Srinivasaramanujan a reminiscence 3

SRINIVASA RAMANUJAN’S

LIFE AND HIS GENIUS

1887-1920

Srinivasa Ramanujan Said:

CONTENT

• Prodigy (5 – 8)Child

• Life Downs & Ups (9 – 14)Struggle

• Acquaintance with Prof.G.H.Hardy (15 – 20)Hardy

• Properties of Taxicab number ( 21 )1729

• Misperception (22,23)View

• Family life (24 – 29)Personal

• To him (30 – 32)God

• Quotes about Ramanujan (33 – 41)Quotes

• Ramanujan Eponyms (42, 43)Eponyms

• Lessons learnt / unlearnt (44 – 48)Lessons

• Ramanujan and Magic Square (49 – 76)Magic Squares

• References / Further Readings (77 – 79) References12/22/2014

VEERARAGAVAN C S [email protected]

SRINIVASA RAMANUJAN

Born - 22 December 1887

Kumbakonam, Madras Presidency British India

Died - 26 April 1920

Chetput, Madras, British India

College - Government Arts College

Pachaiyappa’s College

Cambridge University

Academic Advisors - G.H.Hardy

J.E.Littlewood12/22/2014VEERARAGAVAN C S [email protected]

4

CHILD PRODIGY

Learned college level mathematics by age 11

Bernoulli numbers by age 13 (rediscovering Euler’s identity)

While in school, he was gifted George SchoobridgeCarr’s Synopsis of Pure and Applied Mathematics.

This book listed 4865 formulae in algebra, trigonometry, analytical geometry and calculus without proof.

Ramanujan not only proved himself each but derived many new results and recorded them.

12/22/2014VEERARAGAVAN C S [email protected]

JACOB

1655-1705

JOHANN

1667-1748

NICOLAUS I

1687-1759

NICOLAUS II

1695-1726

DANIEL

1700-1782

JOHANN Ii

1710-1790

JOHANN III

1744-1807

JACOB II

1759-1789

A THOUGHT OF A 7 YEAR OLD

Teacher: nn

= 1, for every integer n.

Ramanujan: “Is zero divided by zero is also one?”

Teacher : ??????

Ramanujan’s Explanation !

“Zero divided by zero may be anything.

The zero of the numerator may be several times the zero of the denominator and vice versa”.

(Thinking of limits and limiting process)


6

Which number is

greater than infinity?

EARLY LIFE

Born in Erode to K. Srinivasa Iyengar and Komalathammal

Lived in Sarangapani Street in Kumbakonam

Went school first on 1.10.1892.

Had to switch primary school 3 times due to circumstances.

Completed Math exam in half the allotted time.

Stood District First at Kumbakonam High School (1898)

Carr’s synopsis of Elementary Results in Pure and Applied Mathematics. Book acknowledged in awakening the genius of Ramanujan.

Left college without a degree and pursued research in Mathematics.


ADULTHOOD IN INDIA

After High school, he passed a competitive exam in English & Maths and secured the Scholarship.

Due to his preoccupation with Maths, he could not pass in English & Sanskrit and not promoted to Senior F.A. Class and lost Scholarship

Married to a 9 year old bride Janaki Ammal on 14 July 1909

Went door to door for job from 18 to age 24.

Tutored college students



ATTENTION FROM MATHEMATICIANS

Met V. Ramaswamy Aiyer, founder of Indian Mathematical Society

“I was struck by the extraordinary mathematical results contained in it [the notebooks]. I had no mind to smother his genius by an appointment in the lowest rungs of the revenue department”

Introduced to R. Ramachandra Rao, secretary of the Indian Mathematical Society

Impressed by Ramanujan but doubted his integrity.

Continued Mathematical Research with Rao’s financial help



22-12-2014

VEERARAGAVAN C S, APTITUDE

TRAINER, [email protected] 12

N

1 1 1

2 1 2 1

3 1 3 3 1

4 1 4 6 4 1

5 1 5 10 10 5 1

Eg: P = 1000, R=10 %, and N=3 years.What is CI & Amount?

Step 1:10% of 1000 = 100.10% of 100 = 1010% of 10 = 1Since n = 3, three times calculation.Step 2:Amount after 3 years = 1 * 1000 + 3 * 100 + 3 * 10 + 1 * 1 = Rs.1331Step 3:CI after 3 years = 3*100 + 3*10 + 1*1 = Rs.331 (leaving out first term in step 2).

1 2 1 ஒரு பேருந்தி

யிருமலர் தவிசில்

ஒருமுறைஅயறை யனீ்ைறை

1 2 3 2 1 ஒரு முறை

இரு சுடர் மீதிைிலியங்கா

மும்மதில்இலங்றக

யிரு கால்வறைய

ஒருசிறல

1 2 3 4 3 2 1 ஒன்ைிய ஈரெயிற்ைழல்வாய் வாைியில் அட்டறை

மூவடி நாைிலம்பவண்டி

முப்புரிரலாடு மானுரி யிலங்கும் மார்விைில்

FIRST CONTRIBUTION

Published his work in Journal of Indian Mathematical Society at the age of 23, first full paper (15 pages) on “Some properties of Bernoulli Numbers”.

First problem which he posed

He then formulated an equation to solve the infinitely nested radicals problem.

Wrote his 1st formal paper for the journal on the properties of Bernoulli Numbers


RAMANUJAN PAPERS/ram01.pdf

1 + 2 1 + 3 1 + ⋯ . RAMANUJAN HIMSELF

SUPPLIED THE SOLUTION TO THIS PROBLEM

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 36

= 1 + 2 1 + 3 1 + ⋯ .


WORK

In early 1912 he got a job in the MadrasAccountant Generals office with a salary of Rs 20per month.

Later he applied for a position under the Chief

Accountant of the Madras Port Trust

Accepted as a Class III, Grade IV accounting clerk

making 30 rupees per month

Spent spare time doing Mathematical Research


INTRODUCTION WITH

G.H.HARDY

G.H. Hardy was an academician at Cambridge University

He was a prominent English mathematician, known for his achievements in number theory and mathematical analysis.

Later on Ramanujan wrote to G.H.Hardy

Hardy recognised some of his formulae but other “seemed scarcely possible to believe”. Some of them were –

Relating to infinite series -


RECOGNITION OF HIS

GENIUS

Initially, G. H. Hardy thought that the works ofRamanujan were fraud because most of them wereimpossible to believe.

But eventually ,he was convinced and interested in histalent.


G.H.HARDY’S RECOGNITION

Hardy invited Ramanujan to Cambridge University butRamanujan refused.

Hardy then enlisted E.H.Neville to bring him toEngland.

With his parents supporting him he agreed this time.


CONTACTING ENGLISH

MATHEMATICIANS

M. J. M. Hill of University College Londonargued that though Ramanujan had taste forMathematics he lacked the proper educationalbackground and foundation

He refused to take Ramanujan as student

But gave him professional advice on his work


CALCULUS &

NUMBER

THEORY


LIFE INENGLAND

Ramanujan boarded the S.S.Nevasa on 17 March 1914 and arrived in London on 14th April

Ramanujan began working with Hardy and Littlewood

Hardy received 120 theorems from him in 1st 2 letters but there were many more results in his notebook

Ramanujan spent nearly 5 years in Cambridge

Ramanujan was awarded the B.A degree by Research in March 1916 at an age of 28 years for his work on Highly Composite Numbers.


RAM VOL 1/letter to hardy 5.jpg

LIFE IN ENGLAND

He was elected a Fellow of the Royal Society of London in February 1918 at an age of 30 years.

He was the second Indian to become FRS.( First one was in 1841).

He was elected to a Trinity College Fellowship as the FIRST INDIAN.

During his five years stay in Cambridge he published twenty one research papers containing theorems.


The first Indian

citizen to be

elected to the

Fellowship of the

Royal Society

was Ardaseer

Cursetjee.

He was part of the

same

industrialisation

processes which

inspired the GTS,

introducing both

gas lighting and

steam pumps to his

native town of

Bombay.

The Royal Society

and India | Royal

Society

https://royalsociet

y.org/exhibitions/20

07/india/

https://royalsociety.org/exhibitions/2007/india/

RAMANUJAN - HARDY NUMBER

1729

Hardy arrived in a cab numbered 1729

He commented that the number was uninteresting or dull.

Instantly Ramanujan claimed that it was the smallest natural number which can be written as sum of cubes in 2 ways

1729 = 13 + 123 = 93 + 103

1729 = 7 x 13 x 19 product of primes in A.P

1729 divisible by its sum of digits.

1729 = 19 x 91

1729 is a sandwich number or HARSHAD number.12/22/2014VEERARAGAVAN C S [email protected]

TAXICAB NUMBERS



MISPERCEPTIONS

Ramanujan recorded the bulk of his results in four notebooks of loose leaf paper (About 4000 theorems)

These results written up without any derivations.

Since paper was very expensive, He would do most of his work (derivations) on SLATE and transfer just the results to paper.

Hence the perception that he was unable to prove his results and simply thought up the final result directly is NOT CORRECT


MISPERCEPTION

Professor Bruce C.Berndt of University ofIllinois, who worked on Ramanujan notebooks, stated that “Over the last 40 years,nearly all of Ramanujan’s theorems havebeen proven right”.

Also Mathematicians agreed unanimously onthe point that it was not possible for someoneto imagine those results without solving /proving.


ILLNESS & RETURN TO INDIA

Ramanujan's health worsened in England

Diagnosed with Tuberculosis and Vitamin deficiency

Returned to Kumbakonam in 1919 and died soon thereafter at the age of 32

In 1994 Dr. D.A.B. Young analysed his records and concluded he had hepatic amoebiasis


RAMANUJAN’S NOTEBOOKS

Recorded his work in 4 notebooks of loose leaf paper

Results were written without derivation

Mathematician Bruce C. Berndt says that Ramanujan was able to make the proofs but chose not to.

Might have worked on slate

Or may be influenced by G.S Carr’s book which stated results without proofs

Mathematicians such as Hardy, G.N. Watson, B.M. Wilson and Bruce Berndt created papers from his work


OTHER MATHEMATICIANS’ VIEWS

OF RAMANUJAN

J.H. Hardy was highly impressed by Ramanujan

Hardy said that the solutions were "arrived at by a process of mingled argument, intuition, and induction, of which he was entirely unable to give any coherent account”

On the basis of pure talent

Hardy rated himself a score of 25 out of 100,

J.E. Littlewood 30, David Hilbert 80 and

Ramanujan 100 !

Physicist Jayant Narlikar appreciated Ramanujan’s discoveries


D:/wiki/David_Hilbert

RECOGNITION

Tamil Nadu celebrates 22 December as ‘State IT Day’

Stamp released by the Govt. in 1962

22nd December celebrated as Ramanujan Day in Govt Arts College, Kumbakonam

National Symposium On Mathematical Methods and Applications (NSMMA)

SASTRA Ramanujan Prize


IN POPULAR CULTURE

A play ‘First Class Man’ is centered around Ramanujan

Book by Robert Kanigel titled ‘The Man Who Knew Infinity: A Life of the Genius Ramanujan’ is his biography

In the famous film ‘Good Will Hunting’ the main character is compared to Ramanujan

‘A Disappearing Number’, a show by British Stage Production is about Ramanujan and Hardy

Character Amita Ramanujan in the show Numb3rs is named after him

Roger Spottiswoode is working on a movie on mathematical genius Srinivasa Ramanujan starring Rang De Basanti actor Siddharth. Titled The First Class Man, the film's scripting has been completed and shooting is being planned .



PERSONALITY AND SPIRITUAL LIFE

A person with a somewhat shy and quiet disposition

A dignified man with pleasant manners

Ramanujan credited his success to his family Goddess, Namagiri of Namakkal

He claimed to receive visions of scrolls of complex mathematical content unfolding before his eyes

"An equation for me has no meaning, unless it represents a thought of God.”


D:/wiki/Kuladevata

D:/wiki/Namagiri

D:/wiki/Namakkal

SPRITUALITY

For example, 2n – 1 will denote the primordial GOD.

When n is zero, the expression denotes ZERO.

He spoke of “ZERO” as the symbol of the absolute (Nirguna– Brahmam) of the extreme monistic school of philosophy)

The reality to which no qualities can be attributed, which no qualities can be a

When n is 1, it denotes UNITY, the Infinite GOD.

When n is 2, it denotes TRINITY.

When n is 3, it denotes SAPTHA RISHIS and so on.12/22/2014VEERARAGAVAN C S [email protected]

SPRITUALITY

He looked “infinity” as the totality of all possibilities which was capable of becoming manifest in reality and which was inexhaustible.

According to Ramanujan, The product of infinity and zero would supply the whole set of finite numbers.

Each act of creation, could be symbolized as a particular product of infinity and zero, and from each product would emerge a particular individual of which the appropriate symbol was a particular finite number.

As narrated by Prof.Prasantha Chandra, Statistician, contemporary and good friend of Ramanujan at Cambridge.


QUOTES

PROF. RICHARD ASKEY –UNIVERSITY OF W ISCONSIN - MADISON

“Try to imagine the quality of Ramanujan’s mind, onewhich drove him to work unceasingly while deathly ill,and one great enough to grow deeper while his bodybecame weaker.

I stand in awe of his accomplishments; understandingis beyond me.

We would admire any mathematician whose life’s workwas half of what Ramanujan found in the last year ofhis life while he was dying”.


BRUCE C BERNDTPROFESSOR, UNIVERSITY OF ILLINOIS

Hardy’s personal ratings of mathematicians:

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


ROBERT KANIGELAUTHOR OF “THE MAN WHO KNEW INFINITY:

A LIFE OF THE GENIUS RAMANUJAN”

Sheer intuitive brilliance coupled to long, hard hourson his slate made up for most of his educationallapse.

This ‘poor and solitary Hindu pitting his brainsagainst the accumulated wisdom of Europe’ asHardy called him, had rediscovered a century ofmathematics and

Made new discoveries that would captivatemathematicians for next century.


S.CHANDRASEKHAR

INDIAN ASTROPHYSICIST,

NOBEL LAUREATE 1983

“I think it is fair to say that almost all themathematicians who reached distinctionduring the three or four decades followingRamanujan were directly or indirectly inspiredby his example.

Even those who do not know aboutRamanujan’s work are bound to be fascinatedby his life.”


S.CHANDRASEKHAR

INDIAN ASTROPHYSICIST,

NOBEL LAUREATE 1983 “The fact that Ramanujan’s early years were spent in a

scientifically sterile atmosphere,

that his life in India was not without hardships that undercircumstances that appeared to most Indians as nothingshort of miraculous,

He had gone to Cambridge, supported by eminentmathematicians, and

Had returned to India with very assurance that he would beconsidered,

in time as one of the most original mathematicians of thecentury.


PROF. HARDY

“I have to form myself, as I have never really formed beforeand try to help you to form, some of the reasoned estimateof the most romantic figure in the recent history ofmathematics,

a man whose career seems full of paradoxes andcontradictions,

who defies all cannons by which we are accustomed tojudge one another and

about whom all of us will probably agree in one judgementonly,

that he was in some sense a very great mathematician.”12/22/2014VEERARAGAVAN C S [email protected]

BERTRAND ARTHUR WILLIAM RUSSELL

BRITISH PHILOSOPHER & MATHEMATICIAN,

NOBEL LAUREATE

I found Hardy and Littlewood in a stateof wild excitement because they believe,they have discovered a second Newton,

A Hindu Clerk in Madras… He wrote toHardy telling of some results he has got,which Hardy thinks quite wonderful.”


PROF. ATLE SELBERG

NORWEGIAN MATHEMATICIAN

“Ramanujan’s conjectures formulated and their latergeneralization, have come to play a more central rolein the mathematics of today, serving as a kind of focusfor the attention of quite a large group of the bestmathematicians of our time.

The estimates of Ramanujan’s nature in mathematicscertainly have been growing over the years.

There is no doubt about that.”


PROF.JAYANT NARLIKAR

INDIAN ASTROPHYSICIST

IN HIS BOOK SCIENTIC EDGE

S.Ramanujan, discovered by the Cambridgemathematician Hardy, whose great mathematicalfindings were beginning to be appreciated from 1915to 1919.

His achievements were to be fully understood muchlater, well after his untimely death in 1920.

For example, his work on the highly compositenumbers started a whole new line of investigations inthe theory of such numbers.



EPONYMS

Magic Square

Brocard –Ramanujan Diophatineequation

Dougall –Ramanujan

identity

Hardy –Ramanujan

number

Landau –Ramanujan

constant

Ramanujan’scongruences

Ramanujan –Nagell equation

Ramanujan –Peterssenconjecture

Ramanujan –Skolem’stheorem

Ramanujan –Soldner constant

Ramanujan summation

Ramanujan theta function

Ramanujan graph

Ramanujan’s tau function

Ramanujan’sternary quadratic

form

Ramanujan’sprime

Ramanujan’scostant

Ramanujan’ssum

Rogers –Ramanujan’s

identity

12/22/2014VEERARAGAVAN C S

[email protected] 49

LESSONS LEARNT &/

UNLEARNT

Despite the hardship faced by Ramanujan, herose to such a scientific standing andreputation.

No Indians has enjoyed before, should beenough for young Indians to comprehend that

If they are deserving and can work hard, theycan perhaps soar the way what Ramanujanhad.


Even today in India, Ramanujan cannot get alectureship in a school / college because he hadno degree.

Many researchers / Universities will pursuestudies / researches on his work but he willhave to struggle to get even a teaching job.

LESSONS LEARNT &/

UNLEARNT


Even after more than 90 years of the death ofRamanujan, the situation is not very differentas far the rigidity of the education system isconcerned.

Today also a ‘Ramanujan’ has to clear alltraditional subjects’ exams to get a degreeirrespective of being genius in one or moredifferent subjects.

LESSONS LEARNT &/

UNLEARNT


He was offered a chair in India only after becoming a Fellow of the Royal Society.

But it is disgraceful that India’s talent has to wait for foreign recognition to get acceptance in India or else immigrate to other places.

Many of those won international recognition including noble prizes had no other option but to migrate for opportunities & recognition.(Ex. Karmerkar)

The process of this brain drain is still continuing.

LESSONS LEARNT &/

UNLEARNT


The most important lesson that we should draw from Ramanujan’s life about the condition of

The Educational systems / provisions should be made to support specially gifted children with very strong interests in one direction, at all stages of the educational system. (like SUPER 30, INSPIRE AWARD ETC.,

LESSONS LEARNT &/

UNLEARNT


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?




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.?




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..?




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…?




22 12 18 87

88 17 9 25

10 24 89 16

19 86 23 11

Sum of corner

numbers is also 139.

Interesting?




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..?




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..?




22 12 18 87

88 17 9 25

10 24 89 16

19 86 23 11

Look at these central

squares.

Interesting…?




22 12 18 87

88 17 9 25

10 24 89 16

19 86 23 11

Can you try these

combinations?

Interesting…..?




22 12 18 87

88 17 9 25

10 24 89 16

19 86 23 11

Try these

combinations also?

Interesting.…..?




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

BE A PROUD INDIAN



“When food is the problem, how can I find money for paper? I may require four

reams of paper every month.”

Deplorable Condition of

Ramanujan

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

The last three Books of Ramanujan



Calculations of Ramanujan in his

own handwriting









Mock Theta Functions



TOUGH LIFE IN ENGLANDPure vegetarian meals was not

available.

Too busy with calculations and

very often neglected food and

spent till late night.

The cold and damp climate

disturbed his health.

He was attacked by

Tuberculosis.

He returned to India.

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 Miss a Great

Mathematician

Recognition by Govt.of

IndiaThe Prime Minister of India,

Dr. Manmohan Singh 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

REFERENCES

12/22/2014

