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UGC Seminar-Ramanujan.pptx Maths Decoded
1. JS
University Grants Commission’s National Seminari
JSS Mahavidyapeetha
Sri Jayachamarajendra
College of Engineering
welcomes you for a
presentation
and Greets you on
the celebration of
Indian Mathematics
3. In 2011, on the 125th
anniversary of his birth, the
Indian Government declared
that 22 December will be
celebrated every year as
National Mathematics Day
Hailed as an all-time great
mathematician, like Euler, Gauss or
Jacobi, for his natural genius, has
left behind 4000 original theorems,
despite his lack of formal education
and a short life-span.
4. Father: Srinivasa Iyengar
Mother: Komalata Ammal
Born in: Hindu Brahmin Family
Wife: Janaki Ammal
Siblings: Sadagopan
Date of Birth: December 22nd,
1887
Place of Birth: Erode in Tanjore
District
Death: April 26th,1920
At Chetpet-Madras
5. Remembering Ramanujan
• He was born in a smallish town in
India on December 22, 1887 His
family was of the Brahmin
(priests, and by age 10 Ramanujan
stood out by scoring top in his
district in the standard exams. He
also was known as having an
exceptional memory, and being
able to recite digits of numbers
like pi as well as things like roots
of Sanskrit words. When he
graduated from high school at age
17 he was recognized for his
mathematical prowess, and given
a scholarship for college.
6. Mother sends a letter to
“Hindu” paper
about missing
Ramanujan
By the time Ramanujan got to college, all he wanted to do was
mathematics—and he failed his other classes, and at one point
ran away, causing his mother to send a missing-person letter to
the newspaper:
7. Ramanujan moved to
Madras, tried different
colleges, had medical
problems, and continued his
independent math research.
In 1909, when he was 21, his
mother arranged (in keeping
with customs of the time) for
him to marry a then-10-year-
old girl named Janaki, who
started living with him a
couple of years later.
8. 1903-1914, he kept a record of
the final results of his original
research work in Note Books.
he showed to Dewan Bahadur
Ramachandra Rao (Collector of
Nellore), V. Ramaswamy Iyer
(Founder of Indian Mathematical
Society), R. Narayana Iyer
(Treasurer of IMS and Manager,
Madras Port Trust)
Ramanjan writes to Prof. G.H.
Hardy of Trinity College,
Cambridge, which contained
about 100 theorems,
Founder the Indian Mathematical Society, and his
jaw dropped when he saw Ramanujan’s work.
”I was struck by the extraordinary mathematical
results contained in it. I had no mind to smother his
genius by an appointment in the lowest rungs of the
revenue department.”
V. RAMASWAMY AIYER, 1871 – 1936
Mathematician
9. When Ramanujan’s mathematical friends
didn’t succeed in getting him a scholarship,
Ramanujan started looking for jobs, and
wound up in March 1912 as an accounting
clerk—or effectively, a human calculator—
for the Port of Madras (which was then, as
now, a big shipping hub). His boss, the
Chief Accountant, happened to be
interested in academic mathematics, and
became a lifelong supporter of his.
The head of the Port of Madras was a rather
distinguished British civil engineer, and
partly through him, Ramanujan started
interacting with a network of technically
oriented British expatriates.
Gets a job in Port of Madras
10. On about January 31, 1913 a mathematician
named G. H. Hardy in Cambridge, England
received a package of papers with a cover letter
: “Dear Sir, I beg to introduce myself
to you as a clerk in the Accounts
Department of the Port Trust Office
at Madras on a salary of only £20 per
annum. I am now about 23 years of
age….” and went on to say that its
author had made “startling” progress
on a theory of divergent series in
mathematics, and had all but solved
the longstanding problem of
the distribution of prime numbers.
11. G. H. Hardy was born in 1877 to schoolteacher parents
based about 30 miles south of London. He was from the
beginning a top student, particularly in mathematics.
Cambridge undergraduate mathematics was at the time very
focused on solving ornately constructed calculus-related
problems as a kind of competitive sport—with the final event
being the Mathematical Tripos exams, which ranked everyone
from the “Senior Wrangler” (top score) to the “Wooden Spoon”
(lowest passing score). The way the British academic system
worked at that time—and basically until the 1960s—was that as soon as they
graduated, top students could be elected to “college fellowships” Hardy was at Trinity
College—the largest and most scientifically distinguished college at Cambridge
University—and when he graduated in 1900, he was duly elected to a college
fellowship that could last the rest of their lives.
Hardy’s first research paper was about doing integrals .
12. By the age of 12, he had mastered trigonometry and
developed many theorems on his own with no
assistance. Ramanujan spent nearly five years in
Cambridge collaborating with Hardy and Littlewood,
and published part of his findings there. Ramanujan
was awarded a Bachelor of Science degree by
research (this degree was later renamed PhD) in
March 1916 for his work on highly composite
numbers, the first part of which was published as a
paper in the Proceedings of the London
Mathematical Society. Throughout his life,
Ramanujan was plagued by health problems. His
health worsened in England; possibly he was also
less resilient due to the difficulty of keeping to the
strict dietary requirements of his religion in England
and wartime rationing during 1914-1918. He was
diagnosed with tuberculosis was confined to a
sanatorium.
In 1919 he returned to Kumbakonam, Madras
Presidency, and soon thereafter, in 1920, died at the
age of 32
13. Madras University, offered
him the first research
scholarship of the University
in May 1913 ; then offered
him a scholarship of 250
pounds a year for five years
with 100 pounds for passage
by ship and for initial outfit
to go to England in 1914 ;
Soon after his triumphant
return from Cambridge
granted Ramanujan 250
pounds a year as an
allowance for 5 years
commencing from April
1919 which he could never
utilize
14. “While asleep, I had an unusual
experience. There was a red
screen formed by flowing blood
formed by flowing blood, as it
were. I was observing it.
Suddenly a hand began to write
on the screen. I became all
attention. That hand wrote a
number of elliptic integrals.
They stuck to my mind. As soon
as I woke up, I committed them
to writing.”
15. Ramanujan was awarded in 1916 the
B.A. Degree by research of the
Cambridge University. He was elected
a Fellow of the Royal Society of
London in Feb. 1918 being a
``Research student in Mathematics
Distinguished as a pure mathematician
particularly for his investigations in
elliptic functions and the theory of
numbers'' and he was elected to a
Trinity College Fellowship, in Oct.
1918 (- a prize fellowship worth 250
pounds a year for six years with no
duties or condition, which he was not
destined to avail of).
16. The ``Collected Papers of Ramanujan'' was
edited by Profs. G.H.Hardy, P.V. Seshu
Aiyar and B.M. Wilson and first published
by Cambridge University Press in 1927
(later by Chelsea, 1962 ; and by Narosa,
1987), seven years after his death. His
`Lost' Notebook found in the estate of Prof.
G.N. Watson in the spring of 1976 by Prof.
George Andrews of Pennsylvania State
University, and its facsimile edition was
brought out by Narosa Publishing House in
1987, on the occasion of Ramanujan's birth
centenary.
18. Major Contributions in the field of Mathematics
Mathematical analysis, Number theory, Infinite series and Continued fractions
Landau-Ramanujan Constant
Ramanujan Conjeture
Ramanujan Prime
Ramanujan Slodner Constant
Ramanujan Theta function
Srinivasa Ramanujan had almost no formal training
in pure mathematics
During his short life, Srinivasa Ramanujan independently
compiled nearly 3,900 results (mostly identities and
equations). 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
Srinivasa Ramanujan
19. In his notebooks, Ramanujan wrote down 17 ways
to represent 1/pi as an infinite series. Series
representations have been known for centuries.
For example, the Gregory-Leibniz series,
discovered in the 17th century is pi/4 = 1 - ⅓ + ⅕ -
1/7 + … However, this series converges extremely
slowly; it takes more than 600 terms to settle
down at 3.14, let alone the rest of the number.
Ramanujan came up with something much more
elaborate that got to 1/pi faster: 1/pi =
(sqrt(8)/9801) * (1103 + 659832/24591257856 + …).
This series gets you to 3.141592 after the first term
and adds 8 correct digits per term thereafter. This
series was used in 1985 to calculate pi to more
than 17 million digits even though it hadn’t yet
been proven.
Get pi fast
20. What was the Ramanujan who arrived
in Cambridge like? He was described as
enthusiastic and eager, though
diffident. He made jokes, sometimes at
his own expense. He could talk about
politics and philosophy as well as
mathematics. He was never
particularly introspective. In official
settings he was polite and deferential
and tried to follow local customs. His
native language was Tamil, and earlier
in his life he had failed English exams,
but by the time he arrived in England,
his English was excellent.
21. He liked to hang out with other Indian
students, sometimes going to musical
events, or boating on the river.
Physically, he was described as short and
stout—with his main notable feature
being the brightness of his eyes. He
worked hard, chasing one mathematical
problem after another. He kept his living
space sparse, with only a few books and
papers. He was sensible about practical
things, for example in figuring out issues
with cooking and vegetarian ingredients.
And from what one can tell, he was
happy to be in Cambridge.
22. In 1940, Hardy gave all the letters he had from Ramanujan to
the Cambridge University Library, but the original cover letter for what
Ramanujan sent in 1913 was not among them—so now the only record we
have of that is the transcription Hardy later published. Ramanujan’s three
main notebooks sat for many years on top of a cabinet in the librarian’s
office at the University of Madras, where they suffered damage from
insects, but were never lost. His other mathematical documents passed
through several hands, and some of them wound up in the incredibly
messy office of a Cambridge mathematician—but when he died in 1965 they
were noticed and sent to a library, where they languished until they were
“rediscovered” with excitement as Ramanujan’s lost notebook in 1976.
23. When Ramanujan died, it took only days for his
various relatives to start asking for financial
support. There were large medical bills from
England, and there was talk of selling
Ramanujan’s papers to raise money.
Ramanujan’s wife was 21 when he died, but as
was the custom, she never remarried. She lived
very modestly, making her living mostly from
tailoring. In 1950 she adopted the son of a friend
of hers who had died. By the 1960s, Ramanujan
was becoming something of a general Indian
hero, and she started receiving various honors
and pensions. Over the years, quite a few
mathematicians had come to visit her—and she
had supplied them for example with the passport
photo that has become the most famous picture
of Ramanujan.
She lived a long life, dying in 1994 at the age of
95, having outlived Ramanujan by 73 years.
Died at a young
age of 32
in London on 26
April 1920.
24. To be fair, however, Hardy wrote the
book at a low point in his own life,
when he was concerned about his
health and the loss of his
mathematical faculties. And perhaps
that explains why he made a point of
explaining that “mathematics… is a
young man’s game”. (And in an
article about Ramanujan, he wrote
that “a mathematician is often
comparatively old at 30, and his death may be less of a catastrophe
than it seems.”) I don’t know if the sentiment had been expressed
before—but by the 1970s it was taken as an established fact, extending
to science as well as mathematics. Kids I knew would tell me I’d
better get on with things, because it’d be all over by age 30.
25. Ramanujan seems to have
supported himself by
doing math tutoring—but
soon became known
around Madras as a math
whiz, and began
publishing in the recently
launched Journal of the
Indian Mathematical
Society. His first paper—
published in 1911—was on
computational properties
of Bernoulli numbers (the
same Bernoulli numbers
that Ada Lovelace had
used in her 1843 paper on
the Analytical Engine).
26. For a generation yearning
for independence and
progress, Ramanujan was
an inspiration.
Subramanyam
Chandrasekhar, later to
be a Nobel laureate, got
inspired in his childhood
days when his mother
recounted Ramanujan’s
story to him.
27. The Man
Who Knew
Infinity
April 2016
A film by
Manjul Bhargava and Ken Ono
While in high school Ramanujan had started studying
mathematics on his own—and doing his own research
notably on the numerical evaluation of Euler’s
constant, and on properties of the Bernoulli numbers.
30. In a famous anecdote, Hardy took a cab to visit
Ramanujan. When he got there, he told Ramanujan
that the cab’s number, 1729, was “rather a dull one.”
Ramanujan said, “No, it is a very interesting number. It
is the smallest number expressible as a sum of two
cubes in two different ways. That is, 1729 = 1^3 +
12^3 = 9^3 + 10^3. This number is now called the
Hardy-Ramanujan number, and the smallest numbers
that can be expressed as the sum of two cubes
in n different ways have been dubbed taxicab
numbers.
Prof.G.H.Hardy Comes in a taxi to meet Mr.Ramanujam
34. The next number in the
sequence, the smallest number
that can be expressed as the
sum of two cubes in three
different ways, is 87,539,319.
Can you find a new taxi with this number?
38. 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.The
number is 139
44. RAMANUJAN’S MAGIC SQUARE
22 12 18 87
88 17 9 25
10 24 89 16
19 86 23 11
Do you know
the mystery in
this magic
square
created by
Srinivasa
Ramanujan?
46. RAMANUJAN’S MAGIC SQUARE
22 12 18 87
88 17 9 25
10 24 89 16
19 86 23 11
Yes. It is
22.12.1887
BE A PROUD INDIAN
47. . His bust was commissioned by Professors
R. Askey, S. Chandrasekhar, G.E. Andrews,
Bruce C. Berndt (`the gang of four'!) and
`more than one hundred mathematicians
and scientists who contributed money for
the bust' sculpted by Paul Granlund in 1984
49. G.H.Hardy came up with a scale of mathematical
ability that went from 0 to 100. He put himself at
25. David Hilbert, the great German
mathematician, was at 80. Ramanujan was 100.
When he died in 1920 at the age of 32, Ramanujan
left behind three notebooks and a sheaf of papers
(the “lost notebook”). These notebooks contained
thousands of results that are still inspiring
mathematical work decades later.
100/100
G.H.Hardy
50. “Ramanujan is a role model for
the possible,” says Ken Ono, a
Professor of Mathematics and
Computer Science at Emory
University and also an advisor
and associate producer of a
recent film on Ramanujan, The
man who knew infinity. He has
set up a project called ‘Spirit of
Ramanujan project’ that aims to
support research initiatives of
emerging engineers,
mathematicians and scientists
and in particular those who, like
Ramanujan, lack traditional
institutional support, and are
young in the age group of 12
and 18 years.
Prof.Ken Ono
Mathematician
51. That was the wonderful
thing about Ramanujan. He
discovered so much, and yet
he left so much more in his
garden for other people to
discover.” FREEMAN DYSON, BORN 1923
Mathematician and Physicist
52. … each of the 24 modes in the
Ramanujan function corresponds to a
physical vibration of a string. Whenever
the string executes its complex motions in
space-time by splitting and recombining,
a large number of highly sophisticated
mathematical identities must be satisfied.
These are precisely the mathematical
identities discovered by Ramanujan.”
MICHIO KAKU, BORN 1947
Theoretical Physicist
53. “For my part, it is difficult for
me to say what I owe to
Ramanujan – his originality has
been a constant source of
suggestion to me ever since I
knew him, and his death is one
of the worst blows I have ever had.”
G. H. HARDY, 1877 – 1947
54. Suppose that we rate mathematicians
on the basis of pure talent on a scale
from 0 to 100. Hardy gave himself a
score of 25, Littlewood 30, Hilbert 80
and Ramanujan 100.”
PAUL ERDŐS, 1913 – 1996
Mathematician
55. What a genius
Ramanujam was?
We should be proud
of this Great man
who belongs to our
Motherland-BHARATH
