Contribution and Life History of SA Ramanujan School Project
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Transcript of Contribution and Life History of SA Ramanujan School Project
CONTRIBUTION AND LIFE HISTORY
OF SRINIVASA AYANGAR
RAMANUJAN (1887-1920)
THE GREATEST NUMBER
CRUNCHER THAT EVER
LIVED!!
THE GREATEST EVER NUMBER CRUCHER??
EARLY LIFE
Born : 22 December 1887 in
Erode Tamil Nadu
Father, K. Srinivasa Iyengar, a
clerk in a sari shop
Mother, Komalatammal
housewife
Lived in town of Kumbakonam
Suffered but survived
smallpox in December 1889
Moved to Kanchipuram, near
Madras (now Chennai) in later
life
EARLY EDUCATION
Educated : local school.
In March 1894, moved to a Telugu
medium school Kanchipuram, then
enrolled in the Kangayan Primary School.
Passed primary examination and stood first
in the district at Town High School
Kumbakonam (1898).
Mastered advanced trigonometry written by
S. L. Loney at the age of 13 years.
EARLY SIGNS OF A GENIUS AT
7 YEARS?
Once his teacher said that when zero is divided by any number, the result is zero.
Ramanujan immediately asked his teacher, whether zero divided by zero gives zero;
This shows early signs of his genius!
EARLY SIGNS OF MATHEMATICAL
GENIUS
•Completed mathematical exams in half the allotted
time, and showed a familiarity with geometry and
infinite series
• He was shown how to solve cubic equations.
Developed his own method to solve the quadratic
equations
• In 1903 when he was 16, Ramanujan obtained from a
friend a library-loaned copy of a book by G. S. Carr.
• The book was titled A Synopsis of Elementary Results
in Pure and Applied Mathematics and was a collection
of 5000 theorems.
• Independently developed and investigated
the Bernoulli numbers and had calculated the Euler–
Mascheroni constant up to 15 decimal places.
ADULTHOOD
He was a self-taught Mathematician.
But when he took his exam, he passed in Maths, but failed in other subjects because of his disinterest.
So, he couldn’t enter the university of Madras for further studies.
He married a nine years old girl named Janaki Ammal at the age of 22 but he did not live with his wife till she reached the age of 12.
With his extraordinary talent, people around him helped to take his achievements known to other Internationally renowned mathematicians .
BECOMING A MATHEMATICIAN
Ramanujan met deputy collector V.
Ramaswamy Aiyer, who had recently
founded the Indian Mathematical Society.
Ramanujan, wishing for a job at the
revenue department where Ramaswamy
Aiyer worked, showed him his mathematics
notebooks.
SUPPORT FROM FRIENDS
Ramanujan's friend, C. V. Rajagopalachari,
persisted with Ramachandra Rao for
discussions and support .
Ramanujan discussed elliptic integrals, hyper-
geometric series, and his theory of divergent
series
Rao asked him what he wanted, Ramanujan
replied that he needed some work and
financial support
Rao consented and sent him to Madras.
EARLY HURDLES
In the spring of 1913, Narayana Iyer, Ramachandra Raoand E. W. Middlemast tried to present Ramanujan'swork to British mathematicians.
He said that although Ramanujan had "a taste for mathematics, and some ability“
Although Hill did not offer to take Ramanujan on as a student, he did give thorough and serious professional advice on his work.
With the help of friends, Ramanujan drafted letters to leading mathematicians at Cambridge University.[59]
The first two professors, H. F. Baker and E. W. Hobson, returned Ramanujan's papers without comment.
One of the theorems Hardy found so incredible was found on the bottom of page three (valid for 0 < a < b + 1/2):
ACHIEVEMENTS
In mathematics, there is a distinction between having an insight and having a proof.
Ramanujan's talent suggested a plethora of formulae that could then be investigated in depth later.
Ramanujan's discoveries are unusually rich and that there is often more to them than initially meets the eye.
Examples of the most interesting of these formulae include the intriguing infininteseries for π, which he calculated.
NUMBER’S BEST FRIEND!!
SRINIVASA RAMANUJAN
AND HIS MAGIC
SQUARE
RAMANUJAN’S MAGIC SQUARE
22 12 18 87
88 17 9 25
10 24 89 16
19 86 23 11
This square looks like any
other normal magic
square. But this is formed
by great mathematician of
our country – Srinivasa
Ramanujan.
What is so great in it?
RAMANUJAN’S MAGIC SQUARE
22 12 18 87
88 17 9 25
10 24 89 16
19 86 23 11
Sum of numbers of
any row is 139.
What is so great in
it.?
RAMANUJAN’S MAGIC SQUARE
22 12 18 87
88 17 9 25
10 24 89 16
19 86 23 11
Sum of numbers of
any column is also
139.
Oh, this will be there in any
magic square.
What is so great in it..?
RAMANUJAN’S MAGIC SQUARE
22 12 18 87
88 17 9 25
10 24 89 16
19 86 23 11
Sum of numbers of
any diagonal is also
139.
Oh, this also will be there
in any magic square.
What is so great in
it…?
RAMANUJAN’S MAGIC SQUARE
22 12 18 87
88 17 9 25
10 24 89 16
19 86 23 11
Sum of corner
numbers is also
139.
Interesting?
RAMANUJAN’S MAGIC SQUARE
22 12 18 87
88 17 9 25
10 24 89 16
19 86 23 11
Look at these
possibilities. Sum
of identical
coloured boxes is
also 139.
Interesting..?
RAMANUJAN’S MAGIC SQUARE
22 12 18 87
88 17 9 25
10 24 89 16
19 86 23 11
Look at these
possibilities. Sum
of identical
coloured boxes is
also 139.
Interesting..?
RAMANUJAN’S MAGIC SQUARE
22 12 18 87
88 17 9 25
10 24 89 16
19 86 23 11
Look at these
central squares.
Interesting…?
RAMANUJAN’S MAGIC SQUARE
22 12 18 87
88 17 9 25
10 24 89 16
19 86 23 11
Can you try these
combinations?
Interesting…..?
RAMANUJAN’S MAGIC SQUARE
22 12 18 87
88 17 9 25
10 24 89 16
19 86 23 11
Try these combinations
also?
Interesting.…..?
RAMANUJAN’S MAGIC SQUARE
22 12 18 87
88 17 9 25
10 24 89 16
19 86 23 11
It is 22nd Dec 1887.
Yes. It is
22.12.1887
WE SHOULD BE
PROUD TO BE AN
INDIAN
QUICK PROBLEM SOLVING ABILITY
Compared to Heegner numbers, which have class number 1 and yield similar formulae, Ramanujan'sseries for π converges extraordinarily rapidly (exponentially) and forms the basis of some of the fastest algorithms currently used to calculate π..
One of his remarkable capabilities was the rapid solution for problems. He was sharing a room with P. C. Mahalanobis who had a problem
Imagine that you are on a street with houses marked 1 through n. There is a house in between (x) such that the sum of the house numbers to left of it equals the sum of the house numbers to its right. If n is between 50 and 500, what are n and x?"
This is a bivariate problem with multiple solutions. Ramanujan thought about it and gave the answer with a twist: He gave a continued fraction
.
Ramanujan himself supplied the
solution to the problem
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3 9 1 8 1 2.4
1 2 16 1 2 1 15
1 2 1 3.5 1 2 1 3 25
1 2 1 3 1 24 1 2 1 3 1 4.6
1 2 1 3 1 4 1 35 ..........
1 2 1 3 1 4 1 5 1 .....
RAMANUJAN PETERSSON
CONJECTURE
Although there are numerous statements that could
bear the name Ramanujan conjecture, there is one
statement that was very influential on later work.
In particular, the connection of this conjecture with
conjectures of André Weil in algebraic geometry
opened up new areas of research. That Ramanujan
conjecture is an assertion on the size of the Tau-
function
It was finally proved in 1973, as a consequence
of Pierre Deligne's proof of the Weil conjectures. The
reduction step involved is complicated.
Deligne won a Fields Medal in 1978 for his work on
Weil conjectures.
RAMANUJAN HARDY NUMBER
The number 1729 is known as the Hardy–Ramanujan number after a famous anecdote of the British mathematician G. H. Hardy regarding a visit to the hospital to see Ramanujan. In Hardy's words
“I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen.
1729 = 13 + 123 = 93 + 103.Generalizations of this idea have created the notion of "taxicab numbers". Coincidentally, 1729 is also a Carmichael number.
Actual Taxi photo
RAMANUJAM HARDY NUMBER
THE SMALLEST NATURAL NUMBER CAN BE
REPRESENTED IN TWO DIFFERENT WAYS AS
A SUM OF TWO CUBES:
1729=13 +123
=93 +103
IT IS ALSO INCIDENTALLY THE PRODUCT OF
THREE PRIME NUMBERS
LARGEST KNOWN SIMILAR NUMBER
IS
885623890831
=75113 +77303
=87593+59783
RAMANUJAN WAS INDEED A
FRIEND OF NUMBERS.
RAMANUJAM HARDY
NUMBER
CONTRIBUTION TO THE
THEOREY OF PARTITIONS
N No. of
PARTITIONS
1 1
2 2
3 3
4 5
5 7
6 11
A partition of a natural number ‘n’ is a sequence of non-decreasing positive integers whose sum is ‘n’.
EXAMPLE:
FOR N=4,PARTITIONS ARE
4 = 4
=1+3
=2+2
=1+1+2
=1+1+1+1
P(4)=5,WHETHER P IS A PARTITION FUNCTION
The highest highly composite number listed by
Ramanujan is 6746328388800
Having 10080 factors
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BOOKS BY RAMANUJAM
Srinivasa Ramanujan, G. H. Hardy, P. V. Seshu Aiyar, B. M. Wilson, Bruce C. Berndt (2000). Collected Papers of Srinivasa Ramanujan. AMS. ISBN 0-8218-2076-1. Originally published in 1927 after Ramanujan's death. It contains the 37 papers published in professional journals by Ramanujan during his lifetime
S. Ramanujan (1957). Notebooks (2 Volumes). Bombay: Tata Institute of Fundamental Research. These books contain photocopies of the original notebooks as written by Ramanujan.
S. Ramanujan (1988). The Lost Notebook and Other Unpublished Papers. New Delhi: Narosa. ISBN 3-540-18726-X. This book contains photo copies of the pages of the "Lost Notebook".Problems posed by Ramanujan, Journal of the Indian Mathematical Society.
S. Ramanujan (2012). Notebooks (2 Volumes). Bombay: Tata Institute of Fundamental Research.This was produced from scanned and microfilmed images of the original manuscripts
THE BOOKS OF RAMANUJAN
CALCULATIONS OF RAMANUJAN IN HIS OWN
HANDWRITING
12/11/2013 Source: Confidential
35
12/11/2013 Source: Confidential
36
12/11/2013 Source: Confidential
37
12/11/2013 Source: Confidential
38
MOCK THETA FUNCTIONS
SRINIVASA RAMANUJAN ONCE SAID:
Con
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40
TOUGH LIFE IN ENGAND
Pure Vegetarian meals was not easily available
He once said of his own condition” When food
is a problem, how may I find money for paper?
I may require 4 reams of paper every month.”
Too busy with calculations and very often
neglected food and spent working till late night
The cold and damp weather affected his health
severely.
He suffered from Tuberculosis.
He returned to India after that.
RAMANUJAN SAILED TO
INDIAN ON 27 FEBRUARY
1919 AND ARRIVED ON 13
MARCH
HOWEVER HIS HEALTH WAS
VERY POOR.
HE PASSED AWAY ON 26TH
APRIL 1920 AT
KUMBAKONAM(TAMIL
NAIDU)
We will always miss this Great Mathematician
Sou
rce:
Con
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RECOGNITION BY GOVT OF INDIA
The Prime Minister of India, Dr. ManmohanSingh has declared the year 2012 as the “National
Mathematical Year” and the date December 22, being the birthday of Srinivasa Ramanujan has been declared as the
National Mathematics
day” to be celebrated every year
SUMMARY
Ramanujan had found the method to find the
value of π upto millions of decimal places
Ramanujam found new and quick ways to solve
mathematical problems which have shown the
way to other mathematicians around the world.
The path he has shown helped design
algorithms currently being developed over 100
years after he passed away.
STILL SHOWING US THE WAY IN THE FIELD OF
MATHEMATICS WITH HIS AMAZING CONTRIBUTIONS!!