Ramanujan As a Genius Par Excellence

Ramanujan -- a Genius Par Excellence

                                                                                                                        -Dr.S.SelvaRani

                                                                                                                  

Srinivasa  Ramanujan  has    hailed  as  a  natural  mathematical genius  from  our land.  He made substantial contributions  to the analytical  theory of  numbers and worked on elliptic  functions,  continued  fractions  and infinite series.

Ramanjuan was born on December 22^nd 1887.   In 1900 he began to work on his own on mathematics summing geometric and arithmetic series.  During his high school studies at Kumbakonam, Ramanujan came across a mathematics book by G S Carr called ‘Synopsis of elementary results in pure mathematics’. This book, with its very concise style, allowed Ramanujan to teach himself mathematics. The book contained theorems, formulae and short proofs.   It Contained  about 4865 formulas.

His Journey towards Mathematics

Professor G.H.Hardy  was his friend, philosopher and guide. In 1940, Hardy gave two lectures at Yale University, which were subsequently published as a book entitled: "Ramanujan: Twelve Lectures inspired by his life and work". Earlier, in 1927, Hardy, along with Dewan Bahadur Ramachandra Rao and P V Seshu Iyer, brought out the "Collected papers of Srinivasa Ramanujan", which have been more recently reprinted, in 1999, by the American Mathematical Society and the London Mathematical Society.  This reprinting of the two volumes at the dawn of this century clearly is an indication of the intrinsic worth of the work of Ramanujan in his brief life span of 32 years, 4 months and 4 days, of which he spent five years, 1914-919, at the Trinity College, Cambridge University.

By 1904 Ramanujan had begun to undertake deep research. He investigated the series ∑(1/n) and calculated Euler's constant to 15 decimal places. He began to study the Bernoulli numbers. In 1906 Ramanujan went to Madras where he entered Pachaiyappa's College.  Continuing his mathematical work Ramanujan studied continued fractions, hyper geometric series and  divergent series.

Ramanujan continued to develop his mathematical ideas and began to pose problems and solve problems in the Journal of the Indian Mathematical Society. He devoloped relations between elliptic modular equations in 1910. After publication of a brilliant research paper on Bernoulli numbers in 1911 in the Journal of the Indian Mathematical Society he gained recognition for his work. Despite his lack of a university education, he was becoming well known in the Madras area as a mathematical genius.

The man near to Infinity :

In 1914, Hardy brought Ramanujan to Trinity College, Cambridge, to begin an extraordinary collaboration.   Right from the start Ramanujan's collaboration with Hardy led to important results.

On 16 March 1916,  Ramanujan graduated from Cambridge with a Bachelor of Science by Research (the degree was called a Ph.D. from 1920).  Ramanujan's dissertation was on ‘Highly composite numbers’ and consisted of seven of his papers published in England.

Ramanujan fell seriously ill in 1917 and spent most of his time in various nursing homes.   On February 18, 1918,  Ramanujan was elected a fellow of the Cambridge Philosophical Society and then three days later, the greatest honour that he would receive, his name appeared on the list for election as a fellow of the Royal Society of London.    On 10 October 1918 he was elected a Fellow of Trinity College Cambridge, the fellowship to run for six years.

Gliding glimpses of Ramanujan;s Work :

The letters Ramanujan wrote to Hardy in 1913 had contained many fascinating results. Ramanujan worked out the Riemann series, the elliptic integrals, hypergeometric series and functional equations of the zeta function.   Ramanujan independently discovered results of Gauss, Kummer and others on hypergeometric series. Ramanujan's own work on partial sums and products of hypergeometric series have led to major development in the topic.

In a joint paper with Hardy, Ramanujan gave an asymptotic formula for p(n). Ramanujan left a number of unpublished notebooks filled with theorems that mathematicians have continued to study. G N Watson, Mason Professor of Pure Mathematics at Birmingham from 1918 to 1951 published 14 papers under the general title Theorems stated by Ramanujan and in all he published nearly 30 papers which were inspired by Ramanujan's work. Hardy passed on to Watson the large number of manuscripts of Ramanujan that he had, both written before 1914 and some written in Ramanujan's last year in India before his death.

Ramanujan's infinite Series

One of Ramanujan's infinite series is now the basis for methods used to compute Pi:

Ramanujan's equation arrives at values of Pi to large numbers of decimal places more rapidly than just about any other known series. Each extra term in the summation adds around eight digits to the decimal expantion of  π.  Ramanujan calculated the value of   π   to 14 decimal places. i.e. 3.141592653589793.

Ramanujan also gave 14 other series for 1/π   but offers little explanation of where they came from. Even now, with the help of a more theoretical understanding aided by mathematical tools such as computer software for manipulating algebric expressions, mathematicians still find it hard to generate the kind of identities that Ramanujan already found.

Ramanujan and Euler's Constant

Ramanujan was evidently fascinated with Euler’s Constant

In his second note book, Ramanujan gave many beautiful formulas for π and  1/ π  and Euler's constant   γ  = 0:57721566 … , which occurs in many well-known formulas involving the Gamma function, the Riemann zeta function, the divisor function d(n), etc.

From  1970  onwards Mathematicians discovered the importance of Ramanujan’s  work  on gamma, Riemann Zeta functions  & Euler’s constant  in the application of Computer Science,  particularly,   in the field of security of internet,  etc.,

Magic square

A magic square is an arrangement of numbers (usually integers) in a square grid, where the numbers in each row, and in each column, and the numbers that run diagonally in both directions, all add up to the same number.

Magic square of order 3

The constant that is the sum of every row, column and diagonal is called the magic constant or magic sum, M. Every normal magic square has a unique constant determined solely by the value of n, which can be calculated using this formula:

For example, if n = 3, the formula says M = [3 (32 + 1)]/2, which simplifies to 15. For normal magic squares of order n = 3, 4, 5, 6, 7, and 8, the magic constants are, respectively: 15, 34, 65, 111, 175, and 260.

A construction of a magic square of order 4 :

Go left to right through the square counting and filling in on the diagonals only. Then continue by going left to right from the top left of the table and fill in counting down from 16 to 1. as shown below.

M = Order 4 1 4 6 7 10 11 13 16

M = Order 4 1 15 14 4 12 6 7 9 8 10 11 5 13 3 2 16

An extension of the above example for Orders 8 and 12 First generate a "truth" table, where a '1' indicates selecting from the square where the numbers are written in order 1 to n2 (left-to-right, top-to-bottom), and a '0' indicates selecting from the square where the numbers are written in reverse order n2 to 1. For M = 4, the "truth" table is as shown below, (third matrix from left.)

M = Order 4 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

M = Order 4 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

M = Order 4 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1

M = Order 4 1 15 14 4 12 6 7 9 8 10 11 5 13 3 2 16

Continued Fraction

Let us take P / Q as general form of  a continued fraction.  Where P and Q are whole, positive numbers.  Then expressing in the form of a continued fraction,                                                             

            =a+1/(b+1)/(c+1/(d+…….)))  =a+

Where a, b, c, d, e, etc are all whole numbers.  If P/Q is less than 1, then the first number a, will be 0.

Ramanujan developed a number of interesting closed-form expressions for non-simple continued fractions.  These include the almost integers.

1)                                                        

2)                                    

   

                                                                         

The Hardy-Ramanujan Number :

The following is a famous story told by G.H. Hardy about S.Ramanujan.  Once, in a taxi from London,  Hardy noticed its number, 1729.    When Hardy met Ramanujan in a hospital, he declared ‘1729’ as a dull number,.  Immediately,  Ramanujan answered, “No Hardy, it is a very interesting number.  It is the smallest number,  expressible as the sum of two cubes in two different ways.”

The number  1729 has since become known as   Hardy-Ramanujan number,  “the sum of two positive cubes in two different ways” 

Ramanujan published 39 papers in all, of which 5 were in collaboration with Hardy. He also proposed 59 questions and/or Answer to Questions in the JIMS, during his short life span of 32 years, 4 months and 4 days. 

The  Mathematic genius achieved infinite things during his short life span, Today, the world of Mathematics remembers him,  honours  him and several Departments  and Institutes have been named  after him.