Indian mathematicians have made anumber of contributions to mathematicsthat have significantly influencedscientists and mathematicians in themodern era. These include place-valuearithmetical notation, the ruler, theconcept of zero, and most importantly,the arabic-hindu numerals predominantlyused today.
Aryabhata was born in taregna, which is a smalltown in bihar, India, about 30 km from Patna(then known as pataliputra), the capital city ofbihar state. Evidences justify his birth there.In taregna aryabhata set up an astronomicalobservatory in the sun temple 6th century.There is no evidence that he was born outsidepatliputra and traveled to magadha, the centreof instruction, culture and knowledge for hisstudies where he even set up a coachinginstitute. However, early buddhist textsdescribe ashmakas as being further south, indakshinapath or the deccan, while other textsdescribe the ashmakas as havingfought alexander.
It is fairly certain that, at some point, hewent to kusumapura for advanced studiesand that he lived there for some time. Averse mentions that aryabhatta was thehead of an institution at kusumapura, and,because the university of nalanda was inpatliputra at the time and had anastronomical observatory, it is speculatedthat aryabhata might have been the headof the nalanda university as well.Aryabhata is also reputed to have set upan observatory at the sun templein taregana, bihar.
The place-value system, first seen in the 3rdcentury bakhshali manuscript, was clearly inplace in his work. While he did not use asymbol for zero, the Frenchmathematician Georges if rah explains thatknowledge of zero was implicit inAryabhatas place-value system as a place holderfor the powers of ten with null coefficientsHowever, aryabhata did not use the brahmi numerals.Continuing the sanskrit tradition from vedictimes, he used letters of the alphabet to denotenumbers, expressing quantities, such as the table ofsines in a mnemonic form.
In ganitapada 6, aryabhata gives the area of atriangle astribhujasya phalashariram samadalakotibhujardhasamvargahthat translates to: for a triangle, the result of aperpendicular with the half-side is the area.Aryabhata discussed the concept of sine in his workby the name of ardha-jya. Literally, it means "half-chord". For simplicity, people started calling it jya.When arabic writers translated his works fromsanskrit into arabic, they referred it as jiba
In aryabhatiya aryabhata provided elegant resultsfor the summation of series of squares and cubes:AND
Aryabhata worked on the approximation for pi andmay have come to the conclusion that it is irrational.In the second part of the aryabhatiyam , he writes:Caturadhikam satamastagunam vasastistathasahasranam.Ayutadvayavis kambhasyasanno taparinahah."Add four to 100, multiply by eight, and then add62,000. By this rule the circumference of a circle witha diameter of 20,000 can be approached."This implies that the ratio of the circumference tothe diameter is ((4 + 100) 8 + 62000)/20000= 62832/20000 = 3.1416, which is accurate tofive significant figures.
The aryabhatta numeration is A system of numeralsbased on sanskrit phonemes. It was introduced in theearly 6th century by aryabhatta, in the first chaptertitled gitika padam of his aryabhatiya. It attributes Anumerical value to each syllable of the form consonantvowel possible in sanskrit phonology, from ka = 1 up tohau = 10
Aryabhaṭtas sine table is a set of twenty-four ofnumbers given in the astronomical treatisearyabhatiya composed by the fifth century Indianmathematician and astronomer aryabhatta (476–550 CE), for the computation of the half-chordsof certain set of arcs of a circle. It is not a tablein the modern sense of a mathematical table; thatis, it is not a set of numbers arranged into rowsand columns.
The second section of aryabhatya titled ganitapādacontains a stanza indicating a method for thecomputation of the sine table. There are severalambiguities in correctly interpreting the meaning of thisverse. For example, the following is a translation of theverse given by katz wherein the words in squarebrackets are insatz"when the second half- chord partitioned is less than thefirst half-chord, which is approximately equal to thecorresponding arc, by a certain amount, the remainingsine-differences are less than the previous ones
Basic Information about RamanujanBorn: 22 december 1887 in erode, british ındiaDied : 26 april 1920 in chetput,british ındia because ofhepatic amoebiasis(a parasitic infection of the liver)His mother was a housewife and his father worked as aclerk in a sari shopHe could not spent a stable childhood because of hispoor family and their life standartsAbout his talent,g.H. Hardy, who was known a bigmathematician and one of ramanujan’s academicadvisors with J.E. Littlewood, said only a few giantmathematicians like euler,gauss,newton had the sametalent which ramanujan had.
His Early Life By age 12,he mastered anadvanced trigonometry bookwritten by S.L. Loney by himselfAfter his graduation from highschool,he could not get a degreefrom both colleges he entered atdifferent times(governmentcollege,pachaiyappa’s college) dueto his unwillingness about subjectsexcept mathematics and he couldnot enter any universityHe has become seriously ill fromtime to time and they took somuch time to be recovered.
AdulthoodHe was married with a nine year old girl named janaki annal when hewas 22 but he did not live with his wife till she was 12.Despite the fact that he was not educated well he was known to theuniversity mathematicians by his works and growing fame inmadras,where he had his second college experience in.Ramanujan has been publishing his works with the help of people whoadmired his talent in journal of ındian mathematical societyHe got a temporary job in madras accountant general’s office,afterthat he was accepted as a clerk in chief accountant of the madrasport trust. He did easily what he was given and he spent his sparetime with mathematical research,which his boss encouraged himabout.Since he has showed his supernatural talent by himself,again peoplearound him tried to connect with big english matematicians aboutramanujan.G.H. Hardy thought at first it could be fraud becausemost of ramanujan’s works were impossible to believe.Buteventually,they were convinced and interested in his talent.
He was invited england to improve his worksby G.H. Hardy and J.E. Littlewood,who weretwo of big mathematicans at this time.Hardy and ramanujan had two oppositepersonalities.As hardy was an atheist andbelieves mathematical proof andanalysis,ramanujan was a deeply religiousguy and he believed in his trustworthyintuition.Hardy had hard times on hiseducation without giving any damage on hisself confidence and his values.He was elected to the london mathematicalsociety and he became a fellow of the royalsociety.Life in England
He had his entire life with health problems but his health has been worse inengland due to stress,lack of vegetarian food and being far away from home.Ramanujan returned ındia in 1919 and after a short while he died in ındiadespite medical treatment.About him..He was a religious man.He often said, “an equation for me has no meaning,unless it represents a thought of god.”People said that he also had an obsession about vegetarian food.Hardy and littlewood had so many troubles while they have been educatinghim.Once littlewood said about it, “it was extremely difficult because everytime some matter, which it was thought that ramanujan needed to know, wasmentioned, Ramanujans response was an avalanche of original ideas whichmade it almost impossible for littlewood to persist in his original intention’’Back to India
Ramanujan has left a number of theorems and his notebookswhich have still been being worked on.Ramanujan found the mistery in the number,1729,while he wasin his bed when he was sick. Hardy was asked about 1729 whathe thought about it and he said it has nothinginteresting.Then ramanujan stated that 1729 is the smallestnumber which could be represented as in two different waysas a sum of twu cubes. After that,1729 have been called“ramanujan-hardy number”.According to the big mathematicians and specialists lived inthat time,ramanujan’s talent was reminded themgauss,jacobi,euler.In memoriam of ramanujan,books have been written andmovies were made since he died.An example could be themovie named the man who knew infinity: A life of the geniusramanujan based on the book.