INFINITYEditorial Board

Editor-in-ChiefAditya V. Gupta

Chief-of-ProductionAviral Gupta

EditorsUjjwal DahujaShivam Goyal

Associate EditorsDevesh Sharma

Udbhav Agarwal

Correspondents Abhinav Kejriwal

Anmol JainHarshvardhan Singh

Pranjalya Shukl Siddharth SuriSabhya Katia

Yash Dhandhania

Faculty AdvisorsAnjan Chaudhary

Chandan GhughtyalMona Khanna

ContributorsAzan Brar

Rishith AgarwalShrey Aryan

Special ThanksPurnima Dutta

Editorial

As I look back over this term and the progress that the ‘In�nity’ has made within the span of the term, a satisfactory feeling rises within me giving me the strength to hope for an even better second term. This term the ‘In�nity’ evolved from a mere publication into a society of sorts with regu-lar meetings and active debates. The number of people working with the publication too increased with a few keen ‘C’ Formers and ‘D’ Formers attending regularly. These meetings did not follow the conventional manner of operation where a single master or student droned on about some concept or ‘interesting’ theory while the others looked on with an outwardly convincing expression of understanding on their face but a deep-seated desire within to escape. These meetings were instead conducted in the manner of an informal discussion where everybody had the freedom to raise their doubts and express their thoughts at any point of time. The sessions were made even livelier with our masters, Mr. Chan-dan Singh Ghugtyal and Mrs. Mona Khanna, demonstrating a practical method to explain and memorize formulae such as the square of two terms, which otherwise seem to be very mundane and confusing with all the letters and numbers indistinguishably intertwined amongst each other. On the one hand as we had these discussions going on for the math-enthusiasts, we also had our work progressing on the publication. The �rst ten minutes of each meeting were dedicated to getting an update on the work done by the students and as the discussion proceeded more work was assigned, often related to the discussion itself. As the issue suggests a major topic of discussion was India’s mathemati-cal pride itself, Srinivasa Ramanujan. The topic was �rst introduced by Mr.Anjan Chaudhary and almost, as if magically, (much like Ramanujan himself ) it roused the interest of a number of people which resulted in the topic soon snowballing to become the theme for our �rst issue. Every single member had something or the other that he knew about Ramanu-jan or had read somewhere thus providing for a very passionate and informative discussion. In this edition of the ‘In�nity’ a majority of the articles focus on the prodigious Ramanujan, his short life and his unparal-leled achievements.The issue also contains a section talking about this year being declared as the National Year of Mathematics, its importance and three gifted Indian mathematicians who today represent India’s slowly dimming presence in the �eld of mathematics. There is also a puzzle towards the end for those interested in challenging themselves getting their grey cells to work. And to counter of allegations of printing content that many say is ‘OHT’, we as a team have worked hard to make this issue one that can be understood by all and still retain its mathemati-cal perspective. So I do hope that you enjoy reading what we have to present, and that it leaves a few thinking, if not all.

May 2012, Issue No. 10

Spring 2012 • Infinity 1

2 Infinity • Spring 2012

Ramanujan’s LifePranjalya Shukla and Sidhartha Suri take us down memory lane and give us glimpses of Ramanujan’s life.

K. Srinivasa Rao - , "Suppose we rate mathematicians on the basis of pure talent on a scale from 0 to 100, Hardy will be given a score of 25, J.E. Littlewood 30, David Hilbert 80 and Ramanujan 100."Such were the achievements of Ramanujan. He was India’s greatest mathematical genius. He made substantial contributions to the analytical theory of numbers and also worked on elliptic functions, continued fractions and in�nite series.Srinivasa Ramanujan was born in his grandmother’s house in Erode, a small village near Madras in 22 December 1887. When Ramanujan was about one year old, his mother took him to the town of Kumba-konam. His father worked in the town as a clerk in a cloth merchant’s shop and his mother was a house-wife, and sang at a local temple. When Ramanujan was �ve years old, he was enrolled in a local primary school but for the next �ve years he continued to shift schools. In January 1898, when Ramanujan was 10 years old, he entered the town higher Secondary School. This is where he encountered formal math-ematics for the �rst time. At 11, Ramanujan had started advanced trigonometry.

His headmaster had stated, “Ramanujan is an outstanding student and he deserved scores higher than the maximum possible marks.”

He started solving questions in books written by S.L. Looney. Ramanujan had started taking more informa-tion from two college boys who used to stay at his home as boarders. He had become a master of trigo-nometry by the age of 13 (the age at which others are introduced to trigonometry). He had also started discovering his own formulae. He would �nish all his mathematics exams in half the allotted time. His mathematical talent was distinctly visible. When he was 16 he took a copy of’A Synopsis of Elementary Results in Pure and Applied Mathematics’. It had 5000 theorems. Ramanujan studied the contents of the book in detail and this book was the key element in awakening the genius in Ramanujan. The next year,

he developed and investigated the Bernoulli num-bers and had calculated Euler’s constant up to 15 decimal places. At that time his peers commented that they "rarely understood him" and "stood in respectful awe" of him.

Although Hardy was one of the foremost mathematicians of his day and an expert in a number of �elds , he commented that, "many of them (Ramanujan’s theorems) defeated me completely.”

When Ramanujan had graduated from Town Higher Secondary School in 1904, at the age of 17, he was awarded the K. Ranganatha Rao prize for mathemat-ics by his school’s Headmaster, Krishnaswami Iyer. His headmaster had stated, “Ramanujan is an outstand-ing student and he deserved scored higher than the maximum possible marks.” Ramanujan had got a scholarship in Government Arts College, Kumbako-nam but he was so focussed on mathematics that he had failed the rest of the subjects, which resulted in his losing the scholarship. In August 1905, when he was 18, he ran away from home heading towards Vishakapatnam. Later, he enrolled at Pachaiyappa’s College in Madras. There also, he excelled in math-ematics but performed poorly in the other subjects. Thus, he failed the Fine Arts degree exam in Decem-ber 1906 and left college (without a degree). Since then, he continued to pursue independent research on mathematics. At this time of his life, he was living in extreme poverty and was on the brink of starva-tion.In 1908, Ramanujan, now 21, continued his math-ematical research by studying continued fractions and divergent series. Around this time he also fell seriously ill and had to undergo an operation in April 1909, from which it took him considerable time to recover. He married on the 14th of July 1909, at the age of 22, when his mother found a ten year old girl, S. Janaki Ammal, for him. However, he didn’t live with her until two years after their marriage. In late 1912 and early 1913, Ramanujan sent his theo-

rems and his formulae derivations to three Cambridge masters: H.F. Baker, E.W. Hobson, and G.H. Hardy. The �rst two professors laughed at his idiotic theorems while G.H. Hardy saw the amazing talent hidden in Ramanujan. G.H. Hardy then said, “Not one (theorem) could have been set in the most advanced math-ematical examination in the world.” Although Hardy was one of the foremost mathematicians of his day and an expert in a number of �elds that Ramanujan was writing about, he commented that, "many of them (theorems) defeated me completely; I had never seen anything in the least like them before. A single look at them is enough to show that they could only be written down by a mathematician of the highest class.”Hardy now was bringing about plans to bring Ramanujan to Cambridge, this wasn’t an easy task as it was Ramanujan’s 1st trip abroad and he was a pure and chaste brahmin and he would have a hard time settling. However, his well-wishers wanted him to go to England and a few of his friends met the governor of Madras to plead the case so that Ramanujan could go to Cambridge. A sum of 10,000 rupees was collected for his travel besides the 5 year scholarship he received to publish some of his mathematical �ndings there. Ramanjuan didn’t change his working habits at all while at Cambridge. He used to work for about 17 hours a day, sleep for 6 hours and continue to work from where he left o�. As said by Professor Littlewood, “Ramanujan lived with numbers.” While at Cambridge, Ramanujan used his intuition to prove theories and solve problems. He was then advised to attend a class with Arthur Berry, a mathematics tutor and Mr. Berry stated, “I was working out some formulas on the blackboard. I was looking at Ramanujan from time to time to see whether he was following what I was doing. At one stage Ramanujan’s face was beaming and he appeared to be greatly excited. He then got up from his seat, went to the blackboard and wrote some of the results which I had not yet proved. Ramanujan must have reached these results by pure intuition. ... Many of the results apparently came to his mind without any e�ort.”Ramanujan got his B.A. degree on the 16th of March, 1916, at age 29, for his high contribution in the �elds of composite numbers, elliptic functions and theory of numbers. Early in 1918, he tried to commit suicide by lying on the rail tracks and luckily he failed as the driver saw him and stopped the train. After this Ramnujan’s health started failing but he never let his family know. He had high persistent fever and a bad food situation. His health kept worsening in England due to the unbearable stress and the lack of vegetar-ian food during the First World War. He also became lonely and struggled with depression. He was later diagnosed with Tuberculosis and extreme Vitamin de�ciency. Despite the odds in the midst of all this, he discovered the Hardy-Ramanujan Number : 1729.Ramanujan’s health continued to worsen. In 1919, at the age of 32, he returned to India and was put under medical attention. Unfortunately, he passed away on 26th of April, 1920. He was concluded with diseases like Hepatic Amoebiasis, Dysentry, etc. which couldn’t be cured. Post Ramanujan’s death, G.H. Hardy said, “Ramanujan as a person was somewhat shy and quiet disposition, a digni�ed man with pleasant manners and great modesty. He was also known to be extremely sensitive. On one occasion, he had prepared a bu�et for a number of guests, and when one guest politely refused to taste a dish he had prepared, he left immediately and took a taxi to Oxford. He also lived a rather Spartan life while at Cambridge.”On discovering a new theorem or after reconstructing it he often said, "An equation for me has no mean-ing, unless it represents a thought of God.Narlikar also goes on to say that his work was one of the top-ten achievements of twentieth-century Indian science and "could be considered in the Nobel Prize class."

Spring 2012 • Infinity 3

rems and his formulae derivations to three Cam-bridge masters: H.F. Baker, E.W. Hobson, and G.H. Hardy. The �rst two professors laughed at his idiotic theorems while G.H. Hardy saw the amazing talent hidden in Ramanujan. G.H. Hardy then said, “Not one (theorem) could have been set in the most advanced mathematical examination in the world.” Although Hardy was one of the foremost mathematicians of his day and an expert in a number of �elds that Ramanu-jan was writing about, he commented that, "many of them (theorems) defeated me completely; I had never seen anything in the least like them before. A single look at them is enough to show that they could only be written down by a mathematician of the highest class.”Hardy now was bringing about plans to bring Ramanujan to Cambridge, this wasn’t an easy task as it was Ramanujan’s 1st trip abroad and he was a pure and chaste brahmin and he would have a hard time settling. However, his well-wishers wanted him to go to England and a few of his friends met the governor of Madras to plead the case so that Ramanujan could go to Cambridge. A sum of 10,000 rupees was collected for his travel besides the 5 year scholarship he received to publish some of his mathematical �ndings there. Ramanjuan didn’t change his working habits at all while at Cambridge. He used to work for about 17 hours a day, sleep for 6 hours and continue to work from where he left o�. As said by Professor Little-wood, “Ramanujan lived with numbers.” While at Cambridge, Ramanujan used his intuition to prove theories and solve problems. He was then advised to attend a class with Arthur Berry, a mathematics tutor and Mr. Berry stated, “I was working out some formu-las on the blackboard. I was looking at Ramanujan from time to time to see whether he was following what I was doing. At one stage Ramanujan’s face was beaming and he appeared to be greatly excited. He then got up from his seat, went to the blackboard and wrote some of the results which I had not yet proved. Ramanujan must have reached these results by pure intuition. ... Many of the results apparently came to his mind without any e�ort.”

He continued to pursue independent research on mathematics. At this time of his life, he was living in extreme poverty and on the brink of starvation.

Ramanujan got his B.A. degree on the 16th of March,

1916, at age 29, for his high contribution in the �elds of composite numbers, elliptic functions and theory of numbers. Early in 1918, he tried to commit suicide by lying on the rail tracks and luckily he failed as the driver saw him and stopped the train. After this Ramnujan’s health started failing but he never let his family know. He had high persistent fever and a bad food situation. His health kept worsening in England due to the unbearable stress and the lack of vegetar-ian food during the First World War. He also became lonely and struggled with depression. He was later diagnosed with Tuberculosis and extreme Vitamin

de�ciency. Despite the odds in the midst of all this, he discovered the Hardy-Ramanujan Number : 1729.Ramanujan’s health continued to worsen. In 1919, at the age of 32, he returned to India and was put under medical attention. Unfortunately, he passed away on 26th of April, 1920. He was diagnosed with diseases like Hepatic Amoebiasis, Dysentry, etc. which couldn’t be cured. Post Ramanujan’s death, G.H. Hardy said, “Ramanujan as a person was somewhat shy and of a quiet disposition, a digni�ed man with pleasant man-ners and great modesty. He was also known to be extremely sensitive. On one occasion, he had prepared a bu�et for a number of guests, and when one guest politely refused to taste a dish he had prepared, he left immediately and took a taxi to Oxford. He also lived a rather Spartan life while at Cambridge.”On discovering a new theorem or after reconstruct-ing it he often said, "An equation for me has no mean-ing, unless it represents a thought of God.Narlikar also goes on to say that his work was one of the top-ten achievements of twentieth-century Indian science and "could be considered in the Nobel Prize class."

4 Infinity • Spring 2012

‘1729’Sabhya Katia examines Ramanujan’s number and discusses its importance and ‘versatility’ as a number.

Once, the famous mathematician G.H. Hardy had gone to visit his dear friend S.K. Ramanujan in the hospital. He had come in a taxi cab numbered 1729, a number which he had found quite dull. Conse-quently, Hardy went on to tell Ramanujan about the dullness of the number 1729 as well. Almost instantly, Ramanujan interestingly pointed out that 1729 was actually, as Hardy later put it, the smallest number expressible as the sum of two cubes in two di�erent ways’. Coincidental luck with numbers, or sheer math-ematical genius? That’s for you to decide.Anyway, coming to 1729, the two ways by which it can be expressed as the sum of two cubes are:

1729 = 13 + 123 = 93 + 103

But Ramanujan had overlooked a small detail, that both bases have a positive value each. There are smaller numbers which can be expressed as the sum of the cubes of two di�erent numbers but in those cases one of the two bases needed to be negative. For Eg:

91 = 63 + (−5)3 = 43 + 33

So we can conclude that 1729 is the smallest integer that can be expressed as the sum of cubes of two di�erent positive numbers in two di�erent ways.However, there are smaller numbers than this which are negative. For example: -91,-189, -1729, etc. The smallest numbers that can expressed in this form have been termed as “taxicab numbers”. Again, these numbers have been found in one of Ramanujan’s note books made years before the incident in the hospital had happened, and was noted by Frenicle de Bassy in 1657.

So we can conclude that 1729 is the smallest integer that can be expressed as the sum of cubes of two di�erent positive numbers in two di�erent ways.

1729 has also been de�ned as the �rst number in the sequence of “Fermat near misses”(sequence A050794 in OEIS) which explains that the numbers that can be

expressed in the form of 1 + z3 can also be expressed in the form of a sum of cubes of two numbers. Inter-estingly 1729 is also the third Carmichael number and the �rst absolute Euler pseudoprime. It is also a spenic number, a Zeisel number, a centered cube number, a dodecagonal number, a 24-gonal as well as an 84-gonal number.Meanwhile, while investigating pairs of distinct inte-ger- valued quadratic forms that represent every inte-ger the same number of times Schiemann discovered that these quadratic equations must have four or more variables, and the least possible discriminant of these type of four- variable pair is, yes, 1729.

It is also a spenic number, a Zeisel number, a centered cube number, a dodecagonal number, a 24-gonal as well as an 84-gonal number.

Continuing with the list, 1729 is a Harshad number, which is a number divisible by the sum of its digits in base 10. It also holds this property in other number systems like in the octal system {1729 = 33018, 3 + 3 + 0 + 1 = 7} and in the hexadecimal system {1729 = 6C116, 6 + C + 1 = 1910}.1729 has another mildly interesting property: the 1729th decimal place is the beginning of the �rst occurrence of all ten digits consecutively in the deci-mal representation of the transcendental number e {2.718…}.Looking at the numerous properties that 1729 holds, I had a fairly strenuous time deciding which one deserved to conclude more than the others. Anyway, I chose something that I thought would appeal most to our readers.Masahiko Fujiwara stated that the number 1729 is one of the positive integers {the other numbers being 81, 1458 and 1} whose digits when added together, produce a number which when multiplied by its reverse, yield the original number.

1 + 7 + 2 + 9 = 19

19 × 91 = 1729

Man has discovered a myriad things in the past, which have aided humanity to a great extent: Be it the discovery of �re or a place like America. However, G.H. Hardy discovered a mathematician for the world in the form of the legendary Ramanujan.On the other hand, I ‘discovered’ Ramanujan through his discoverer’s book, ‘A Mathematician’s Apology.’ It was in this book that I �rst came across the unique mathematician who had completely transformed the 20th century mathematics. Any study, be it a research or a biography of Ramanujan is incomplete without a study of the relationship between Ramanujan and Hardy. The fact is that, Ramanujan and Hardy, together did change the world with their intelligence. There are countless theories Ramanujan formulated. The depressing part was that most of his prestigious works were lost and only few have survived till date. Ramanujan sailed to England in 1914 and between 1914-1917, Hardy and Ramanujan worked together, taking mathematics to a whole new level. During these years, Ramanujan worked on the theory of a partition function(used to measure the frequency of decomposition of a natural number) and published his results in a manner Hardy termed as ‘most inter-esting’.

In other works on functions, Hardy and Ramanujan also worked to create the ‘circle-method’(widely used today) using which an asymptotic formula of a func-tion can be arrived at. Using the circle method, Ramanujan also made breakthroughs in solving the Goldbach’s and Waring’s conjecture.

When Ramanujan sent his �rst letter to Hardy, the English mathematician was taken aback by the abso-lute genius; the results that Ramanujan had commu-nicated to Hardy were stunning. Hardy, on receiving the letter, discussed with J.E. Littlewood, another mathematical sensation of the time which were all the results Ramanujan had put forward. In the end, he concluded, “They must be true, because if they were not true, no one would have had the imagination to invent them.'' That letter contained 120 theorems.

When Ramanujan sent his �rst letter to Hardy, the English mathematician was taken aback by the absolute genius; the results that Ramanujan had communicated to Hardy were stunning.

However, the landmark paper from Ramanujan came in 1916 titled “On certain mathematical functions.” In this paper, Ramanujan worked on the properties of Fourier co-e�cients, a research that formed the foun-dation of major mathematical research for the years to come. In the process, Ramanujan also proved three conjectures, which guided him to the proof of his proposed theory. Even Andrew Wiles, who solved the Fermat’s last theorem. This said that the discoveries of Ramanujan had helped him solved the most evasive mathematical problem. It was the last three conjec-tures of Ramanjuan that took the mathematical fraternity by storm, one of which is called the ‘Ramanujan conjecture’. In his last letter to G.H. Hardy, Ramanujan wrote about his work on mock theta functions, a �eld of mathematics being researched vigorously even today. G.H. Hardy most aptly describes Ramanujan when he says that Ramanujan was in the same league of mathematicians as Gauss and Erdos in terms of natural genius. Though the legend died at the age of 32, he is also worthy enough of being called in the extraordinary league of Newton and Einstein. Certainly, it was Ramanujan who revolutionized mathematics, a subject which �nds application in our daily lives.

Spring 2012 • Infinity 5

Ramanujan & HardyUjjwal Dahuja discusses the famous and all-important Ramanujan Hardy relationship which helped Ramanujan in shaping his life.

Man has discovered a myriad things in the past, which have aided humanity to a great extent: Be it the discovery of �re or a place like America. However, G.H. Hardy discovered a mathematician for the world in the form of the legendary Ramanujan.On the other hand, I ‘discovered’ Ramanujan through his discoverer’s book, ‘A Mathematician’s Apology.’ It was in this book that I �rst came across the unique mathematician who had completely transformed the 20th century mathematics. Any study, be it a research or a biography of Ramanujan is incomplete without a study of the relationship between Ramanujan and Hardy. The fact is that, Ramanujan and Hardy, together did change the world with their intelligence. There are countless theories Ramanujan formulated. The depressing part was that most of his prestigious works were lost and only few have survived till date. Ramanujan sailed to England in 1914 and between 1914-1917, Hardy and Ramanujan worked together, taking mathematics to a whole new level. During these years, Ramanujan worked on the theory of a partition function(used to measure the frequency of decomposition of a natural number) and published his results in a manner Hardy termed as ‘most inter-esting’.

In other works on functions, Hardy and Ramanujan also worked to create the ‘circle-method’(widely used today) using which an asymptotic formula of a func-tion can be arrived at. Using the circle method, Ramanujan also made breakthroughs in solving the Goldbach’s and Waring’s conjecture.

When Ramanujan sent his �rst letter to Hardy, the English mathematician was taken aback by the abso-lute genius; the results that Ramanujan had commu-nicated to Hardy were stunning. Hardy, on receiving the letter, discussed with J.E. Littlewood, another mathematical sensation of the time which were all the results Ramanujan had put forward. In the end, he concluded, “They must be true, because if they were not true, no one would have had the imagination to invent them.'' That letter contained 120 theorems.

When Ramanujan sent his �rst letter to Hardy, the English mathematician was taken aback by the absolute genius; the results that Ramanujan had communicated to Hardy were stunning.

However, the landmark paper from Ramanujan came in 1916 titled “On certain mathematical functions.” In this paper, Ramanujan worked on the properties of Fourier co-e�cients, a research that formed the foun-dation of major mathematical research for the years to come. In the process, Ramanujan also proved three conjectures, which guided him to the proof of his proposed theory. Even Andrew Wiles, who solved the Fermat’s last theorem. This said that the discoveries of Ramanujan had helped him solved the most evasive mathematical problem. It was the last three conjec-tures of Ramanjuan that took the mathematical fraternity by storm, one of which is called the ‘Ramanujan conjecture’. In his last letter to G.H. Hardy, Ramanujan wrote about his work on mock theta functions, a �eld of mathematics being researched vigorously even today. G.H. Hardy most aptly describes Ramanujan when he says that Ramanujan was in the same league of mathematicians as Gauss and Erdos in terms of natural genius. Though the legend died at the age of 32, he is also worthy enough of being called in the extraordinary league of Newton and Einstein. Certainly, it was Ramanujan who revolutionized mathematics, a subject which �nds application in our daily lives.

6 Infinity • Spring 2012

Indian MathematiciansAzan Brar, Parth Khanna and Rishith Agarwal brie�y take us through the lives of some Indian mathemticians who have made the country proud with their work in mathematics.

Kannan Soundararajan

Kannan Soundararajan is an Indian mathematician. He currently is a professor of mathematics at Stanford University. Before moving to Stanford in 2006, he was a faculty member at University of Michigan where he pursued his undergraduate studies. His main research interest is in Number theory especially L-functions and multiplicative number theory. Soundararajan grew up in Chennai and was a student at Padma Seshadri High School in Nungambakkam in Chennai. Soundararajan joined the University of Michigan, Ann Arbor, in 1991 for undergraduate studies, and gradu-ated with highest honours in 1995. He won the inau-gural Morgan Prize in 1995 for his work in analytic number theory whilst an undergraduate at the University of Michigan, where he later served as professor. He joined Princeton University in 1995 and did his PhD under the guidance of Professor Peter Sarnak. As a graduate student at Princeton, he held a presti-gious Sloan Foundation Fellowship. After his Ph.D. he received the �rst �ve year fellowship from the Ameri-can Institute of Mathematics, and held positions at Princeton University, the Institute for Advanced Study, and the University of Michigan. He moved to Stanford University in 2006 where he is currently a Professor of Mathematics and the Director of the Mathematics Research Center at Stanford. He received the Salem Prize in 2003 "for contributions to the area of Dirichlet L-functions and related character sums". In 2005, he won the $10,000 SASTRA Ramanu-jan Prize, shared with Manjul Bhargava, awarded by SASTRA in Thanjavur, India, for his outstanding contri-butions to number theory. In 2011, he was awarded the Infosys Prize.

Dr. Vashishta Narayan Singh

Dr. Vashishta Narayan Singh was born in Bihar to a family that was not very well o�. His father was a constable with the Bihar Police and Dr. Vashishta studied at Neharat Vidyalaya and at the Science College. After �nishing the Bihar Board in Matricula-tion Examination and Bihar Intermediate Education Council for Intermediate Examinations Science

College, he was confronted by a professor from University of California, Berkley. The professor, impressed by his past performances and hoping for the unimaginable, presented him with �ve of the toughest problems that he had come across in his lifetime. It was predictable that Mr. Singh would solve them, but solve them each in two di�erent ways, made the professor sure that he deserved a place at his college. Shy at �rst, Dr. Singh had a mind of his own. He was one of those resolved and con�ned thinkers. Making wise use of his scholarship, he �nished his PhD with �ying colors, and went on to work for NASA. While on one side of the world his life was at its best, on the other side his family was waiting for him to get married. Dr. Singh then left for his homeland to do his country proud. Unfortunately, by then he had become a drug-addict. His parents married him to an Army O�cer’s daughter, happy with the enormous dowry. His marriage continued for quite some time, until his wife realized his love a�air with the powder. Ultimately, she left him. In late 1970’s he was admit-ted at the Meerut Hospital, and was diagnosed with schizophrenia, which resulted in him losing every-thing. Currently, one can �nd him in his village, picking up rags or staring at thin air.

C.S.Seshadri FRS

C.S.Seshadri FRS is an eminent Indian mathematician. He is currently Director-Emeritus of the Chennai Mathematical Institute. The Seshadri constant is named after him. He is also a recipient of the presti-gious Padma Bhushan, which he recieved in 2009, the third highest civilian honor in the country.Seshadri's has worked mainly in algebraic geometry. His work with M S Narasimhan on unitary vector bundles and the Narasimhan–Seshadri Theorem has been exceptional. His work on Geometric Invariant Theory and on Schubert varieties is also widely recog-nized. Apart from the great amount of research done by him, Seshadri's contribution includes the incep-tion of Chennai Mathematical Institute, which is one of the leading institutes for study in mathematics in India.

Ramanujan’s Brilliance Udbhav Agarwal justi�es as to why Ramanujan was distinguished as one of the greatest mathematicians.

Spring 2012 • Infinity 7

AchievementsPranjalya Shukla lists down a few of Ramanujan’s many achievements that distinguished him as one of the greatest mathematicians.

Ramanujan worked on the Gauss’ and Kummer’s hypergeometric series. Later, his own work on partial sums and products of hypergeometric series resulted in a huge development in that topic. He worked on the number p(n) of the partitions of an integer ‘n’ and this was his most famous work. He has also made remarkable contri-butions to the analytical theory of number, his main work being on his number : 1729. Also, during his time in Eng-land he worked on elliptic functions continued fractions and in�nite series. Ramanujan has also proved over 3,000 theorems, identities and equations which regard the properties of highly composite numbers and mock theta func-tions. He also performed well in the areas considering gamma functions, modular forms, divergent series

and prime number theory. Ramanujan identi�ed many e�cient and converging in�nite series for the calculations of the value of pi (π).

At the age of 12, he had also re-discovered the Euclid’s iden-tity in his own way. Ramanujan also compiled all of his work in four loose notebooks and when this notebook was found Berndt said: "The discovery of this 'Lost Notebook' caused roughly as much stir in the mathematical world as the discovery of Beethoven’s tenth symphony would cause in the musical world."In 1919, Ramanujan published

a new proof of Bertrand's postulate which, as he notes, was �rst proved by Chebyshev. At the end of the two-page published paper, Ramanujan derived a generalized result to �nd a prime number, a monu-mental discovery of the time.

It has been a century and an half; one-�fty years have passed since the death of someone so important that importance itself would shy o�. Nothing is more worthy of him than the words of E.H. Neville, ‘he was a mathematician so great that his name transcends jealousies, the one superlatively great mathematician whom India has produced in the last thousand years”; Srinivasa Ramanujan.Marc Karc once said, ‘An ordinary genius is a fellow that you and I would be just as good as, if we were only many times better. There is no mystery as to how his mind works. Once we understand what he has done, we feel certain that we, too, could have done it. It is di�erent with the magicians,’ and this was exactly what was so profound with Mr. Ramanujan, his style

was simple yet so perfect. His notebooks re�ected such exquisiteness which was so unique that made him extraordinary. It was hard to believe that a clerk in the Accountant General's o�ce in Madras in the year1912, a post got after much trial and tribulation, would be counted among the most in�uential men just 100 years afterwards.A pen, few sheets of paper, an envelope, an address, a postage stamp and this exquisite brain; was all that was required to bring him to the podium where he stands today. It was in the year 1914 when using �ve things of other men and one of his own that Ramanu-jan wrote and posted his �rst letter to Hardy, creating history. The University of Madras had already recog-nized this talent among the pigeons the previous

year and was now fully responsible for him. They �nanced him to study math on the same table on which Newton and Einstein once sat; in Trinity College, Cambridge; until his return to India in 1919. Sadly so, he had only a year left to him when he returned. The Royal Society had already awarded him the Fellowship of the Royal Society in 1918, making him the �rst Indian mathematician to be awarded the same, as if they knew about his untimely demise.

In his short life, Mr. Ramanujan was quite an unpublished mathematician. All people knew of him was either through Mr. Hardy or the Madras University. It was after his death that his works were really recognized, not directly by the people but by his contemporaries. Hardly had the �rst book on Ramanujan, ‘Collected Papers of Srinivasa Ramanujan’ released in 1927, when everyone turned their eyes to this accountant turned scholar. The pioneers had dug gold.

Ramanujan was a thinker( in the words of John Littlewood, ‘Every positive integer was one of Ramanujan’s personal friends’), his works were scattered all over; on lose sheets, in his three notebooks, on boards. So much so that in 1973, about �fty years after his death, Prof. George E. Andrews found the ‘Lost Notebook of Ramanujan’ on the estate of G. N. Watson, which could have been lying there since 1923. In that year, Prof. Hardy spent a few months in editing a chapter in the Note-books of Ramanujan, on hyper-geometric series, and found it so daunting a task that he felt that he would not be able to do any other research work if he continued to look into these Notebooks. This task was then procrastinated to the Univer-sity of Madras, where the same G.N. Watson took it up, to which the misplacement and consequent discovery of the ‘lost’ notebook could be linked to. This notebook was nothing but a compilation of loose sheets �eshing some 600 theorems on, what Ramanujan called, mock ‘theta’ functions.

Till date thousands of research papers have incorporated Ramanujan’s works. Around three journals: The Hardy - Ramanu-jan Journal (since 1975); the Journal of the Ramanujan Mathematical Society (since 1985); and Ramanujan Journal (since 1997), a tribute be�tting the greatest. But one of the most interesting pieces of his works was ‘The 59 Questions’ that he had posed in the Journal of Indian Mathematical Society (JIMS). Each question was di�erent in its own way. But each had that stark similarity as explained in Karc’s statement. They were so subtle, so �gurative that those who could not solve the ques-tions at �rst, mocked at the answers when they were revealed.

Despite the fact that they were fabulous questions in themselves, Ramanujan had contributed more than he could ever imagine by giving his proofs as problems in his inimitable style. He had made a distinct and solid approach to mathematics, unlike any of his contemporaries. It was justi�able that why on a scale of 1 to 100, Hardy had rated himself on 25 and Ramanujan at 100. As said by Prof. Atle Selberg, on ‘The 59 Questions’, “Ramanujan had proven that a felicitous but unproved conjecture may be of much more consequence than the proof of many a respectable theorem.”

On ending notes, he was one of the most complex characters ever to pop up in history. At such a young age he had achieved so much. If ever you feel dismal or feel lowly of the world, take a stroll in Ramanujan’s garden. Be wondered at the wonderment around, laugh at the inexplicable subtlety, think whatever you can but do not forget that what is this life if not �lled with such intricacies? Intricacies portrayed best by you, by us... and people like Ramanujan.

8 Infinity • Spring 2012

year and was now fully responsible for him. They �nanced him to study math on the same table on which Newton and Einstein once sat; in Trinity College, Cambridge; until his return to India in 1919. Sadly so, he had only a year left to him when he returned. The Royal Society had already awarded him the Fellowship of the Royal Society in 1918, making him the �rst Indian mathematician to be awarded the same, as if they knew about his untimely demise.

A pen, few sheets of paper, an envelope, an address, a postage stamp and this exquisite brain; was all that was required to bring him to the podium where he stands today.

In his short life, Mr. Ramanujan was quite an unpub-lished mathematician. All people knew of him was either through Mr. Hardy or the Madras University. It was after his death that his works were really recog-nized, not directly by the people but by his contem-poraries. Hardly had the �rst book on Ramanujan, ‘Collected Papers of Srinivasa Ramanujan’ released in 1927, when everyone turned their eyes to this accountant turned scholar. The pioneers had dug gold.Ramanujan was a thinker( in the words of John Little-wood, ‘Every positive integer was one of Ramanujan’s personal friends’), his works were scattered all over; on lose sheets, in his three notebooks and on boards. So much so that in 1973, about �fty years after his death, Prof. George E. Andrews found the ‘Lost Note-book of Ramanujan’ on the estate of G. N. Watson, which could have been lying there since 1923. In that year, Prof. Hardy spent a few months in editing a chapter in the Notebooks of Ramanujan, on hyper-geometric series, and found it so daunting a task that he felt that he would not be able to do any other research work if he continued to look into these Note-books.

It was justi�able that on a scale of 1 to 100, Hardy had rated himself on 25 and Ramanujan at 100.

This task was then transferred to the University of Madras, where the same G.N. Watson took it up, to which the misplacement and consequent discovery of the ‘lost’ notebook could be linked to. This note-book was nothing but a compilation of loose sheets �eshing some 600 theorems on, what Ramanujan

called, mock ‘theta’ functions.Till date thousands of research papers have incorpo-rated Ramanujan’s works. Around three journals: The Hardy - Ramanujan Journal (since 1975); the Journal of the Ramanujan Mathematical Society (since 1985); and Ramanujan Journal (since 1997), a tribute be�t-ting the greatest. But one of the most interesting pieces of his works was ‘The 59 Questions’ that he had posed in the Journal of Indian Mathematical Society (JIMS). Each question was di�erent in its own way. But each had that stark similarity as explained in Karc’s statement. They were so subtle, so �gurative that those who could not solve the questions at �rst, mocked at the answers when they were revealed. Despite the fact that they were fabulous questions in themselves, Ramanujan had contributed more than he could ever imagine by giving his proofs as prob-lems in his inimitable style. He had made a distinct and solid approach to mathematics, unlike any of his contemporaries. It was justi�able that why on a scale of 1 to 100, Hardy had rated himself on 25 and Ramanujan at 100. As said by Prof. Atle Selberg, on ‘The 59 Questions’, “Ramanujan had proven that a felicitous but unproved conjecture may be of much more conse-quence than the proof of many a respectable theo-rem.”

Marc Karc once said, ‘An ordinary genius is a fellow that you and I would be just as good as, if we were only many times better. There is no mystery as to how his mind works. Once we understand what he has done, we feel certain that we, too, could have done it. It is di�erent with the magicians,’ and this was exactly what was so profound with Mr. Ramanujan, his style was simple yet so perfect.

On ending notes, he was one of the most complex characters ever to pop up in history. At such a young age he had achieved so much. If ever you feel dismal or feel lowly of the world, take a stroll in Ramanujan’s garden. Be wondered at the wonderment around, laugh at the inexplicable subtlety, think whatever you can but do not forget that what is this life if not �lled with such intricacies? Intricacies portrayed best by you, by us... and people like Ramanujan.

Spring 2012 • Infinity 9

National Year of Mathematics

Abhinav Kejriwal talks about how and why 2012 is a very signi�cant year for mathematics in India.

The year 2012 has a lot of tags attached to it. It has been declared as one of the cyclic Chinese dragon years, the year of changes by God, and most infamously, the year of the apocalypse. But this article is out to discuss 2012 as something much more relevant, as the year of Mathematics.Our Prime Minister rightly declared it “The National Mathematics Year.” In fact, besides India, Nigeria too is celebrating 2012 as The Mathematics Year. Now, the �rst question that should arise is, what does the year 2012 have to do with Mathematics? Well, the reason for India to celebrate this year as the Year of Math-ematics is to pay a tribute to the Legendary Math-ematician, Srinivasa Ramanujan(1887-1920). The irony, however, lies in the fact that Ramanujan’s birth-day is only a day after the world will “supposedly” end, that is 22 December. The Nigerian government, how-ever, celebrates 2012 as The Mathematics Year to promote and further the scope of mathematics in its country.The Indian PM has appropriately followed Nigeria’s decision by declaring 2012 as the National Math-ematics Year, because it is truly a need of the hour. India has contributed majorly in the past to math-ematics, but the current generations have undoubt-edly not been living up to those standards. The coun-try has not been able to contribute e�ectively over the recent years and many countries have overtaken us in the �eld. Looking at the situation, the declara-tion was a must. While mathematics �nds application in our daily lives, be it in calculating the budget of a Pizza treat or the National Budget, we have not gone much further since Ramanujan’s time. It is correct to say Mathematics is the Mother of all Sciences and hence if we Indians wish to uphold our status in the world of science, we must completely understand the worth of this declaration.Firstly, PM Manmohan Singh said, “A genius like Ramanujan would shine bright even in the most adverse of circumstances, but we should be geared to encourage and nurture good talent which may not be of the same calibre as that of Ramanujan.” The

country lacks top class mathematicians and this is a matter of serious concern. The ultimate need today is that of taking mathematics to a whole new level. Today’s youth has the power to in�uence and change. We can do the same in implementing real mathemat-ics in our lives. This year must give us more incentive to do so, and make a change, one that will be indel-ible.Secondly, there is a common misconception that mathematics does not give way to a prosperous and lucrative career range. People often say that math-ematics restricts one only to teaching. However, the tfact is that the world is in grave need of mathemati-cians. Besides, Mathematical Sectors like actuarial science, computer science, biomathematics, cryptog-raphy and �nance form the core of a massive array of career option. The Indian Brain is considered by far the most intelligent of its kind. Mathematics will open myriad ways to opportunities and this would not only enrich mathematics as we know, but societies that revolve around it as well. Unfortunately, China has overtaken India in the �eld of science. It is a must for us to do something about it, more importantly during and post the National Mathematics Year. Moreover, a major problem that needs to be solved is “Math-Phobia.” This fear of math terri�es most of the students and this is the root cause of younger Indian generations not being able to contribute to the area of mathematics. Einstein rightly said, “Everything should be made as simple as possible, but not simpler.” Mathematics will only be simple if we make it simple.So considering the fact that our country is facing a threat from the decline of mathematics, it is every Indian’s duty to understand and eliminate this threat.

News from the Mathematics Department

The team is planning to visit the The Ramanujan Insti-tute for Advanced study in Mathematics to com-memorate the 125th birth anniversary of Srinivas ramanujan.This will be in the �rst week of December.

10 Infinity • Spring 2012

PuzzledGet puzzled with Shivam Goyal’s and Anmol Jain’s set of puzzles here in this section as pub-lished originally written by Margaret C. Edmiston.

PUZZLE 1: APPLES FOR ALL

While out exploring, a group of boys came upon an apple tree whose fruits were ripe for the picking. One of the boys climbed the tree and picked enough apples for each boy to have three, with none left over. Then along came three more boys, making it impos-sible to divide the picked apples evenly.However, after picking one more apple and adding it to the total, every boy had two apples, with none left over. How many apples were divided among how many boys?

SOLUTION: 16 apples, 8 boys. Let x = the original number of boys, so that 3x= the number of apples �rst picked. The �nal number of boys was x + 3 and the �nal number of apples was 3x + 1. Thus, (3x + 1) ÷ (x + 3) = 2. Simpli�ed, this equation becomes 3x + 1 = 2x + 6. So x = 5 and 3x = 15. After three more boys appeared, there were eight boys to divide sixteen apples, so each boy got two apples.

PUZZLE 2: MEASURING TWO GALLONS OF CIDER

“I want 2 gallons of cider for me and my pals, “said Mongo to the pub owner.The pub owner replied, “I have a 3-gallon container and a 4-gallon container. Will you use one of those and guess at the amount?”“I don’t need to guess,” said Mongo. “I can measure exactly 2 gallons using the containers you have.”How can Mongo do this?

SOLUTION: Fill the 3-gallon container with cider and empty it into the 4-gallon container. Fill the 3-gallon container a second time and pour it into the 4-gallon container. When the 4-gallon container is full, 2 gallons remain in the 3-gallon container.

PUZZLE 3: HOW MUCH FOR A KNIFE, SWORD, OR ARROW?

In Pymm, thirty-six coins will buy one knife, one sword and nine arrows. Two swords can be traded for one knife and four arrows.What is the price for each item purchased separately?

SOLUTION: 2 coins for a knife, 3 coins for an arrow, 7 coins for a sword. Say x, y and z are the cost in coins of one arrow, one knife and one sword, in that order. Then (1) 36 =y + z + 9x; (2) 2z = y + 4x, or y = 2z – 4x. Substituting 2z – 4x for y in equation (1) leads to 36 = 2z – 4x + z + 9x. This simpli�es to equation (3): x = (36 – 3z) + 5.Normally, solving one equation containing two variables will not produce a unique answer. However, in this case we know that x, y and z are all positive integers, and, from equation (3), the quantity 36-3z is evenly divisible by 5(since x is positive integer).Trial and error quickly sows that the positive integral values of z that make 36 – 3z evenly divisible by 5, and x a positive integer, are z = 2 and z = 7. If z = 2, then x = 6; but if x = 6, then from equation (1) or (2) y would be -20 which is not possible. So z = 7; hence x = 3, y = 2.

PUZZLE 4

Start form the entrance of the maze. Follow the arrows and keep on adding the numbers in such a way that you should get 1000 before exiting the maze.

National Year of Mathematics

Abhinav Kejriwal talks about how and why 2012 is a very signi�cant year for mathematics in India.