HISTORY OF RAMANUJAN

This is the presentation which was presented in PPT CONTEST on TCP PRESENTO 2020 organized by THIAGARAJAR COLLEGE OF PRECEPTORS, MADURAI.

1. PEDAGOGY OF MATHEMATICS Annabel Catherine Juliana J. Stella Matutina College of Education Histor y of Ramanujan TCP PRESENTO 2020, THIAGARAJAR COLLEGE OF PRECEPTORS, MADURAI .

2. Content Quotes Introduction Honors Ramanujan Prime Number Theory Magic Square Partition Taxicab Number Ramanujan Summation Nested and De-Nested Ramanujan Sato Series Conclusion TCP PRESENTO 2020, THIAGARAJAR COLLEGE OF PRECEPTORS, MADURAI.

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4. INTRODUCTION TCP PRESENTO 2020, THIAGARAJAR COLLEGE OF PRECEPTORS, MADURAI.

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8. Srinivasa Ramanujan Iyenger was an Indian Mathematics. One of the most talented mathematicians in recent history. Had no formal training in mathematics. Made a large contribution to 1 Number Theory 2 Infinite Series 3 Continued Fractions. TCP PRESENTO 2020, THIAGARAJAR COLLEGE OF PRECEPTORS, MADURAI.

9. Srinivasa Ramanujan was mentored by G.H.Hardy in the early 1910’s.After getting his degree at cambridge , Ramanujan did his own work. He compiled over 3500 identities and equations in his life. Some of the identities were found in his “Lost Notebooks.” His formulas are now being used in Crystallography and String Theory. In 2011, Ramanujan’s birthday was made an annual“National Mathematics Day” by Prime Minister Dr.Manmohan Singh . TCP PRESENTO 2020, THIAGARAJAR COLLEGE OF PRECEPTORS, MADURAI.

10. In March 1916, Graduated Bachelor of Science degree by research. ( later called PhD ) In1917, Elected to London Mathematics Society. In 1918, become a Fellow of the Royal society. Again in 1918, Fellow of Trinity College, Cambridge . TCP PRESENTO 2020, THIAGARAJAR COLLEGE OF PRECEPTORS, MADURAI.

11. Ramanujan Prime Ramanjuan prime is a prime number that satisfies a result proven by Srinivasa Ramanujan relating to the prime- counting function. In 1919,Ramanujan published a new proof of Bertrand’s postulate which as he notes, was first proved by chebyshen . Ramanujan primes are the least integers Rn for which there are at least n primes between x and x/2. The first prime of Ramanujan are 2,11,17,29,41. TCP PRESENTO 2020, THIAGARAJAR COLLEGE OF PRECEPTORS, MADURAI.

12. List of Ramanujan Primes TCP PRESENTO 2020, THIAGARAJAR COLLEGE OF PRECEPTORS, MADURAI.

13. Number Theory MAGIC SQUARE Ramanujan created a super magic square. The top row is his birthdate ( December 22,1887 ). This is not the super magic squares because not only do the rows ,columns and diagonals add up to the same number, but the four corners, the four middle squares( 17,9,24,89 ),the first and last rows two middle numbers( 12,18,86,23 ) and the first and last columns two middle numbers( 88,10,25,16 ) all add up to the sum of 139. TCP PRESENTO 2020, THIAGARAJAR COLLEGE OF PRECEPTORS, MADURAI.

14. Number Theory TCP PRESENTO 2020, THIAGARAJAR COLLEGE OF PRECEPTORS, MADURAI.

15. The magic squares form the nucleus of the theory of partitions developed by Srinivasa Ramanujan . His fascination for magic squares led him in his later life to work on this theory. Let p(n) denote the partition function n. For example, 1 has the partition 1; 2 has the partition 2, 1+1; 3 has the partition 3,2+1,1+1+1. and so on…. Theory of Partitions TCP PRESENTO 2020, THIAGARAJAR COLLEGE OF PRECEPTORS, MADURAI.

16. For example, 6 has the partition 6, 5+1,4+2,4+1+1,3+3,3+2+1,3+1+1+1,2+2+2,2+2+1+1,2+1+1+1+1,1+1+1+1+1+1. Theory of Partitions TCP PRESENTO 2020, THIAGARAJAR COLLEGE OF PRECEPTORS, MADURAI.

17. Taxi Cab Number TCP PRESENTO 2020, THIAGARAJAR COLLEGE OF PRECEPTORS, MADURAI.

18. A Taxi Cab Number is the smallest number that can be expressed as the sum of 2 positive cubes in ‘n’ distinct way. It has nothing to do with TAXIS, but the name comes from a well-known conversation that took place between 2 famous mathematician G.H.Hardy Srinivasa Ramanujan TCP PRESENTO 2020, THIAGARAJAR COLLEGE OF PRECEPTORS, MADURAI.

19. The Story of G.H. Hardy Taxi One day G.H.Hardy went to visit a friend, the brilliant young Indian Mathematician Srinivasa Ramanujan , who was ill. Both men were mathematics and liked to think about numbers. When Ramanujan heard that Hardy had come in a taxi he asked what the number of the taxi was. Hardy said that it was a boring number 1729. Ramanujan explained that it was the “( Smallest Number that could be expressed by the sum of 2 cubes in 2 different ways) .” 1729 is called “Hardy- Ramanujan Numbers or Taxi Number. TCP PRESENTO 2020, THIAGARAJAR COLLEGE OF PRECEPTORS, MADURAI.

20. Ramanujan Summation Ramanujan Summation is a technique invented by the mathematician Srinivasa Ramanujan for assigning a value to divergent infinite series. It states that if you add all the natural numbers that is 1,2,3,4, and so on to infinity you will find that is equal to -1/12. ( i.e ) 1+2+3+4+----+infinity= -1/12 . TCP PRESENTO 2020, THIAGARAJAR COLLEGE OF PRECEPTORS, MADURAI.

21. Nested Radical TCP PRESENTO 2020, THIAGARAJAR COLLEGE OF PRECEPTORS, MADURAI.

22. Denesting Radical Rewriting a nested radical in this way is called denesting . This is not always possible and even when possible, it is often difficult. TCP PRESENTO 2020, THIAGARAJAR COLLEGE OF PRECEPTORS, MADURAI.

23. Sato Series Rapid Convergent Series A Ramanujan – Sato Series generalizes Ramanujan’s pi formulas such as TCP PRESENTO 2020, THIAGARAJAR COLLEGE OF PRECEPTORS, MADURAI.

24. Conclusion He discovered mock theta function in the last year of his life. For many years these functions were a mystery, but they are now known to be the holomorphic parts of harmonic weak mass forms. Of ramanujan’s published papers -37 in total –professor Bruce C. Berndt reveals that “ a huge portions of his work was left behind in three notebooks and a lost notebook”. Srinivasa Ramanujan was died on 26 April 1920.(at aged 32) TCP PRESENTO 2020, THIAGARAJAR COLLEGE OF PRECEPTORS, MADURAI.

25. THANK YOU TCP PRESENTO 2020, THIAGARAJAR COLLEGE OF PRECEPTORS, MADURAI.