This presentation is about the Indian Mathematician Bhaskara II.
Prepared for B.Ed. Sem. II students of Mathematics pedagogy, of university of Lucknow.
Srinivasa Ramanujan A great INDIAN MATHEMATICIANSchooldays_6531
We Indians are not too great but we have some GREATEST personalities like Aryabhatta -- Who gave the world ZERO
This is a small presentation on life history of Srinivasa Ramanujan.
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This presentation is about the Indian Mathematician Bhaskara II.
Prepared for B.Ed. Sem. II students of Mathematics pedagogy, of university of Lucknow.
Srinivasa Ramanujan A great INDIAN MATHEMATICIANSchooldays_6531
We Indians are not too great but we have some GREATEST personalities like Aryabhatta -- Who gave the world ZERO
This is a small presentation on life history of Srinivasa Ramanujan.
Please LIKE and SHARE.
This ppt is on Srinivasa Ramanujan FRS was an Indian mathematician who lived during the British Rule in India. Though he had almost no formal training in pure mathematics, he made substantial contributions to mathematical
Srinivasa Ramanujan Date Of Birth 22.12.1887Padma Lalitha
In last slide I have mentioned Srinivasa Ramanujan D.O.B. as
22.12.1987. I am extremely sorry for that. Please read it as 22.12.1887. Thanks to my friend Smt. Indira, who brought it to my notice.
National Mathematics Day Celebration 22 DecemberRakibulSK3
Srinivasa Ramanujan was a great Indian
mathematician . He was born on 22nd December
1887 in Erode (Tamil Nadu) during British
Government .His full name was Srinivasa Iyenger
Ramanujan . His father name was Kuppuswamy
Srinivasa Iyenger and his mother’s name was
Komalatammal. He was enrolled in the Town
higher Secondary School from 1897-1904 , Wherehe encountered formal mathematics for the first Time.
By age 11, he had exhausted the mathematical knowledge of two college students who were lodgers at his home .He was latter lent a book on advanced
trigonometry written by S.L.Loney. He completely mastered on this book by the age of 13 and discovered
Sophisticated theorems on his own . In July 1909,
Ramanujan married S.Janaki Ammal , who was then
Just 10 years old . The Cambridge mathematician
G.H. Hardy arranged for Ramanujan to visit Trinity
College in Cambridge .Ramanujan arrived in Cambridge in 1914 and He completed his graduation from Cambridge University ,London . He made a lot of his theories which are very popular in the world and That is why His theories still get used in lots of countries . He wrote many books Comprising his theories and formulas . He is famous for his contribution to number theory and infinite Series .His birthday is celebrated as National Mathematics Day in India every year .He was died on 26th April 1920 ,at the age of 32 years in Madras . He Worked for a very short period but his teaching are Still alive in many people’s mind and text books .His Contribution in the Field of mathematics has been immense and will be remembered forever.
During his short life, Ramanujan independently compiled nearly 3,900 results (mostly identities and equations).[6] Many were completely novel; his original and highly unconventional results, such as the Ramanujan prime, the Ramanujan theta function, partition formulae and mock theta functions, have opened entire new areas of work and inspired a vast amount of further research.[7] Of his thousands of results, all but a dozen or two have now been proven correct.[8] The Ramanujan Journal, a scientific journal, was established to publish work in all areas of mathematics influenced by Ramanujan,[9] and his notebooks—containing summaries of his published and unpublished results—have been analysed and studied for decades since his death as a source of new mathematical ideas. As late as 2012, researchers continued to discover that mere comments in his writings about "simple properties" and "similar outputs" for certain findings were themselves profound and subtle number theory results that remained unsuspected until nearly a century after his death.[10][11] He became one of the youngest Fellows of the Royal Society and only the second Indian member, and the first Indian to be elected a Fellow of Trinity College, Cambridge. Of his original letters, Hardy stated that a single look was enough to show they could have been written only by a mathematician of the highest Ramanujan.
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Ramanujan’s knowledge of mathematics (most of which he had worked out for himself) was startling. Although he was almost completely unaware of modern developments in mathematics, his mastery of continued fractions was unequaled by any living mathematician. He worked out the Riemann series, the elliptic integrals, hypergeometric series, the functional equations of the zeta function, and his own theory of divergent series.
Pythagoras of Samos (c. 570 – 495 BCE) was a Greek philosopher and mathematician. He is best known for proving Pythagoras’ Theorem, but made many other mathematical and scientific discoveries.
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4. 4
SREENIVASA IYENGAR RAMANUJAN
Born on 22 December 1887 into a Tamil Brahmin Iyengar
family in Erode, Madras Presidency (now Tamil Nadu),
at the residence of his maternal grandparents.
Indian Mathematician and autodidact lived during the
British Raj.
Substantial contributions to mathematical analysis, number
theory, infinite series, and continued fractions including
solutions to mathematical problems considered to be
unsolvable.
Academic advisors were G. H. Hardy and J. E. Littlewood.
5. FAMOUS HISTORY
One day a primary school teacher of 3rd
form was telling to his students, “If three fruits
are divided among three persons, each would
get one”. Thus generalized that any number
divided by itself was unity. This made a child
of that class jump and ask, “Is zero divided by
zero also unity? If no fruits are divided to
nobody, will each get one? This little boy was
none other than Ramanujan.
5
7. 1. Hardy - Ramanujan Number:
Once Hardy visited to Putney were Ramanujan was
hospitalized. He visited there in a taxi cab having number
1729. Hardy was very superstitious due to his such nature
when he entered into Ramanujan’s room, he quoted that he
had just came in a taxi cab having number 1729 which
seemed to him an unlucky number, but at that time,
Ramanujan promptly replied that this was a very interesting
number as it is the smallest number which can be expressed
as the sum of cubes of two numbers in two different ways as
gen below:
Later some theorems were established in theory of elliptic
curves which involves this fascinating number.
7
8. Infinite Series for π
Sreenivasa Ramanujan also discovered some
remarkable infinite series of π around 1910.The
series,
Computes a further eight decimal places of π with
each term in the series. Later on, a number of
efficient algorithms have been developed by number
theorists using the infinite series of π given by
Ramanujan.
8
9. 2. Goldbach’s Conjecture:
Goldbach’s Conjecture is one of the important
illustrations of Ramanujan contributions
towards the proof of the conjecture. The
statement is every even integer greater than 2
is the sum of two primes, that is 6 = 3 + 3.
Ramanujan and his associates had shown that
every large integer could be written as the sum
of at most four.
9
10. 3. Theory of Equations:
Ramanujan was shown how to solve cubic
equations in 1902 and he went on to find his
own method to solve the quadratic. He derived
the formula to solve the biquadratic equations.
The following years, he tried to provide the
formula for solving quintic but he couldn’t as
he was not aware of the fact that quintic could
not be solved by radicals.
10
11. 4. Ramanujan -Hardy Asymptotic Formula:
Ramanujan’s one of the major works was
in the partition of numbers. In a joint paper
with Hardy, Ramanujan gave an
asymptotic formulas for p(n). In fact, a
careful analysis of the generating function
for p(n) leads to the Hardy – Ramanujan
asymptotic formula given by,
11
12. • In their proof, they discovered a new
method called ‘circle method’ which made
the Hardy – Ramanujan formula that p(n)
has exponential growth. It had the
remarkable property that it appeared to
give the correct value of p(n) and this was
later proved by Rademacher using special
functions and than Ken one gave the
algebraic formula to calculate partition
function for any natural number n
12
14. In his 1919 paper, he gave proof for the
first 2 congruence using the following
identities using proch hammer symbol
Notation. After the death of Ramanujan,
in 1920, the proof of all above
congruences extracted from his
unpublished work.
14
15. • For example n = 36 is highly composite
because it has d(36) = 9 and smaller natural
numbers have less number of divisors.
•
is the prime factorization of a highly
composite number n , then the primes 2, 3, …,
p form a chain of consecutive primes where
the sequences of exponents is decreasing.
and the final exponent is 1, except for n = 4
and n = 36 15
16. 6. Highly Composite Numbers:
• A natural number n is said to be highly
composite number if it has more divisors
than any smaller natural number. If we
denote the number of divisors of n by d(n),
then we say is called a highly
composite
16
17. 7 Some other contributions:
• Apart from the contributions mentioned above, he
worked in some other areas of mathematics such as
hypo geometric series, Bernoulli numbers, Fermat’s last
theorem.
• He focused mainly on developing the relationship
between partial sums and products of hyper geometric
series.
• He independently discovered Bernoulli numbers and
using these numbers, he formulated the value of Euler’s
constant up to 15 decimal places.
• He nearly verified Fermat’s last theorem which states
that no their natural numbers x, y, z satisfy the equations
17