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

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MOCK THETA FUNCTIONS

SRINIVASA RAMANUJAN ONCE SAID:

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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:

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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!!