Monday, December 31, 2012
Sunday, December 23, 2012
Thursday, December 6, 2012
Tuesday, October 23, 2012
Some basic definitions in Geometry
Definition — a precise and unambiguous description of the meaning of a mathematical term. It characterizes the meaning of a word by giving all the properties and only those properties that must be true.
Theorem — a mathematical statement that is proved using rigorous mathematical reasoning. In a mathematical paper, the term theorem is often reserved for the most important results.
Lemma — a minor result whose sole purpose is to help in proving a theorem. It is a stepping stone on the path to proving a theorem. Very occasionally lemmas can take on a life of their own (Zorn’s lemma, Urysohn’s lemma, Burnside’s lemma, Sperner’s lemma).
Corollary — a result in which the (usually short) proof relies heavily on a given theorem (we often say that “this is a corollary of Theorem A”).
Proposition — a proved and often interesting result, but generally less important than a theorem.
Conjecture — a statement that is unproved, but is believed to be true (Collatz conjecture, Goldbach conjecture, twin prime conjecture).
Claim — an assertion that is then proved. It is often used like an informal lemma.
Axiom/Postulate — a statement that is assumed to be true without proof. These are the basic building blocks from which all theorems are proved (Euclid’s five postulates, ZermeloFraenkel axioms, Peano axioms).
Identity — a mathematical expression giving the equality of two (often variable) quantities (trigonometric identities, Euler’s identity).
Paradox — a statement that can be shown, using a given set of axioms and definitions, to be both true and false. Paradoxes are often used to show the inconsistencies in a flawed theory (Russell’s paradox). The term paradox is often used informally to describe a surprising or counterintuitive result that follows from a given set of rules (BanachTarski paradox, Alabama paradox, Gabriel’s horn).
Thursday, September 20, 2012
Monday, August 20, 2012

Mathematics In India
Past Present Future






BY S.SATISH S.A(Maths),ZPHS NUNNA
మానవ
జీవితంలో గణిత శాస్త్రానికి ప్రతేక
స్ధానం ఉంది. మానవ జీవిత పరిణామ
ప్రస్ధానంలో గణిత శాస్త్ర
అనువర్తనాలతో ఎన్నో ముఖ్యమైన
సంఘటనలు మానవ జీవనగతినే మార్చివేశాయి
గణిత శాస్త్రం అభివృది భారతదేశ
స్ధాయిలో గత, వర్తమాన కాలాలో ఎలాఉంది,
భవిష్యతులో ఎలా ఉండబోతుందో ఒక సారి పరిశీలిద్దాం.
పూర్వకాలంలో
గణిత శాస్త్రాNNIన్ని సహాయక
అనువర్తిత అవసరాలకు వినియోగించేవారు. హారప్పా నాగరికత కాలంలో
ప్రజా ఉపయోగ కరమైన కట్టడాలు,
నిర్మాణ సమస్యలు పరిష్కరించడానికి వినియోగించేవారు. ఖగోళ శాస్త్రం, జ్యోతిష్య శాస్త్రం మరియు వేదకాలంలో
హామగుండాల నిర్మాణంలో
బౌధాయనుడు ఆయన శిష్యులు శుల్బ సూత్రాలను వినియోగించారు.
క్రీస్తు పూర్వం
5 లేదా 6 వ శతాబ్ధాల వరకు గణిత శాస్త్ర అధ్యయనం, జ్ఞాన సముపార్జనకు మరియు
ఇతర విజ్ఞాన శాస్త్ర శాఖల అవసరాల కోసం జరిగేది.
వేదాలలో భాగంగా ‘4’ శుల్బ సుత్రాలు చాలా ప్రాముఖ్యత వహిస్తాయి. ఈ సూత్రాలు
క్రీస్తుపుర్వం 800 నుంచి 200 సంవత్సరాలకు
చెందినవి. ఈనాలుగు సూత్రాలను వాటి రచయితల
పేర్లమీదుగా పిలుస్తారు. బౌధాయన, మానవ, ఆపస్ధంభ, కాత్యాయనుడు ఈ సూత్రాల రచయితలు .
శుల్బ సుత్రాలు ప్రస్తుతం పైదాగరస్ పేరున ఉన్న సిద్దాంతాన్ని కలిగి ఉండడం ప్ర్రాచిన భారతీయులకు గణిత పరిజ్ఞానం ఎంత
ఉందో తెలియజేస్తోంది. అకరణియ సంఖ్యల భావనని కూడా
శుల్బ సుత్రాలు పరిచయం చేశాయి. ఆధునిక
గణితంలోని శ్ర్రేణి విస్తరణకు కూడా ఈ శుల్బ సూత్రాలలో మార్గం చూపబడినది.
క్రీపూ 600 నుంచి 500 సంవత్సరాల మధ్య
జైన పండితుల కృషితో ‘అనంతం’ అనే భావన గణితంలో అభివృద్ధీ చెందింది.
సమితుల భావనలలో కార్డినల్ సంఖ్య అంటే
సమితి లోని మూలకాల సంఖ్య పూర్వ కాలంలోనే
భారతీయ గణితంలో అభివృద్ది చెందింది. ఇటివల 19 వ శతాబ్దంలో జార్జీ
కాంటర్ కాలంలో మాత్రమే యూరోపియన్ గణితానికి
కార్డినల్ సంఖ్యా భావన గురించి తెలిసింది. భారతీయ సంఖ్యా విధానం, స్ధానవిలువలు, ’శున్యం’ భావన భారతీయ గణితానికి
ప్రపంచంలో ఆధిక్యం తెచ్చి పెట్టాయి అనడంలో అతిశ యోక్తి లేదు. క్రీప్రూ 300 సంవత్సరాల క్రితమే ఈనాడు ఉపయోగిస్తున్న సంఖ్యలను మనం బ్రహ్మి
సంఖ్యలుగా చూడవచ్చు. బ్రహ్మి సంఖ్యలు, గుప్తుల కాలంలో క్రీపూ 400 సంవత్సరాల కాలంలో,
తదనంతరం క్రీపూ 600 నుంచి 1000 సంవత్సరాల మధ్య
దేవనాగరి సంఖ్యా విధానంగా మార్పు చెందింది. క్రీపూ 600 సంవత్సర కాలం నాటికీ
భారత దేశంలో స్థాన విలువలు విధానం పూర్తిగా అభివృది చెంది వాడుకలోకి వచ్చింది. ‘పది గుర్తుల ద్వారా
అంకెలను సూచించడం, వాటి వాస్తవ విలువలను, స్థాన
విలువల విధానం, అంకెల స్థాన విలువలను
తెలియచేసే విధానాన్ని మనకు అందించిన భారతదేశానికి సర్వదా రుణపడి ఉండాలని’ ప్రముఖ గణిత శాస్త్రవేత
లాప్లాస్ చెప్పారు. ఈ విధానం ద్వారా గణన ప్రక్రియను సులభంగా వేగంగా నిర్వహించడం
సాధ్యమవుతుంది. ఇంకా ఆసక్తి కరమైన విషయం ఏమిటంటే 17 వ శాతాబ్ధం వరకు
యురప్ లో ‘0’ వాడకం లేదంటే చాలా
ఆశ్చర్యంగా ఉంటుంది.
క్రీ .శ .500 నుంచి 1200 సంవత్సరాల మధ్య భారతదేశంలో సంప్రదాయక గణితం భాగా అభివృది
చెందింది. ఈ కాలంలో గణితంలో చాలా ప్రముఖమైన పండితుల పేర్లు వినవచ్చేవి.
వీరిలో మొదటి ఆర్యభట్ట, బ్రహ్మ గుప్త, మొదటి భాస్కర, మహావీర, రెండవ ఆర్యభట్ట,
భాస్కరా చార్య, 2 వ భాస్కరుడు ప్రముఖమైన గణిత పండితులు.
ax + by =c అనే రేఖీయ సమీకరణం మూలాలు కనుగొనే పద్ధతిని
ఆర్యభట్ట కనుగొన్నాడు. ఈ విధానానికి కట్టక లేదా పల్వరైజర్ పద్ధతి అని పేరు పెట్టాడు. అదే విధంగా
కి 4 దశాంశ స్ధానాల వరకు విలువ కనుగొనడం, త్రికోణమితిలో సైన్ ప్రమేయానికి విలువలు కనుగొనడం వంటి చాలా
ముఖ్యమైన ఆవిష్కరణలు చేసాడు.
ఇక ఆధునిక గణిత విషయానికి వస్తే సంఖ్యా శాస్త్రం  number theory లో అత్యంత ముఖ్యమైన ఆకర్షణియమైన పలితాలు రాబట్టిన
రామానుజన్ పాత్ర చాలా ముఖ్యమైనది. ఈయన కృషితో ఆధునిక అంకగణిత సిద్ధాంతం(మాడ్యులర్ రూపం), బీజీయ రేఖా గణితం లో ప్రధాన స్ధానం సాధించింది.
ప్రస్తుతం దైనందిన జీవితంలో గణితం చాలా ప్రముఖ పాత్ర వహిస్తోంది.
P.C మహలనోబిస్ భారత గణాంక
పరిశోధనా కేంద్రం స్ధాపించి, ప్రపంచ ప్రఖ్యాతిగాంచిన జాతీయ నమునాసేకరణ విధానాన్నీ ప్రారంభిచాడు.
C. R. రావు ధియరీ ఆప్
ఎస్టిమేషన్ ద్వారా భారత గణిత ప్రజ్ఞను
ప్రపంచానికి చాటి చెప్పాడు.
సంఖ్యా వాదంలో మరో ప్రపంచ ప్రఖ్యాతిగన్న శాస్త్రవేత, కప్రేకర్ 6174 కప్రేకర్ స్దిరాంకం ద్వారా ప్రసిద్ధి చెందాడు. హరీష్ చంద్ర ఇన్ఫినెట్ డైమైన్షనల్
గ్రూప్ రిప్రిసెంటేషన్ సిద్ధాంతం
ద్వారా ఉన్నత స్ధాయి గణితంలో విస్తృత సేవలందించాడు.
శకుంతలా దేవి వంటి మహిళా గణిత మేధావులు దేశ కీర్తి
ప్రతిష్టలను సమున్నత స్థానం లో నిలిపారు
విమానాశ్రయలు, కమ్యునికేషన్, వేర్ హౌస్ లలో
సమస్యలు సాధించడానికి నూతన అల్గరిధమ్ ను
రూపొందించి నరేంద్ర కమలాకర్ ప్రపంచ ప్రసిద్ది గాంచాడు.
భారతదేశంలో గణిత శాస్త్రం ప్రస్తుత పరిస్థితిని
తెలుసుకోవాలంటే దేశంలోని విశ్వవిద్యాలయాల్లో విద్యార్ధులు
ప్రస్తుతం ఎంచుకుంటున్న కోర్సులను
పరిశీలిస్తే ఒక అవగాహనకు రాగలం.
డిల్లి విశ్వవిద్యాలయం గణిత శాస్త్ర శాఖ ప్రధాన ఆచార్యులు B. K. దాస్ గారి అభిప్రాయం ప్రకారం ఆధునిక కాలంలో అభివృద్ది చెందుతున్న గణన, సాంకేతిక అంశాల వల్ల గణిత
శాస్త్ర ప్రాధాన్యం తిరిగి పెరుగుతోందని
తెలిపారు. గత కొద్ది సంవత్సరాలుగా గణిత
శాస్త్రాన్ని ఎంచుకుంటున్న విద్యార్ధుల సంఖ్యా భాగా పెరుగుతోందని, వ్యాపార గణితం,
భౌతిక శాస్త్ర గణితం, రేఖీయ కార్యక్రమ విధానం, గేమ్స్ థియరి వంటి గణిత శాఖలు భాగా
ప్రాచుర్యం పొందుతున్నాయి అని చెప్పారు . కమలా నెహ్రు కాలేజిలోని గణిత శాస్త్ర అధ్యాపకులు
రీటా మల్హోత్ర ప్రస్తుత యువత వృత్తి పర
మైన కోర్సులు ఎంచుకుంటున్న తరుణంలో గణితంలో అపార ఉపాధి అవకాశాలున్న గేమ్స్ థియరి, mathematical finance వంటి రంగాల్లో ఎక్కువ అభివృ ద్ది జరుగుతుందని తెలిపారు.
ఇక భవిష్యతులో
భారతదేశ గణిత శాస్త్ర భవిష్యత్తు గురించి వివరించాలన్నా ఆలోచించాలన్నా ప్రస్తుతం దేశంలోని విశ్వవిద్యాలయాల్లో జరుగుతున్న గణిత
పరిశోధనాంశాలను పరిశీలించాలి. ప్రస్తుతం దేశంలోని విశ్వవిద్యాలయాల్లో మోడలింగ్ పార్టికల్ మూవ్ మెంట్, ఏనిమల్
నావిగేషన్, కంపారిషన్ ఆప్ న్యుమెరికల్
ఇంటిగ్రేటర్స్ పర్ సిమ్యులేటింగ్ ధి సోలార్ సిస్టమ్, మాధమెటికల్ పిజియోలజి ఇన్ జనరల్ ( సెల్యులార్ పిజియోలజీ, ఆర్గాన్ మోడల్స్ )
పార్స్షి యల్ డిపరెన్షియల్ ఈక్వేషన్స్, ధియరీ ఆప్ కంప్యుటేషన్స్ , నావెల్
అప్రోచ్ టు ధి న్యూమెరికల్ సోల్యుషన్ ఆప్
ఆర్డినరి డిపరెన్షియల్ ఈక్వేషన్స్ వంటి అంశలలో పరిశోధనలు సాగుతున్నాయి.
ఈ అంశాలతో వైద్య, అంతరిక్ష, వ్యాపార, జీవ భౌతిక రంగాలలో అనుప్రయుక్తంగా గణిత
శాస్త్ర అభివృద్ది జరగనుంది. భారత
ప్రభుత్వం కూడా రామానుజన్ శత జయంతి
సందర్భంగా ఈ సంవత్సరాన్ని గణిత శాస్త్ర సంవత్సరంగా ప్రకటించింది. గణిత
శాస్త్ర అభివృద్ది కి విశేష కృషి చేస్తున్నందున గణిత శాస్త్ర
రంగంలో భారత దేశం భవిష్యతులో పూర్వ వైభవాన్నీ, అగ్ర స్ధానాన్నీ అలంకరిస్తుందని ఆశిద్దాం
Friday, August 3, 2012
RESULTS OF SEMINAR ON
IN VIJAYAWADA DIVISION ,KRISHNA DIST.
FIRST PLACE : V PRASANTH, ZPH SCHOOL NUNNA,VIJAYAWADA RURAL MANDAL.
SECOND PLACE : V.SANTHOSH KUMARI, ZPH SCHOOL ,NIDAMANURU,VIJAYAWADA (R) MANDAL
THIRD PLACE:J BHARATHI,ZPH SCHOOL (G) PATAMATALANKA,VIJAYAWADA URBAN
Tuesday, July 31, 2012
Indian scientist proposes solution to math problem

Shyam Ranganathan
 CHENNAI: A mathematics problem with a $1million prize attached to it and one which has major implications in computer applications, like security and machine intelligence, is claimed to have been solved by an Indian working in the United States.Vinay Deolalikar, a scientist working at HP Labs in California, has proposed a solution to the problem, commonly called by mathematicians as ‘Is P=NP?,' in a paper he has published online. The problem is one of the seven listed by the Clay Mathematics Institute for the Millennium Prize worth $1 million, which will be awarded to the successful solver of each problem.The problem refers to the possible equivalence of two classes of problems. ‘NP' problems refer to those that may take different amounts of time to solve, based on the size of the data. But each solution suggested can be checked easily. An example given on www.nature.com is the solution of jigsaw puzzles — it is easy to verify if a solution is correct but to solve the jigsaw puzzle itself may be very difficult.‘P' problems are those that scale in polynomial time with the size of the data. An example is the problem of alphabetically sorting a list of names. A computer can be programmed to sort the names very fast; and adding many more names will make the task difficult only to an extent and it will still be possible for the computer to handle it.Mr. Deolalikar's solution, a 100page proof published online, says ‘P' is in fact not the same as ‘NP.' This means computer security systems in their current form may be pretty robust to conventional computerbased attacks, but this also means some problems cannot be solved by simply throwing a lot of computer power at them.The proof has generated a lot of interest among mathematicians, and some have started looking at the proof to see whether it will hold. But as Richard Lipton points out, quoting mathematician Yuri Manin, on his blog rjlipton.wordpress.com: “A proof becomes a proof after the social act of accepting it as a proof,” and Mr. Deolalikar has to wait for the paper to be published in a refereed journal after mathematicians vet it.Interestingly, in March this year, another of the problems — the Poincare conjecture — was declared solved by Grigoriy Perelman, but the mathematician refused to accept the $1million award.
Saturday, July 14, 2012
Reflexive Property  A quantity is congruent (equal) to itself. a = a 
Symmetric Property  If a = b, then b = a. 
Transitive Property  If a = b and b = c, then a = c. 
Addition Postulate  If equal quantities are added to equal quantities, the sums are equal. 
Subtraction Postulate  If equal quantities are subtracted from equal quantities, the differences are equal. 
Multiplication Postulate  If equal quantities are multiplied by equal quantities, the products are equal. (also Doubles of equal quantities are equal.) 
Division Postulate  If equal quantities are divided by equal nonzero quantities, the quotients are equal. (also Halves of equal quantities are equal.) 
Substitution Postulate  A quantity may be substituted for its equal in any expression. 
Partition Postulate  The whole is equal to the sum of its parts. Also: Betweeness of Points: AB + BC = AC Angle Addition Postulate: m<ABC + m<CBD = m<ABD 
Construction  Two points determine a straight line. 
Construction  From a given point on (or not on) a line, one and only one perpendicular can be drawn to the line. 
Monday, July 9, 2012
new trends in maths
For those aspiring to pursue mathematics at the undergraduate level from the upcoming academic year starting July, the Delhi University
(DU) have a range of options to choose from. About 900 seats in total
are being offered across colleges in the DU for undergraduate programmes
in mathematics.
According to B K Dass, head of department, mathematics, DU, the subject, in the 21st century has gained immense importance. Since technology — essentially computing — has pervaded all aspects of modern society, the use of mathematics and its popularity, have increased enormously.
"Over the last few years, the number of students opting for mathematics has increased. Financial mathematics, mathematical physics, linear programming and games theory are some of the popular combination programmes amongst students," he says.
Dass believes that a large number of students are interested in game theory since it is an emerging industry with tremendous growth prospects. The subject is popular amongst students as it opens avenues for developing applications. Besides, a number of upcoming game development companies require candidates with strong mathematical skills as they need to work on algorithms to develop applications.
Knowledge of maths is applied in all fields of work, from creating software systems to the field of investigation in areas like brain mapping, in liedetector machines, fingerprinting, etc. Also, in art, the application of maths has become important for preservation of historic sculptures, paintings and works of ancient artists.
A lot of avenues have opened for mathematics students, opines Rita Malhotra, faculty, mathematics, Kamla Nehru College. "In today's world students are more careeroriented and are interested in developing applications. With more computing and tactical introduced in every field, mathematics is becoming popular amongst students. Besides, having the subject knowledge helps develop decisionmaking skills of students."
Mathematical finance is a new branch in India with a number of options for maths graduates to get into. Students can make career in game applications, banking, insurance, actuarial sciences, research labs, MNCs or go for an MCA or even management, explains Malhotra.
Courtesy: Myeducationtimes.com
According to B K Dass, head of department, mathematics, DU, the subject, in the 21st century has gained immense importance. Since technology — essentially computing — has pervaded all aspects of modern society, the use of mathematics and its popularity, have increased enormously.
"Over the last few years, the number of students opting for mathematics has increased. Financial mathematics, mathematical physics, linear programming and games theory are some of the popular combination programmes amongst students," he says.
Dass believes that a large number of students are interested in game theory since it is an emerging industry with tremendous growth prospects. The subject is popular amongst students as it opens avenues for developing applications. Besides, a number of upcoming game development companies require candidates with strong mathematical skills as they need to work on algorithms to develop applications.
Knowledge of maths is applied in all fields of work, from creating software systems to the field of investigation in areas like brain mapping, in liedetector machines, fingerprinting, etc. Also, in art, the application of maths has become important for preservation of historic sculptures, paintings and works of ancient artists.
A lot of avenues have opened for mathematics students, opines Rita Malhotra, faculty, mathematics, Kamla Nehru College. "In today's world students are more careeroriented and are interested in developing applications. With more computing and tactical introduced in every field, mathematics is becoming popular amongst students. Besides, having the subject knowledge helps develop decisionmaking skills of students."
Mathematical finance is a new branch in India with a number of options for maths graduates to get into. Students can make career in game applications, banking, insurance, actuarial sciences, research labs, MNCs or go for an MCA or even management, explains Malhotra.
Courtesy: Myeducationtimes.com
Thursday, June 21, 2012
Importance of Geometry
Geometry must be looked at as the consummate, complete and paradigmatic reality given to us inconsequential from the Divine Revelation. These are the reasons why geometry is important: It hones one’s thinking ability by using logical reasoning.
 It helps develop skills in deductive thinking which is applied in all other fields of learning.
 Artists use their knowledge of geometry in creating their master pieces.
 It is a useful groundwork for learning other branches of Mathematics.
 Students with knowledge of Geometry will have sufficient skills abstracting from the external world.
 Geometry facilitates the solution of problems from other fields since its principles are applicable to other disciplines.
 Knowledge of geometry is the best doorway towards other branches of Mathematics.
 It can be used in a wide array of scientific and technical field.
Tuesday, June 12, 2012
Thursday, April 26, 2012
Srinivasa Ramanujan was one of India's greatest mathematical geniuses.
He made contributions to the analytical theory of numbers and worked on
elliptic functions, continued fractions, and infinite series.
Ramanujan was born in his grandmother's house in Erode on December 22, 1887. When Ramanujan was a year old his mother took him to the town of Kumbakonam, near Madras. His father worked in Kumbakonam as a clerk in a cloth merchant's shop.
When he was five years old, Ramanujan went to the primary school in Kumbakonam although he would attend several different primary schools before entering the Town High School in Kumbakonam in January 1898. At the Town High School, Ramanujan did well in all his school subjects and showed himself as a talented student. In 1900 he began to work on his own on mathematics summing geometric and arithmetic series.
Ramanujan was shown how to solve cubic equations in 1902 and he went on to find his own method to solve the quartic.
It was in the Town High School that Ramanujan came across a mathematics book by G. S. Carr called Synopsis of Elementary Results in Pure Mathematics. Ramanujan used this to teach himself mathematics. The book contained theorems, formulas and short proofs. It also contained an index to papers on pure mathematics.
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 numbers, which is entirely his own independent discovery.
Ramanujan, on the strength of his good schoolwork, was given a scholarship to the Government College in Kumbakonam, which he entered in 1904. However the following year his scholarship was not renewed because Ramanujan devoted more and more of his time to mathematics and neglected his other subjects. Without money he was soon in difficulties and, without telling his parents, he ran away to the town of Vizagapatnam. He continued his mathematical work, and at this time he worked on hyper geometric series and investigated relations between integrals and series. He learned later that he had been studying elliptic functions.
In 1906 Ramanujan went to Madras where he entered Pachaiyappa's College. His wanted to pass the First Arts examination that would allow him to be admitted to the University of Madras. He attended lectures at Pachaiyappa's College but became ill after three months study. He took the First Arts examination after having left the course. He passed in mathematics but failed all his other subjects and therefore failed the examination. This meant that he could not enter the University of Madras. In the following years he worked on mathematics developing his own ideas without any help and without any real idea of the then current research topics other than that provided by Carr's book.
Continuing his mathematical work Ramanujan studied continued fractions and divergent series in 1908. At this stage he became seriously ill and underwent an operation in April 1909 after which he took him some considerable time to recover. He married on 14 July 1909 when his mother arranged for him to marry a ten year old girl Janaki Ammal. Ramanujan did not live with his wife until she was twelve years old.
Ramanujan continued to develop his mathematical ideas and began to pose problems and solve problems in the Journal of the Indian Mathematical Society. He developed 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. Even though he lacked a university education, he was becoming well known in the Madras area as a mathematical genius.
In 1911, Ramanujan approached the founder of the Indian Mathematical Society for advice on a job. After this he was appointed to his first job, a temporary post in the Accountant General's Office in Madras. He asked Ramachandra Rao, the founder member of the Indian Mathematical Society who had helped start the mathematics library, about a job in mathematics.
Ramachandra Rao told him to return to Madras and he tried, unsuccessfully, to arrange a scholarship for Ramanujan. In 1912 Ramanujan applied for the post of clerk in the accounts section of the Madras Port Trust.
Ramanujan was appointed to the post of clerk and began his duties on 1 March 1912. Ramanujan was quite lucky to have a number of people working round him with training in mathematics. In fact the Chief Accountant for the Madras Port Trust, S N Aiyar, was trained as a mathematician and published a paper On the distribution of primes in 1913 on Ramanujan's work. The professor of civil engineering at the Madras Engineering College, T. Griffith was also interested in Ramanujan's abilities and, having been educated at University College London, knew the professor of mathematics there, namely M. Hill. He wrote to Hill on 12 November 1912 sending some of Ramanujan's work and a copy of his 1911 paper on Bernoulli numbers.
Hill replied in a fairly encouraging way but showed that he had failed to understand Ramanujan's results on divergent series. The recommendation to Ramanujan that he read Bromwich's Theory of infinite series did not please Ramanujan much. Ramanujan wrote to E. W. Hobson and H. F. Baker trying to interest them in his results but neither replied. In January 1913 Ramanujan wrote to G H Hardy having seen a copy of his 1910 book Orders of infinity.
Hardy studied the long list of unproved theorems which Ramanujan enclosed with his letter.
The University of Madras gave Ramanujan a scholarship in May 1913 for two years and, in 1914, Hardy brought Ramanujan to Trinity College, Cambridge, to begin a surprising collaboration.
Ramanujan sailed from India on March 17, 1914. He arrived in London on April 14, 1914. Right from the beginning Ramanujan had problems with his diet, his religion prevented him from eating the foods available at Cambridge University. The outbreak of World War I made obtaining special items of food harder and it was not long before Ramanujan had health problems.
Right from the start Ramanujan's collaboration with Hardy led to important results. Hardy was, however, unsure how to approach the problem of Ramanujan's lack of formal education.
The war soon took many away on war duty but Hardy remained in Cambridge to work with Ramanujan. Even in his first winter in England, Ramanujan was ill and he wrote in March 1915 that he had been ill due to the winter weather and had not been able to publish anything for five months. What he did publish was the work he did in England, the decision having been made that the results he had obtained while in India, many of which he had communicated to Hardy in his letters, would not be published until the war had ended.
On March 16, 1916 Ramanujan graduated from Cambridge with a Bachelor of Science by Research (the degree was called a Ph.D. in 1920). Ramanujan's thesis was on Highly composite numbers and consisted of seven of his papers published in England.
Ramanujan fell seriously ill in 1917 and his doctors feared that he would die. He did improve a little by September but spent most of his time in various nursing homes
On 18 February 1918 Ramanujan was elected a fellow of the Cambridge Philosophical Society and then three days later, the greatest honor that he would receive, his name appeared on the list for election as a fellow of the Royal Society of London. He had been proposed by an impressive list of mathematicians. His election as a fellow of the Royal Society was confirmed on May 2, 1918, then on October 10, 1918 he was elected a Fellow of Trinity College Cambridge, the fellowship to run for six years.
The honors which were given to Ramanujan seemed to help his health improve a little and he improved his efforts at producing mathematics. By the end of November 1918 Ramanujan's health had greatly improved. .
Ramanujan sailed to India on February 27, 1919 arriving on March 13. However his health was very poor and, despite medical treatment, he died there the following year.
The letters Ramanujan wrote to Hardy in 1913 had contained many fascinating results. Ramanujan worked out the Riemann series, the elliptic integrals, hyper geometric series and functional equations of the zeta function. Despite many brilliant results, some of his theorems on prime numbers were completely wrong.
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. Perhaps his most famous work was on the number p(n) of partitions of an integer n into summands. MacMahon had produced tables of the value of p(n) for small numbers n, and Ramanujan used this numerical data to conjecture some remarkable properties some of which he proved using elliptic functions. Others were only proved after Ramanujan's death.
In a joint paper with Hardy, Ramanujan gave an asymptotic formula for p(n). It had the remarkable property that it appeared to give the correct value of p(n), and that was later proved by Rademacher.
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 was born in his grandmother's house in Erode on December 22, 1887. When Ramanujan was a year old his mother took him to the town of Kumbakonam, near Madras. His father worked in Kumbakonam as a clerk in a cloth merchant's shop.
When he was five years old, Ramanujan went to the primary school in Kumbakonam although he would attend several different primary schools before entering the Town High School in Kumbakonam in January 1898. At the Town High School, Ramanujan did well in all his school subjects and showed himself as a talented student. In 1900 he began to work on his own on mathematics summing geometric and arithmetic series.
Ramanujan was shown how to solve cubic equations in 1902 and he went on to find his own method to solve the quartic.
It was in the Town High School that Ramanujan came across a mathematics book by G. S. Carr called Synopsis of Elementary Results in Pure Mathematics. Ramanujan used this to teach himself mathematics. The book contained theorems, formulas and short proofs. It also contained an index to papers on pure mathematics.
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 numbers, which is entirely his own independent discovery.
Ramanujan, on the strength of his good schoolwork, was given a scholarship to the Government College in Kumbakonam, which he entered in 1904. However the following year his scholarship was not renewed because Ramanujan devoted more and more of his time to mathematics and neglected his other subjects. Without money he was soon in difficulties and, without telling his parents, he ran away to the town of Vizagapatnam. He continued his mathematical work, and at this time he worked on hyper geometric series and investigated relations between integrals and series. He learned later that he had been studying elliptic functions.
In 1906 Ramanujan went to Madras where he entered Pachaiyappa's College. His wanted to pass the First Arts examination that would allow him to be admitted to the University of Madras. He attended lectures at Pachaiyappa's College but became ill after three months study. He took the First Arts examination after having left the course. He passed in mathematics but failed all his other subjects and therefore failed the examination. This meant that he could not enter the University of Madras. In the following years he worked on mathematics developing his own ideas without any help and without any real idea of the then current research topics other than that provided by Carr's book.
Continuing his mathematical work Ramanujan studied continued fractions and divergent series in 1908. At this stage he became seriously ill and underwent an operation in April 1909 after which he took him some considerable time to recover. He married on 14 July 1909 when his mother arranged for him to marry a ten year old girl Janaki Ammal. Ramanujan did not live with his wife until she was twelve years old.
Ramanujan continued to develop his mathematical ideas and began to pose problems and solve problems in the Journal of the Indian Mathematical Society. He developed 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. Even though he lacked a university education, he was becoming well known in the Madras area as a mathematical genius.
In 1911, Ramanujan approached the founder of the Indian Mathematical Society for advice on a job. After this he was appointed to his first job, a temporary post in the Accountant General's Office in Madras. He asked Ramachandra Rao, the founder member of the Indian Mathematical Society who had helped start the mathematics library, about a job in mathematics.
Ramachandra Rao told him to return to Madras and he tried, unsuccessfully, to arrange a scholarship for Ramanujan. In 1912 Ramanujan applied for the post of clerk in the accounts section of the Madras Port Trust.
Ramanujan was appointed to the post of clerk and began his duties on 1 March 1912. Ramanujan was quite lucky to have a number of people working round him with training in mathematics. In fact the Chief Accountant for the Madras Port Trust, S N Aiyar, was trained as a mathematician and published a paper On the distribution of primes in 1913 on Ramanujan's work. The professor of civil engineering at the Madras Engineering College, T. Griffith was also interested in Ramanujan's abilities and, having been educated at University College London, knew the professor of mathematics there, namely M. Hill. He wrote to Hill on 12 November 1912 sending some of Ramanujan's work and a copy of his 1911 paper on Bernoulli numbers.
Hill replied in a fairly encouraging way but showed that he had failed to understand Ramanujan's results on divergent series. The recommendation to Ramanujan that he read Bromwich's Theory of infinite series did not please Ramanujan much. Ramanujan wrote to E. W. Hobson and H. F. Baker trying to interest them in his results but neither replied. In January 1913 Ramanujan wrote to G H Hardy having seen a copy of his 1910 book Orders of infinity.
Hardy studied the long list of unproved theorems which Ramanujan enclosed with his letter.
The University of Madras gave Ramanujan a scholarship in May 1913 for two years and, in 1914, Hardy brought Ramanujan to Trinity College, Cambridge, to begin a surprising collaboration.
Ramanujan sailed from India on March 17, 1914. He arrived in London on April 14, 1914. Right from the beginning Ramanujan had problems with his diet, his religion prevented him from eating the foods available at Cambridge University. The outbreak of World War I made obtaining special items of food harder and it was not long before Ramanujan had health problems.
Right from the start Ramanujan's collaboration with Hardy led to important results. Hardy was, however, unsure how to approach the problem of Ramanujan's lack of formal education.
The war soon took many away on war duty but Hardy remained in Cambridge to work with Ramanujan. Even in his first winter in England, Ramanujan was ill and he wrote in March 1915 that he had been ill due to the winter weather and had not been able to publish anything for five months. What he did publish was the work he did in England, the decision having been made that the results he had obtained while in India, many of which he had communicated to Hardy in his letters, would not be published until the war had ended.
On March 16, 1916 Ramanujan graduated from Cambridge with a Bachelor of Science by Research (the degree was called a Ph.D. in 1920). Ramanujan's thesis was on Highly composite numbers and consisted of seven of his papers published in England.
Ramanujan fell seriously ill in 1917 and his doctors feared that he would die. He did improve a little by September but spent most of his time in various nursing homes
On 18 February 1918 Ramanujan was elected a fellow of the Cambridge Philosophical Society and then three days later, the greatest honor that he would receive, his name appeared on the list for election as a fellow of the Royal Society of London. He had been proposed by an impressive list of mathematicians. His election as a fellow of the Royal Society was confirmed on May 2, 1918, then on October 10, 1918 he was elected a Fellow of Trinity College Cambridge, the fellowship to run for six years.
The honors which were given to Ramanujan seemed to help his health improve a little and he improved his efforts at producing mathematics. By the end of November 1918 Ramanujan's health had greatly improved. .
Ramanujan sailed to India on February 27, 1919 arriving on March 13. However his health was very poor and, despite medical treatment, he died there the following year.
The letters Ramanujan wrote to Hardy in 1913 had contained many fascinating results. Ramanujan worked out the Riemann series, the elliptic integrals, hyper geometric series and functional equations of the zeta function. Despite many brilliant results, some of his theorems on prime numbers were completely wrong.
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. Perhaps his most famous work was on the number p(n) of partitions of an integer n into summands. MacMahon had produced tables of the value of p(n) for small numbers n, and Ramanujan used this numerical data to conjecture some remarkable properties some of which he proved using elliptic functions. Others were only proved after Ramanujan's death.
In a joint paper with Hardy, Ramanujan gave an asymptotic formula for p(n). It had the remarkable property that it appeared to give the correct value of p(n), and that was later proved by Rademacher.
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.
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