Indian mathematician Aryabhata made several important contributions to mathematics in the 5th century CE, including developing a place-value system, trigonometric functions, and approximations of pi. He wrote the Aryabhatiya, which covered topics such as algebra, arithmetic, plane trigonometry, and spherical trigonometry. Some of his key contributions were developing a place-value numeral system, approximating pi as 3.1416, and calculating sine tables and trigonometric formulas. He is considered one of the greatest mathematicians of ancient India.

2. 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.

3. ARYABHATTA

5. Aryabhata 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.

6. 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.

7. ARYABHATTA’S
Contribution in
math’s
Place value
system
Trigonometry
Algebra
Approximation
of pie
Numeration
Indeterminate
equation

8. 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 coefficients
However, 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.

9. In ganitapada 6, aryabhata gives the area of a
triangle as
tribhujasya phalashariram samadalakoti
bhujardhasamvargah
that 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

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

11. 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.

12. 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

14. 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.

15. 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

17. Basic Information about Ramanujan
Born: 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.

18. 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.

19. 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.

20. 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

21. 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

22. 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.

23. Prepared By:
Jay Jiwani Of Class 9

24. THANK YOU