MATHS

PRESENTATION

HOLIDAY HOMEWORK

Indian mathematicians have made a number of contributions to mathematics that have significantly influenced scientists and mathematicians in the modern era. These include place-value arithmetical notation, the ruler, the concept of zero, and most importantly, the arabic-hindu numerals predominantly used today.

MATHEMATICIANS

ARYABHATT

A

AryabhattaAryabhata was born in taregna, which is a

small town in bihar, India, about 30 km from Patna (then known as pataliputra), the capital city of bihar state. Evidences justify his birth there. In taregna aryabhata set up an astronomical observatory in the sun temple 6th century.There is no evidence that he was born outside patliputra and traveled to magadha, the centre of instruction, culture and knowledge for his studies where he even set up a coaching institute. However, early buddhist texts describe ashmakas as being further south, in dakshinapath or the deccan, while other texts describe the ashmakas as having fought alexander.

It is fairly certain that, at some point, he went to kusumapura for advanced studies and that he lived there for some time. A verse mentions that aryabhatta was the head of an institution at kusumapura, and, because the university of nalanda was in patliputra at the time and had an astronomical observatory, it is speculated that aryabhata might have been the head of the nalanda university as well. Aryabhata is also reputed to have set up an observatory at the sun temple in taregana, bihar.

EDUCATION

ARYABHATTA’S

Contribution in math’s

Place value system

Trigonometry

Algebra

Approximation of pie

Numeration

Indeterminate equation

The  place-value  system, first seen in the 3rd century  bakhshali manuscript, was clearly in place in his work. While he did not use a symbol for  zero, the French mathematician  Georges if rah  explains that knowledge of zero was implicit in Aryabhata's  place-value system  as a place holder for the powers of ten with  null  coefficientsHowever, aryabhata did not use the brahmi numerals. Continuing the sanskrit tradition from vedic times, he used letters of the alphabet to denote numbers, expressing quantities, such as the table of sine's in a mnemonic form.

Place value system and zero

In ganitapada 6, aryabhata gives the area of a triangle astribhujasya phalashariram samadalakoti bhujardhasamvargahthat translates to: for a triangle, the result of a perpendicular with the half-side is the area.Aryabhata discussed the concept of sine in his work by the name of ardha-jya. Literally, it means "half-chord". For simplicity, people started calling it jya. When arabic writers translated his works from sanskrit into arabic, they referred it as jiba

TRIGONOMETRY

In aryabhatiya aryabhata provided elegant results for the summation of series of squares and cubes:

AND

Algebra

Aryabhata worked on the approximation for pi  and may have come to the conclusion that it is irrational. In the second part of the aryabhatiyam , he writes:Caturadhikam satamastagunam vasastistatha sahasranam.Ayutadvayavis kambhasyasanno taparinahah."Add four to 100, multiply by eight, and then add 62,000. By this rule the circumference of a circle with a diameter of 20,000 can be approached." This implies that the ratio of the circumference to the diameter is ((4 + 100) × 8 + 62000)/20000 = 62832/20000 = 3.1416, which is accurate to five significant figures.

Approximation of π

The aryabhatta numeration is A system of numerals based on sanskrit phonemes. It was introduced in the early 6th century by aryabhatta, in the first chapter titled gitika padam of his aryabhatiya. It attributes A numerical value to each syllable of the form consonant vowel possible in sanskrit phonology, from ka = 1 up to hau = 10

Aryabhatta numeration

A problem of great interest to indian mathematicians since ancient times has been to find integer solutions to equations that have the form ax + by = c, a topic that has come to be known a diophantine equations. This is an example from bhaskara's commentary on aryabhatiya:that is, find N = 8x+5 = 9y+4 = 7z+1. It turns out that the smallest value for N is 85. They were discussed extensively in ancient vedic text sulba sutras, whose more ancient parts might date to 800 BCE. Aryabhatta method of solving such problems is called the kuttaka method. Kuttaka means "pulverizing" or "breaking into small pieces",

Indeterminate equations

Aryabhaṭta's sine table is a set of twenty-four of numbers given in the astronomical treatise

aryabhatiya composed by the fifth century Indian mathematician and astronomer

aryabhatta (476–550 CE), for the computation of the half-chords of certain set of arcs of a

circle. It is not a table in the modern sense of a mathematical table; that is, it is not a set of

numbers arranged into rows and columns.

Aryabhatta sine table

The second section of aryabhatya titled ganitapāda contains a stanza indicating a method for the computation of the sine table. There are several ambiguities in correctly interpreting the meaning of this verse. For example, the following is a translation of the verse given by katz wherein the words in square brackets are insatz"when the second half- chord partitioned is less than the first half-chord, which is approximately equal to the corresponding arc, by a certain amount, the remaining sine-differences are less than the previous ones

Aryabhata's computational method

Basic Information about RamanujanBorn: 22 december 1887 in erode, british ındia

Died : 26 april 1920 in chetput,british ındia because of hepatic amoebiasis(a parasitic infection of the liver)

His mother was a housewife and his father worked as a clerk in a sari shop

He could not spent a stable childhood because of his poor family and their life standarts

About his talent,g.H. Hardy, who was known a big mathematician and one of ramanujan’s academic advisors with J.E. Littlewood, said only a few giant mathematicians like euler,gauss,newton had the same talent which ramanujan had.

His Early Life By age 12,he mastered an advanced trigonometry book written by S.L. Loney by himself

After his graduation from high school,he could not get a degree from both colleges he entered at different times(government college,pachaiyappa’s college) due to his unwillingness about subjects except mathematics and he could not enter any university

He has become seriously ill from time to time and they took so much time to be recovered.

Adulthood He was married with a nine year old girl named janaki annal when he was 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 the university mathematicians by his works and growing fame in madras,where he had his second college experience in.

Ramanujan has been publishing his works with the help of people who admired his talent in journal of ındian mathematical society

He got a temporary job in madras accountant general’s office,after that he was accepted as a clerk in chief accountant of the madras port trust. He did easily what he was given and he spent his spare time with mathematical research,which his boss encouraged him about.

Since he has showed his supernatural talent by himself,again people around him tried to connect with big english matematicians about ramanujan.G.H. Hardy thought at first it could be fraud because most of ramanujan’s works were impossible to believe.But eventually,they were convinced and interested in his talent.

He was invited england to improve his

works by G.H. Hardy and J.E.

Littlewood,who were two of big

mathematicans at this time.

Hardy and ramanujan had two opposite

personalities.As hardy was an atheist

and believes mathematical proof and

analysis,ramanujan was a deeply

religious guy and he believed in his

trustworthy intuition.Hardy had hard

times on his education without giving

any damage on his self confidence and

his values.

He was elected to the london

mathematical society and he became a

fellow of the royal society.

Life in England

He had his entire life with health problems but his health has been

worse in england 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

ındia despite 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

educating him.Once littlewood said about it, “it was extremely difficult

because every time some matter, which it was thought that ramanujan

needed to know, was mentioned, Ramanujan's response was an

avalanche of original ideas which made it almost impossible for

littlewood to persist in his original intention’’

Back to India

Ramanujan has left a number of theorems and his notebooks which have still been being worked on.

Ramanujan found the mistery in the number,1729,while he was in his bed when he was sick. Hardy was asked about 1729 what he thought about it and he said it has nothing interesting.Then ramanujan stated that 1729 is the smallest number which could be represented as in two different ways as a sum of twu cubes. After that,1729 have been called “ramanujan-hardy number”.

According to the big mathematicians and specialists lived in that time,ramanujan’s talent was reminded them gauss,jacobi,euler.

In memoriam of ramanujan,books have been written and movies were made since he died.An example could be the movie named the man who knew infinity: A life of the genius ramanujan based on the book.

Prepared By:

Jay Jiwani Of Class 9

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