srinivana ramanujan eng

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Dream 2047, December 2012, Vol. 15 No. 3

The year 2012 marks the 125th birthanniversary of Srinivasa Ramanujan,who is considered one of the greatestmathematicians of the twentieth century.Well-known mathematicians Professors G.H. Hardy and J.E. Littlewood comparedRamajuans mathematical abilities and naturalgenius with all-time great mathematicianslike Leonhard Euler, Carl Friedrich Gauss,and Karl Gustav Jacobi.

The influence of Ramanujan onnumber theory is without parallel inmathematics. His papers, problems,and letters would continue to captivatemathematicians for generations to come. Herediscovered a century of mathematics and

made new discoveries.Srinivasa Ramanujan Iyengar (bestknown as Srinivasa Ramanujan) was bornon 22 December 1887, in Erode, about400 km from Chennai (formerly Madras).Ramanujans father Srinivasa Iyengar workedas an accountant for a cloth merchant.Ramanujan was the first child born to hismother Komalatammal.

Ramanujan showed a stronginclination towards mathematics fromearly age and won numerous awards for hiscalculating skills in elementary school. He

passed his primary examination in 1897 andthen joined the Town High School.

While at school, Ramanujan cameacross a book entitledA Synopsis of ElementaryResults in Pure and Applied MathematicsbyGeorge Shoobridge Carr. This book had agreat influence on Ramanujans career. G.H.Hardy (1877 1947), an eminent English

mathematician wrote about the book: He(Carr) is now completely forgotten, even inhis college, except in so far as Ramanujankept his name alive. Ramanujan solvedall the problems in Carrs Synopsis. Whileworking on the problems in the book, hediscovered many other new formulae andprovided results which were not there in

the book. He jotted the results down in anotebook, which he showed to people hethought might be interested. Between 1903and 1914 he had compiled three notebooks.

In 1904, Ramanujan enteredKumbakonams Government College as F.A.student. He was awarded ascholarship. However, afterschool, Ramanujans totalconcentration was focussedon mathematics and heneglected other subjects.As a result he failed and

lost his scholarship. During19061912 Ramanujanwas constantly in searchof a benefactor. Withouta university degree it wasvery difficult for him tofind a suitable job andhad to struggle financially.Unfortunately he did nothave anyone to directhim in his mathematical

research. But that did not deter his passion for

mathematics and he spent most of his timeon mathematics. He noted down his resultsin his notebooks. These notebooks were histreasures. He looked for a job for livelihoodand to support his parents and two brothersHe tutored a few students in mathematicsHowever, because of his unconventionamethods, he was not considered to be a goodteacher. Ramanujans mother Komalatammawas on the lookout for a bride to get her eldesson married. On 14 July, 1909 Ramanujanwas married to Janaki.

In 1910 Ramanujan met Professor

V. Ramaswami Iyer, an ardent scholar omathematics and founder of the IndianMathematical Society. After seeing thenotebooks, Professor Ramaswami wasconvinced that Ramanujan was a giftedmathematician.

Ramanujans earliest contributionwas in the form of question/answer in theJournal of the Indian Mathematical SocietyRamanujan proposed 58 questions andtheir solutions during the period February1911 to October 1911. The first full-lengthresearch paper of Ramanujan, entitled Some

properties of Bernoulli Numbers, appearedin the Journal of the Indian MathematicaSociety in 1911.

In 1912, Ramanujan secured a jobas a clerk in the accounts section of theMadras Port Trust. In the meantime his

Srinivasa Ramanujan

Srinivasa Ramanujan(1887 1920)

Ramanujan (centre) with other scientists at Trinity CollegeG.H. Hardy (1877 1947)

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Rintu NathEmail: [email protected]


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mathematical work caught theattention of other scholars whorecognised his abilities. He wasencouraged to contact Englishmathematicians in the hope that they wouldbe able to assist him. Professor C.L.T.Griffith of Engineering College, Madras,forwarded some of Ramanujans results ondivergent series to Professor M.J.M. Hill ofthe University of London. Unfortunately,Professor Hill could not study the results indetail and suggested a book and gave adviceas to how Ramanujan could get his paperpublished.

In 1913 Ramanujan wrote a letterto the famous English mathematician

G.H.Hardy, who discussed Ramanujans

letter with his collaborator and friend,mathematician John Littlewood (18851977). After studying and discussing theletter, both realised that Ramanujan was aworld-class mathematician and decided tobring Ramanujan to Cambridge. Ramanujanarrived in London on 14 April 1914. For thenext five years, Ramanujan was associatedwith Hardy. Their collaboration representsthe efforts of two great talents. Ramanujanwas awarded the B.A. degree by research,

in March 1916, for his work on highlycomposite numbers. He was the first Indianmathematician to be awarded the prestigiousFellowship of the Royal Society, in February1918. Dr. P.C. Mahalanobis (18931972)was a student at Kings College, Cambridge

during that time and he became agood friend of Ramanujan.

The period of Ramanujansstay in England almost overlappedwith World WarI. Duringhis five-year stay in TrinityCollege, Cambridge, Ramanujanpublished 21 research paper, five

of which were in collaborationwith Hardy. During this timeRamanujan also published shornotes in theJournal of the IndianMathematical Society.

After World War-IRamanujan returned to Indiain 1919. After his return fromEngland his health deterioratedand his wife looked after himEven during those months ofprolonged illness Ramanujan kept

on jotting down his mathematicacalculations and results on sheets of paperIn January 1920, he wrote letter to Hardyand communicated his work on mock thetafunction. Despite all the tender attentionfrom his wife and the best medical attentionfrom doctors, his health deteriorated. He

The remarkable Ramanujan and the golden ratio

Srinivasa Ramanujan was a mathematical genius who had the ability look into the depthof mathematics. He created beautiful equations that became humankinds vast storehouse

of knowledge. Ramanujan was an expert on infinite series, continued fractions andidentities. Ramanujans equations, once comprehended, unfold beautiful mathematicalsymmetry.

The following equation demonstrate his artistry

What is hidden at the right side of the equation is the golden ratio .

Also

Therefore, if is substituted in the equation, we get the following:

The expression includes an infinite continuing fraction, e, p and the golden ratio (). Itis interesting to see how the golden ratio inevitably placed itself in one of the equations of

the greatest mathematician of twentieth century.

Infinite series to calculate pRamanujan discovered some new infinite series formula in1910, but their importance was re-discovered around late1970s, long after his death. One of his elegant formulas waslike this:

Each addition of a term in Ramanujans series could giveapproximately additional eight digits to p. During 1985, about17 million digits of pwere accurately computed by Americanmathematician William Gosper using this formula. So it alsoproved the validity of Ramanujans formula. In 1994, Davidand Gregory Chudnovsky brothers of Columbia Universitycomputed over four billion digits of in a supercomputer usingan algorithm which was also similar in essence to the formulagiven by Ramanujan.

On the occasion of 75th Birthanniversary of Ramanujan, the

Indian Philately Associationbrought out a commemorative

stamp in 1962

Komalatammal,Ramanujans mother

Janaki,Ramanujans wife

National Mathematics Year 2012

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breathed his last on 26 April 1920, at theage of 32.

After Ramanujans death, Hardytried systematic verification of Ramanujansresults from the second notebook. However,it was a daunting task and he persuaded theUniversity of Madras to undertake the task.In 1931, the University of Madras requestedProfessor G.N. Watson to edit the notebooksin a suitable form for publication. Thiswas a formidable task, since the notebookscontained over 300 theorems. Watsonundertook the task of editing the notebookswith Professor B.M. Wilson. Unfortunately,Wilson passed away in 1935, virtuallymarking the end of the efforts to edit thenotebooks.

The collected edition of Ramanujansworks was later edited by Hardy. Thefirst edition of this book was published

in 1927 by Cambridge University Press.This resulted in a flurry of research papersduring the period 192838. In 1999, theAmerican Mathematical society and LondonMathematical Society reprinted the collectedpapers.

Much of Ramanujans mathematicsfalls in the domain of number theory thepurest realm of mathematics. During hisshort lifetime, Ramanujan independentlycompiled nearly 3,900 results (mostlyidentities and equations). He statedresults that were both original and highly

unconventional, such as the Ramanujanprime and the Ramanujan theta function,and these have inspired a vast amount offurther research in mathematics.

As Robart Kanigel says .....few cansay much about his work, and yet somethingin the story of his struggle for the chance topursue his work on his own terms compelsthe imagination, leaving Ramanujan asymbol for genius, for the obstacles it faces,for the burdens it bears, for the pleasure ittakes in its own existence.

References:K. Srinivasa Rao, Relevance of Srinivasa

Ramanujan at the dawn of thenew millennium, The Institute ofMathematical Sciences, Chennai- 600113.

Robart Kanigel, The man who knew infinity,Washington Square Press, New York,1992.

n

and LPAare complementary.The angles CMN and CPN are

subtended at the circumference by the same

arc CN. So, angle CMN = angle CPN. Alsosince the sum of the angles of triangle CNPis 180, and angle CNP is 90, the anglesNPCand NCPare complementary.

Now, refer to Fig. 5. Since pointP lies on the circumcircle of triangleABC, the quadrilateral APCB is cyclic.Therefore opposite angles BAP and BCPare supplementary; i.e., angle BAP + angleBCP = 180.Again, angle BAP + angle

LAP=180.

\Angle BCP = angle LAPNow, refer to Fig. 6. Since angles BCP

and NCPare the same, it follows that angles

Fig 5.

LAPand NCPare equal. Since angles LAPand LPA are complementary, and angleNCPand NPCare complementary. So angle

LPAand NPCare complementary to equaangles, and so angle LPA= angle NPC.

Now, angle LPA= angle LMAAnd angle NPC= angle NMC\angle LMA= angle NMCButACis a straight line\AMCis a straight angle\ angleAMC = 180\ angle LMC = 180 angle LMAOr angle LMC = 180 angle NMC\ angle LMC+ angle NMC = 180\ angle LMN = 180 = A straigh

angle.

Hence the proof.

Fig 6.

Continued from page 31 (Robert Simson and the line in his name)

Cartoon by : V.S.S. Sastri E-mail: [email protected]

National Mathematics Year 2012

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srinivana ramanujan eng

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