Welcome One And All!: Welcome One And All! 1 22nd December 1887 – 26th April 1920 : 22 nd December 1887 – 26 th April 1920 2 TODAY’S RELEVANCE OF INDIAN MATHEMATICS SRINIVASA IYYANGAR RAMANUJAN 125 th Birth Anniversary National Mathematics Year - 2012 Ramanujan’s life and his genius : Ramanujan’s life and his genius 3 1887-1920 PowerPoint Presentation: Srinivasa Ramanujan is known as an all-time great Indian mathematician Born 22 December1887 , Erode, Tamil Nadu, India. 4 PowerPoint Presentation: Once his teacher said that when zero is divided by any number, the result is Zero, Ramanujan immediately asked Him whether zero divided by zero also gives zero; where he showed early signs of his genius! A thought of a 7 year old Srinivasa 5 Srinivasa Iyengar Ramanujan Said: : Srinivasa Iyengar Ramanujan Said: 6 “An equation for me has no meaning unless it expresses a thought of God” School education :-: School education :- Passed primary examination. (1897) Stood first in the district. Town High School- Kumbakonam . (1898) Mastered Advanced Trigonometry written by S. L.Loney at the age of 13 yrs. 7 Favorite Pastime :-: Favorite Pastime :- Sitting with his slate on the window’s ledge Recite formulae for fun. Working out mathematical problems . 8 Early Life: Early Life 9 Ramanujan’s Home Library: Library 10 PowerPoint Presentation: 11 When Ramanujan was in IV Form, his schoolmate C.V. Rajagopalachari ( in Form V) raised the question. " If what are the values of x and y?” Ramanujan's quick answer was: x=9, y=4 This was considered an unexpected achievement by a student of Form IV. Attention from mathematicians: Attention from mathematicians One of the first problems he posed in the journal of the Indian mathematical Society was : Find the values of :- 12 PowerPoint Presentation: Ramanujan himself supplied the solution to the problem. 13 PowerPoint Presentation: He formulated an equation to solve the infinitely nested radicals problem . 14 Its Journal put Ramanujan on the world’s mathematical map . Janaki Ramanujan Wife of Ramanujan: Janaki Ramanujan Wife of Ramanujan July 1909, Ramanujan was married to Janaki . Ramanujan’s father was not present at that time. His mother Komalathammal was solely responsible for this major event in Ramanujan’s life. 15 PowerPoint Presentation: 16 “when food is problem, how can I find money for paper? I may require four reams of paper every month.” PowerPoint Presentation: Government College , kumbhakonam in 1904. Received Scholarship Failed in college examination. Pachaiyappa's College - Madras, for the First Arts examination.(1906) Excelled in mathematics but failed F. A. exam. End of his formal education in India. 17 PowerPoint Presentation: Started working as a clerk in the Accountant-General's office at the Madras Port Trust Office to support himself (1912–1913). Sent samples of his theorems to three academics at the University of Cambridge.(1912–1913) By 1904, investigated the series (1/n). Calculated Euler's constant to 15 decimal places. Developed and investigated Bernoulli numbers and the Euler– Mascheroni constant independently. 18 PowerPoint Presentation: Hardyrecognized the brilliance of his work. Subsequently invited Ramanujan to visit and work with him at Cambridge. Trinity College, Cambridge 19 PowerPoint Presentation: Ramanujan left Madras by the passenger ship S.S. NEVASA (of British India Lines) on 17th March 1914 20 PowerPoint Presentation: Ramanujan was received by E.H. Neville . On 18th April, he went to Cambridge and stayed at Neville’s house on Chester Town Road, for the first two months. 21 PowerPoint Presentation: 22 Srinivas Ramanujan at Cambridge PowerPoint Presentation: 23 Godfrey Harold Hardy: I have done one thing you could never have done, and that is to have collaborated with both Littlewood and Ramanujan on something like equal terms’. Ramanujan number 1729: Ramanujan number 1729 Ramanujan and Hardy arrived at Ramanujan's residence in a cab numbered 1729. Hardy commented that the number 1729 seemed to be uninteresting. Ramanujan said it is actually a very interesting number mathematically. 24 PowerPoint Presentation: 25 PowerPoint Presentation: 26 The smallest natural number representable in two different ways as a sum of two cubes: Generalizations of this idea have created the notion of “taxicab numbers”. PowerPoint Presentation: 27 885623890831=7511 3 +7730 3 =8759 3 +5978 3 Largest known similar number is Ramanujan was indeed a friend of numbers. Contribution to the theory of partitions: : Contribution to the theory of partitions : A partition of a natural number n is a sequence of non-decreasing positive integers whose sum is ‘n’. 28 N NO. OF PARTITIONS 1 1 2 2 3 3 4 5 5 7 6 11 PowerPoint Presentation: 29 EXAMPLE : For n = 4 , partitions are 4 =4 =1+3 = 2+2 =1+1+2 = 1+1+1+1 p(4 ) = 5 ,where p is a Partition Function NO. OF PARTITIONS: NO. OF PARTITIONS 30 The number P(n) increases fast with increasing n. N 10 20 30 40 50 … 600 … 1000 … 5000 P(N) 42 627 5604 37338 204206 (10 21 ) (10 31 ) (10 75 ) PowerPoint Presentation: Can we find p(n) without numerating all the partitions of n? And Ramanujan gave the first answered: Where π is the ratio of the circumference of a circle to its diameter and e = 2.71828. The symbol stand for ‘asymptotically equal to’. 31 PowerPoint Presentation: 32 The highest highly composite number listed by Ramanujan is 6746328388800 (a 13 digit number) having 10080 factors THREE NOTEBOOKS: THREE NOTEBOOKS 1906-1912 Ramanujan continued noting down his results on loose leaf papers Published three note books of pages 212, 352 and 33 (1967). 33 PowerPoint Presentation: 34 Magic Squares, Sums of Series, Combinational Analysis, Polynomials, Number Theory, Analogues of Gamma Functions, Continued Fractions Elliptic integrals Highly composite numbers Properties of primes. CONTRIBUTION’S OF RAMANUJAN IN DIFFERENT AREAS OF MATHEMATICS PowerPoint Presentation: Ratings by Hardy: According to natural talents of mathematicians,. Mathematicians 0 – 100 1. G.H. Hardy 25 2. Littlewood 30 3. Ramanujan 100 4. Hilbert 80 35 Ramanujan Birthday Magic Square : : Ramanujan Birthday Magic Square : 22 12 18 87 59 57 15 8 38 17 50 34 20 53 57 10 36 Date of Birth Of Ramanujan Is 22/12/1887 Sum is 139 = 3 0 /3 1 / 3 2 Hard Life in England : Hard Life in England Vegetarian meals were not prepared. Got too busy and often forgot to eat. The cold, damp climate. He fell ill and no one could find a cure. He decided to return to India . 37 PowerPoint Presentation: 38 Untimely death Ramanujan sailed to India on 27 February 1919 arriving on 13 March. However his health was very poor. Despite of medical treatment, he expired the following year. He passed away on 26 th April 1920 at Kumbakonam ( Tamil nadu ) PowerPoint Presentation: What is your greatest contribution to mathematics ? G. H. Hardy : Discovery of Srinivasa Ramanujan . 39 Thank you For your Patience!!!: BY: 40 Thank you For your Patience!!!