>"Add four to 100, multiply by eight and then add sixty-two thousand. By this rule is the circumference of a circle of diameter 20,000 approximately given"
In other words, , correct to four rounded-off decimal places. The commentator Nilakantha Somayaji, (Kerala School, 15th c.) has argued that the word āsanna (approaching), appearing just before the last word, here means not only that this is 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 (Lambert)."</p>