Top 10 Mathematicians in INDIA

Published May 23, 2013 | By Classteacher

No doubt that the world today is honorable indebted to the contributions made by Indian
mathematicians. One of the most important contribution made by Indian Mathematicians
were the introduction of decimal system as well as the invention of zero. Here are some the
famous Indian mathematicians.

Aryabhata

Aryabhata

Aryabhata worked on the place value system using letters to signify numbers and stating
qualities. Aryabhata discovered the position of nine planets and stated that these planets
revolve around the sun. Aryabhata also stated the correct number of days in a year that is 365.
And Many More things…

Brahmagupta

Brahmagupta

Brahmagupta (597–668 AD) was a Indian mathematician and astronomer who wrote many
important works on mathematics and astronomy. His best known work is the
Brāhmasphuṭasiddhānta (Correctly Established Doctrine of Brahma), written in 628 in
Bhinmal. Its 25 chapters contain several unprecedented mathematical results.

SrinivasaRamanujan
SrinivasaRamanujan

SrinivasaRamanujan is one of the celebrated Indian mathematicians. His important

contributions to the field include Hardy-Ramanujan-Littlewood circle method in number
theory, Roger-Ramanujan’s identities in partition of numbers, work on algebra of
inequalities, elliptic functions, continued fractions, partial sums and products of
hypergeometric series.

P.C. Mahalanobis

P.C. Mahalanobis

Prasanta Chandra Mahalanobis is the founder of Indian Statistical Institute as well as the
National Sample Surveys for which he gained international recognition.
C.R. Rao

C.R. Rao

CalyampudiRadhakrishnaRao, popularly known as C R Rao is a well known statistician,

famous for his “theory of estimation”.
D. R. Kaprekar
D. R. Kaprekar discovered several results in number theory, including a class of numbers and
a constant named after him. Without any formal mathematical education he published
extensively and was very well known in recreational mathematics cricle.

Harish Chandra

Harish Chandra

Harish-Chandra FRS was an Indian American mathematician and physicist who did
fundamental work in representation theory, especially harmonic analysis on semisimple Lie
groups
Satyendranath Bose

Satyendranath Bose
Known for his collaboration with Albert Einstein. He is best known for his work on quantum
mechanics in the early 1920s, providing the foundation for Bose–Einstein statistics and the
theory of the Bose–Einstein condensate.

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Aryabhata is also known as Aryabhata I to distinguish him from the later mathematician of
the same name who lived about 400 years later. Al-Biruni has not helped in understanding
Aryabhata's life, for he seemed to believe that there were two different mathematicians called
Aryabhata living at the same time. He therefore created a confusion of two different
Aryabhatas which was not clarified until 1926 when B Datta showed that al-Biruni's two
Aryabhatas were one and the same person.

We know the year of Aryabhata's birth since he tells us that he was twenty-three years of age
when he wrote Aryabhatiya which he finished in 499. We have given Kusumapura, thought
to be close to Pataliputra (which was refounded as Patna in Bihar in 1541), as the place of
Aryabhata's birth but this is far from certain, as is even the location of Kusumapura itself. As
Parameswaran writes in [26]:-

... no final verdict can be given regarding the locations of Asmakajanapada and
Kusumapura.
We do know that Aryabhata wrote Aryabhatiya in Kusumapura at the time when Pataliputra
was the capital of the Gupta empire and a major centre of learning, but there have been
numerous other places proposed by historians as his birthplace. Some conjecture that he was
born in south India, perhaps Kerala, Tamil Nadu or Andhra Pradesh, while others conjecture
that he was born in the north-east of India, perhaps in Bengal. In [8] it is claimed that
Aryabhata was born in the Asmaka region of the Vakataka dynasty in South India although
the author accepted that he lived most of his life in Kusumapura in the Gupta empire of the
north. However, giving Asmaka as Aryabhata's birthplace rests on a comment made by
NilakanthaSomayaji in the late 15th century. It is now thought by most historians that
Nilakantha confused Aryabhata with Bhaskara I who was a later commentator on the
Aryabhatiya.

We should note that Kusumapura became one of the two major mathematical centres of
India, the other being Ujjain. Both are in the north but Kusumapura (assuming it to be close
to Pataliputra) is on the Ganges and is the more northerly. Pataliputra, being the capital of the
Gupta empire at the time of Aryabhata, was the centre of a communications network which
allowed learning from other parts of the world to reach it easily, and also allowed the
mathematical and astronomical advances made by Aryabhata and his school to reach across
India and also eventually into the Islamic world.

As to the texts written by Aryabhata only one has survived. However Jha claims in [21] that:-

... Aryabhata was an author of at least three astronomical texts and wrote some free stanzas
as well.

The surviving text is Aryabhata's masterpiece the Aryabhatiya which is a small astronomical
treatise written in 118 verses giving a summary of Hindu mathematics up to that time. Its
mathematical section contains 33 verses giving 66 mathematical rules without proof. The
Aryabhatiya contains an introduction of 10 verses, followed by a section on mathematics
with, as we just mentioned, 33 verses, then a section of 25 verses on the reckoning of time
and planetary models, with the final section of 50 verses being on the sphere and eclipses.

There is a difficulty with this layout which is discussed in detail by van der Waerden in [35].
Van der Waerden suggests that in fact the 10 verse Introduction was written later than the
other three sections. One reason for believing that the two parts were not intended as a whole
is that the first section has a different meter to the remaining three sections. However, the
problems do not stop there. We said that the first section had ten verses and indeed Aryabhata
titles the section Set of ten giti stanzas. But it in fact contains eleven giti stanzas and two arya
stanzas. Van der Waerden suggests that three verses have been added and he identifies a
small number of verses in the remaining sections which he argues have also been added by a
member of Aryabhata's school at Kusumapura.

The mathematical part of the Aryabhatiya covers arithmetic, algebra, plane trigonometry and
spherical trigonometry. It also contains continued fractions, quadratic equations, sums of
power series and a table of sines. Let us examine some of these in a little more detail.

First we look at the system for representing numbers which Aryabhata invented and used in
the Aryabhatiya. It consists of giving numerical values to the 33 consonants of the Indian
alphabet to represent 1, 2, 3, ... , 25, 30, 40, 50, 60, 70, 80, 90, 100. The higher numbers are
denoted by these consonants followed by a vowel to obtain 100, 10000, .... In fact the system
allows numbers up to 1018to be represented with an alphabetical notation. Ifrah in [3] argues
that Aryabhata was also familiar with numeral symbols and the place-value system. He writes
in [3]:-

... it is extremely likely that Aryabhata knew the sign for zero and the numerals of the place
value system. This supposition is based on the following two facts: first, the invention of his
alphabetical counting system would have been impossible without zero or the place-value
system; secondly, he carries out calculations on square and cubic roots which are impossible
if the numbers in question are not written according to the place-value system and zero.

Next we look briefly at some algebra contained in the Aryabhatiya. This work is the first we
are aware of which examines integer solutions to equations of the form by = ax + c and by =
ax - c, where a, b, c are integers. The problem arose from studying the problem in astronomy
of determining the periods of the planets. Aryabhata uses the kuttaka method to solve
problems of this type. The word kuttaka means "to pulverise" and the method consisted of
breaking the problem down into new problems where the coefficients became smaller and
smaller with each step. The method here is essentially the use of the Euclidean algorithm to
find the highest common factor of a andb but is also related to continued fractions.

Aryabhata gave an accurate approximation for π. He wrote in the Aryabhatiya the following:-

Add four to one hundred, multiply by eight and then add sixty-two thousand. the result is
approximately the circumference of a circle of diameter twenty thousand. By this rule the
relation of the circumference to diameter is given.

This gives π = 62832/20000 = 3.1416 which is a surprisingly accurate value. In fact π =

3.14159265 correct to 8 places. If obtaining a value this accurate is surprising, it is perhaps
even more surprising that Aryabhata does not use his accurate value for π but prefers to use
√10 = 3.1622 in practice. Aryabhata does not explain how he found this accurate value but,
for example, Ahmad [5] considers this value as an approximation to half the perimeter of a
regular polygon of 256 sides inscribed in the unit circle. However, in [9] Bruins shows that
this result cannot be obtained from the doubling of the number of sides. Another interesting
paper discussing this accurate value of π by Aryabhata is [22] where Jha writes:-

Aryabhata I's value of π is a very close approximation to the modern value and the most
accurate among those of the ancients. There are reasons to believe that Aryabhata devised a
particular method for finding this value. It is shown with sufficient grounds that Aryabhata
himself used it, and several later Indian mathematicians and even the Arabs adopted it. The
conjecture that Aryabhata's value of π is of Greek origin is critically examined and is found
to be without foundation. Aryabhata discovered this value independently and also realised
that π is an irrational number. He had the Indian background, no doubt, but excelled all his
predecessors in evaluating π. Thus the credit of discovering this exact value of π may be
ascribed to the celebrated mathematician, Aryabhata I.

We now look at the trigonometry contained in Aryabhata's treatise. He gave a table of sines
calculating the approximate values at intervals of 90°/24 = 3° 45'. In order to do this he used
a formula for sin(n+1)x - sin nx in terms of sin nx and sin (n-1)x. He also introduced the
versine (versin = 1 - cosine) into trigonometry.
Other rules given by Aryabhata include that for summing the first n integers, the squares of
these integers and also their cubes. Aryabhata gives formulae for the areas of a triangle and of
a circle which are correct, but the formulae for the volumes of a sphere and of a pyramid are
claimed to be wrong by most historians. For example Ganitanand in [15] describes as
"mathematical lapses" the fact that Aryabhata gives the incorrect formula V = Ah/2 for the
volume of a pyramid with height h and triangular base of area A. He also appears to give an
incorrect expression for the volume of a sphere. However, as is often the case, nothing is as
straightforward as it appears and Elfering (see for example [13]) argues that this is not an
error but rather the result of an incorrect translation.

This relates to verses 6, 7, and 10 of the second section of the Aryabhatiya and in [13]
Elfering produces a translation which yields the correct answer for both the volume of a
pyramid and for a sphere. However, in his translation Elfering translates two technical terms
in a different way to the meaning which they usually have. Without some supporting
evidence that these technical terms have been used with these different meanings in other
places it would still appear that Aryabhata did indeed give the incorrect formulae for these
volumes.

We have looked at the mathematics contained in the Aryabhatiya but this is an astronomy
text so we should say a little regarding the astronomy which it contains. Aryabhata gives a
systematic treatment of the position of the planets in space. He gave the circumference of the
earth as 4 967 yojanas and its diameter as 1 5811/24yojanas. Since 1 yojana = 5 miles this
gives the circumference as 24 835 miles, which is an excellent approximation to the currently
accepted value of 24 902 miles. He believed that the apparent rotation of the heavens was due
to the axial rotation of the Earth. This is a quite remarkable view of the nature of the solar
system which later commentators could not bring themselves to follow and most changed the
text to save Aryabhata from what they thought were stupid errors!

Aryabhata gives the radius of the planetary orbits in terms of the radius of the Earth/Sun orbit
as essentially their periods of rotation around the Sun. He believes that the Moon and planets
shine by reflected sunlight, incredibly he believes that the orbits of the planets are ellipses.
He correctly explains the causes of eclipses of the Sun and the Moon. The Indian belief up to
that time was that eclipses were caused by a demon called Rahu. His value for the length of
the year at 365 days 6 hours 12 minutes 30 seconds is an overestimate since the true value is
less than 365 days 6 hours.

Bhaskara I who wrote a commentary on the Aryabhatiya about 100 years later wrote of
Aryabhata:-

Aryabhata is the master who, after reaching the furthest shores and plumbing the inmost
depths of the sea of ultimate knowledge of mathematics, kinematics and spherics, handed
over the three sciences to the learned world.

Article by:J J O'Connor and E F Robertson

SrinivasaRamanujan was one of India's greatest mathematical geniuses. He made
substantial contributions to the analytical theory of numbers and worked on elliptic functions,
continued fractions, and infinite series.

Ramanujan was born in his grandmother's house in Erode, a small village about 400 km
southwest of Madras. When Ramanujan was a year old his mother took him to the town of
Kumbakonam, about 160 km nearer Madras. His father worked in Kumbakonam as a clerk in
a cloth merchant's shop. In December 1889 he contracted smallpox.

When he was nearly five years old, Ramanujan entered the primary school in Kumbakonam
although he would attend several different primary schools before entering the Town High
School in Kumbakonam in January 1898. At the Town High School, Ramanujan was to do
well in all his school subjects and showed himself an able all round scholar. In 1900 he began
to work on his own on mathematics summing geometric and arithmetic series.

Ramanujan was shown how to solve cubic equations in 1902 and he went on to find his own
method to solve the quartic. The following year, not knowing that the quintic could not be
solved by radicals, he tried (and of course failed) to solve the quintic.

It was in the Town High School that Ramanujan came across a mathematics book by G S
Carr called Synopsis of elementary results in pure mathematics. This book, with its very
concise style, allowed Ramanujan to teach himself mathematics, but the style of the book was
to have a rather unfortunate effect on the way Ramanujan was later to write down
mathematics since it provided the only model that he had of written mathematical arguments.
The book contained theorems, formulae and short proofs. It also contained an index to papers
on pure mathematics which had been published in the European Journals of Learned
Societies during the first half of the 19th century. The book, published in 1856, was of course
well out of date by the time Ramanujan used it.

By 1904 Ramanujan had begun to undertake deep research. He investigated the series ∑(1/n)
and calculated Euler's constant to 15 decimal places. He began to study the Bernoulli
numbers, although this was entirely his own independent discovery.

Ramanujan, on the strength of his good school work, was given a scholarship to the
Government College in Kumbakonam which he entered in 1904. However the following year
his scholarship was not renewed because Ramanujan devoted more and more of his time to
mathematics and neglected his other subjects. Without money he was soon in difficulties and,
without telling his parents, he ran away to the town of Vizagapatnam about 650 km north of
Madras. He continued his mathematical work, however, and at this time he worked on
hypergeometric series and investigated relations between integrals and series. He was to
discover later that he had been studying elliptic functions.

In 1906 Ramanujan went to Madras where he entered Pachaiyappa's College. His aim was to
pass the First Arts examination which would allow him to be admitted to the University of
Madras. He attended lectures at Pachaiyappa's College but became ill after three months
study. He took the First Arts examination after having left the course. He passed in
mathematics but failed all his other subjects and therefore failed the examination. This meant
that he could not enter the University of Madras. In the following years he worked on
mathematics developing his own ideas without any help and without any real idea of the then
current research topics other than that provided by Carr's book.

Continuing his mathematical work Ramanujan studied continued fractions and divergent
series in 1908. At this stage he became seriously ill again and underwent an operation in
April 1909 after which he took him some considerable time to recover. He married on 14 July
1909 when his mother arranged for him to marry a ten year old girl S JanakiAmmal.
Ramanujan did not live with his wife, however, until she was twelve years old.

Ramanujan continued to develop his mathematical ideas and began to pose problems and
solve problems in the Journal of the Indian Mathematical Society.He devoloped relations
between elliptic modular equations in 1910. After publication of a brilliant research paper on
Bernoulli numbers in 1911 in the Journal of the Indian Mathematical Society he gained
recognition for his work. Despite his lack of a university education, he was becoming well
known in the Madras area as a mathematical genius.

In 1911 Ramanujan approached the founder of the Indian Mathematical Society for advice on
a job. After this he was appointed to his first job, a temporary post in the Accountant
General's Office in Madras. It was then suggested that he approach RamachandraRao who
was a Collector at Nellore. RamachandraRao was a founder member of the Indian
Mathematical Society who had helped start the mathematics library. He writes in [30]:-

A short uncouth figure, stout, unshaven, not over clean, with one conspicuous feature-shining
eyes- walked in with a frayed notebook under his arm. He was miserably poor. ... He opened
his book and began to explain some of his discoveries. I saw quite at once that there was
something out of the way; but my knowledge did not permit me to judge whether he talked
sense or nonsense. ... I asked him what he wanted. He said he wanted a pittance to live on so
that he might pursue his researches.

RamachandraRao told him to return to Madras and he tried, unsuccessfully, to arrange a

scholarship for Ramanujan. In 1912 Ramanujan applied for the post of clerk in the accounts
section of the Madras Port Trust. In his letter of application he wrote [3]:-

I have passed the Matriculation Examination and studied up to the First Arts but was
prevented from pursuing my studies further owing to several untoward circumstances. I have,
however, been devoting all my time to Mathematics and developing the subject.

Despite the fact that he had no university education, Ramanujan was clearly well known to
the university mathematicians in Madras for, with his letter of application, Ramanujan
included a reference from E W Middlemast who was the Professor of Mathematics at The
Presidency College in Madras. Middlemast, a graduate of St John's College, Cambridge,
wrote [3]:-

I can strongly recommend the applicant. He is a young man of quite exceptional capacity in
mathematics and especially in work relating to numbers. He has a natural aptitude for
computation and is very quick at figure work.

On the strength of the recommendation Ramanujan was appointed to the post of clerk and
began his duties on 1 March 1912. Ramanujan was quite lucky to have a number of people
working round him with a training in mathematics. In fact the Chief Accountant for the
Madras Port Trust, S N Aiyar, was trained as a mathematician and published a paper On the
distribution of primes in 1913 on Ramanujan's work. The professor of civil engineering at the
Madras Engineering College C L T Griffith was also interested in Ramanujan's abilities and,
having been educated at University College London, knew the professor of mathematics
there, namely M J M Hill. He wrote to Hill on 12 November 1912 sending some of
Ramanujan's work and a copy of his 1911 paper on Bernoulli numbers.

Hill replied in a fairly encouraging way but showed that he had failed to understand
Ramanujan's results on divergent series. The recommendation to Ramanujan that he read
Bromwich's Theory of infinite series did not please Ramanujan much. Ramanujan wrote to E
W Hobson and H F Baker trying to interest them in his results but neither replied. In January
1913 Ramanujan wrote to G H Hardy having seen a copy of his 1910 book Orders of infinity.
In Ramanujan's letter to Hardy he introduced himself and his work [10]:-

I have had no university education but I have undergone the ordinary school course. After
leaving school I have been employing the spare time at my disposal to work at mathematics. I
have not trodden through the conventional regular course which is followed in a university
course, but I am striking out a new path for myself. I have made a special investigation of
divergent series in general and the results I get are termed by the local mathematicians as
'startling'.

Hardy, together with Littlewood, studied the long list of unproved theorems which
Ramanujan enclosed with his letter. On 8 February he replied to Ramanujan [3], the letter
beginning:-

I was exceedingly interested by your letter and by the theorems which you state. You will
however understand that, before I can judge properly of the value of what you have done, it is
essential that I should see proofs of some of your assertions. Your results seem to me to fall
into roughly three classes:
(1) there are a number of results that are already known, or easily deducible from known
theorems;
(2) there are results which, so far as I know, are new and interesting, but interesting rather
from their curiosity and apparent difficulty than their importance;
(3) there are results which appear to be new and important...

Ramanujan was delighted with Hardy's reply and when he wrote again he said [8]:-

I have found a friend in you who views my labours sympathetically. ... I am already a half
starving man. To preserve my brains I want food and this is my first consideration. Any
sympathetic letter from you will be helpful to me here to get a scholarship either from the
university of from the government.

Indeed the University of Madras did give Ramanujan a scholarship in May 1913 for two
years and, in 1914, Hardy brought Ramanujan to Trinity College, Cambridge, to begin an
extraordinary collaboration. Setting this up was not an easy matter. Ramanujan was an
orthodox Brahmin and so was a strict vegetarian. His religion should have prevented him
from travelling but this difficulty was overcome, partly by the work of E H Neville who was
a colleague of Hardy's at Trinity College and who met with Ramanujan while lecturing in
India.
Ramanujan sailed from India on 17 March 1914. It was a calm voyage except for three days
on which Ramanujan was seasick. He arrived in London on 14 April 1914 and was met by
Neville. After four days in London they went to Cambridge and Ramanujan spent a couple of
weeks in Neville's home before moving into rooms in Trinity College on 30th April. Right
from the beginning, however, he had problems with his diet. The outbreak of World War I
made obtaining special items of food harder and it was not long before Ramanujan had health
problems.

Right from the start Ramanujan's collaboration with Hardy led to important results. Hardy
was, however, unsure how to approach the problem of Ramanujan's lack of formal education.
He wrote [1]:-

What was to be done in the way of teaching him modern mathematics? The limitations of his
knowledge were as startling as its profundity.

Littlewood was asked to help teach Ramanujan rigorous mathematical methods. However he
said ([31]):-

... that it was extremely difficult because every time some matter, which it was thought that
Ramanujan needed to know, was mentioned, Ramanujan's response was an avalanche of
original ideas which made it almost impossible for Littlewood to persist in his original
intention.

The war soon took Littlewood away on war duty but Hardy remained in Cambridge to work
with Ramanujan. Even in his first winter in England, Ramanujan was ill and he wrote in
March 1915 that he had been ill due to the winter weather and had not been able to publish
anything for five months. What he did publish was the work he did in England, the decision
having been made that the results he had obtained while in India, many of which he had
communicated to Hardy in his letters, would not be published until the war had ended.

On 16 March 1916 Ramanujan graduated from Cambridge with a Bachelor of Science by

Research (the degree was called a Ph.D. from 1920). He had been allowed to enrol in June
1914 despite not having the proper qualifications. Ramanujan's dissertation was on Highly
composite numbers and consisted of seven of his papers published in England.

Ramanujan fell seriously ill in 1917 and his doctors feared that he would die. He did improve
a little by September but spent most of his time in various nursing homes. In February 1918
Hardy wrote (see [3]):-

Batty Shaw found out, what other doctors did not know, that he had undergone an operation
about four years ago. His worst theory was that this had really been for the removal of a
malignant growth, wrongly diagnosed. In view of the fact that Ramanujan is no worse than
six months ago, he has now abandoned this theory - the other doctors never gave it any
support. Tubercle has been the provisionally accepted theory, apart from this, since the
original idea of gastric ulcer was given up. ... Like all Indians he is fatalistic, and it is
terribly hard to get him to take care of himself.

On 18 February 1918 Ramanujan was elected a fellow of the Cambridge Philosophical

Society and then three days later, the greatest honour that he would receive, his name
appeared on the list for election as a fellow of the Royal Society of London. He had been
proposed by an impressive list of mathematicians, namely Hardy, MacMahon, Grace,
Larmor, Bromwich, Hobson, Baker, Littlewood, Nicholson, Young, Whittaker, Forsyth and
Whitehead. His election as a fellow of the Royal Society was confirmed on 2 May 1918, then
on 10 October 1918 he was elected a Fellow of Trinity College Cambridge, the fellowship to
run for six years.

The honours which were bestowed on Ramanujan seemed to help his health improve a little
and he renewed his effors at producing mathematics. By the end of November 1918
Ramanujan's health had greatly improved. Hardy wrote in a letter [3]:-

I think we may now hope that he has turned to corner, and is on the road to a real recovery.
His temperature has ceased to be irregular, and he has gained nearly a stone in weight. ...
There has never been any sign of any diminuation in his extraordinary mathematical talents.
He has produced less, naturally, during his illness but the quality has been the same. ....

He will return to India with a scientific standing and reputation such as no Indian has
enjoyed before, and I am confident that India will regard him as the treasure he is. His
natural simplicity and modesty has never been affected in the least by success - indeed all
that is wanted is to get him to realise that he really is a success.

Ramanujan sailed to India on 27 February 1919 arriving on 13 March. However his health
was very poor and, despite medical treatment, he died there the following year.

The letters Ramanujan wrote to Hardy in 1913 had contained many fascinating results.
Ramanujan worked out the Riemann series, the elliptic integrals, hypergeometric series and
functional equations of the zeta function. On the other hand he had only a vague idea of what
constitutes a mathematical proof. Despite many brilliant results, some of his theorems on
prime numbers were completely wrong.

Ramanujan independently discovered results of Gauss, Kummer and others on

hypergeometric series. Ramanujan's own work on partial sums and products of
hypergeometric series have led to major development in the topic. Perhaps his most famous
work was on the number p(n) of partitions of an integer n into summands. MacMahon had
produced tables of the value of p(n) for small numbers n, and Ramanujan used this numerical
data to conjecture some remarkable properties some of which he proved using elliptic
functions. Other were only proved after Ramanujan's death.

In a joint paper with Hardy, Ramanujan gave an asymptotic formula for p(n). It had the
remarkable property that it appeared to give the correct value of p(n), and this was later
proved by Rademacher.

Ramanujan left a number of unpublished notebooks filled with theorems that mathematicians
have continued to study. G N Watson, Mason Professor of Pure Mathematics at Birmingham
from 1918 to 1951 published 14 papers under the general title Theorems stated by Ramanujan
and in all he published nearly 30 papers which were inspired by Ramanujan's work. Hardy
passed on to Watson the large number of manuscripts of Ramanujan that he had, both written
before 1914 and some written in Ramanujan's last year in India before his death.

The picture above is taken from a stamp issued by the Indian Post Office to celebrate the 75 th
anniversary of his birth.
Article by:J J O'Connor and E F Robertson

CalyampudiRadhakrishnaRao's parents were C Daraiswamy Naidu (1879-1940), a police

inspector, and A Laxmikanthamma. He was the eighth of his parents' ten children, two of
whom died as infants. We should explain that the name Rao is part of his given name - in fact
all the male children in his family were named Rao. His other given name Radhakrishna
comes from the god Krishna (who was the eighth of his parents children and, for that reason,
the custom was to name the eighth child after Krishna). Only the boys in the family were
given a school education, as was the tradition at the time, but this was rather disrupted by
frequent moves that the family made as their father was transferred from one place to another.
CalyampudiRadhakrishna completed his first two years schooling in Gudur, his next two in
Nuzvid, and then grades 6 and 7 in Nandigama, all towns in the state of Andhra Pradesh.
During these years CR, as we will call CalyampudiRadhakrishnaRao throughout this
biography, saw little of his father who was totally absorbed in his work but his mother was a
major influence [11]:-

... in my younger days, [my mother] woke me up every day at four in the morning and lit the
oil lamp for me to study in the quiet hours of the morning when the mind is fresh.

In 1931 CR's father retired and the family settled down in Visakhapatnam, on the coast of
Andhra Pradesh. The family chose this city because of the excellent educational facilities that
were available there for their children. CR studied there for ten years, first at high school,
then mathematics, physics, and chemistry at the intermediate Mrs A V N College before
attending Andhra University. At the intermediate college he won the ChandrasekaraIyer
Scholarship in each of his two years. The School Magazine for 1935 published his picture
with the caption:-

C RadhakrishnaRao who has won the Chandrasekara Scholarship this year. He has had the
unique distinction of knocking off the most coveted prizes throughout his school career.

CR spoke in the interview [5] about the influence of his father:-

When I was 11, I could do complicated arithmetical problems without paper and pencil. My
father appreciated my interest in mathematics and my good performance in school, and he
thought that I should eventually get a degree in mathematics and proceed to do research to
get a doctorate degree. He presented me with a book called 'Problems for Leelavathi', a
collection of problems set by a mathematician for his daughter Leelavathi to solve. He asked
me to work out 5 to 10 problems in the book every day. I enjoyed solving these problems,
which aroused further interest in me to pursue mathematics. Thus, my entry into mathematics
resulted from the encouragement I received from my father and my own interest in solving
mathematical problems.

He graduated M.A. with First Class Honours in Mathematics from Andhra University in
1940. He applied for a research scholarship from Andhra University but his application was
rejected on the grounds that it had been received after the deadline. At this stage, encouraged
by his family, he decided to sit the competitive Indian Civil Service examinations but, being
only twenty years old, he had to wait eighteen months before being allowed to take the
examinations. He applied for job as a mathematician in an army survey unit to fill out the
time before taking the Civil Service examinations. He was called to Calcutta for an interview
but failed to get the job. However, this was a turning point for CR, for he stayed in the South
Indian Hotel before his interview and there he met a young man who was being trained in
statistics at the Indian Statistical Institute. CR had taken a course on probability while
studying for his Master's degree at Andhra University but he had never heard of the Indian
Statistical Institute. The young man took CR to visit the Institute, at that time located in the
Physics Department of Presidency College. It seemed to provide both a job and a chance to
test whether he would like research so CR applied for the one-year training course in
statistics.

The family were in some financial difficulties by this time since CR's father had died in the
previous year. However, one of his brothers and his mother managed to finance him through
the year at the Institute. The training course was rather a disappointment, taught by people
with little understanding of statistical theory. However, there was the head of the Institute P C
Mahalanobis, as well as other top researchers working at the Institute such as K Raghavan
Nair, SamarendraNath Roy and Raj Chandra Bose. CR began undertaking research with Nair
and they published a joint paper Confounded designs for asymmetrical factorial experiments
(1941). In the following year he published six papers, four of them joint publications with
Nair, for example A general class of quasi-factorial designs leading to confounded designs
for factorial experiments and A note on partially balanced incomplete block designs.

A few months after he began training at the Indian Statistical Institute, Calcutta University
announced a new Master's degree in statistics. Lecturers at the University were the same
statisticians working at the Institute and he took courses from K Raghavan Nair,
SamarendraNath Roy and Raj Chandra Bose. He was awarded the degree in 1943 with First
Class and the gold medal. His thesis was a major piece of work in four areas: the design of
experiments, linear models, multivariate analysis, and the characterization of probability
distributions. These would be the topics he continued to study throughout his career. As well
as research in statistics, CR began to look at combinatorial problems with R C Bose and
number theory problems with SarvadamanChowla.

CR was appointed as a Technical Apprentice at the Indian Statistical Institute beginning in

November 1943 and, a few months later, in June 1944, he also worked as a part-time lecturer
at Calcutta University. By the end of 1946 he had over thirty papers in print [5]:-
I continued my research on combinatorics with reference to design of experiments and wrote
a number of papers, some jointly with R C Bose and S D Chowla. I developed a general
theory of least squares without any assumptions on the concomitant variables. I found a test
for redundancy of a specified set of variables in multivariate analysis.

The most significant result CR obtained during this period is now called the Cramér-Rao
inequality and gives a bound for the variance of an unbiased estimate of a parameter. It
appears in his paper Information and accuracy attainable in the estimation of statistical
parameters (1945). The significance of this paper can be seen from the fact that it was
republished in S Kotz and N Johnson (eds.), Breakthroughs in Statistics: 1889-1990
(Springer Verlag, New York, Berlin, 1991). In August 1946 CR boarded a ship sailing from
Calcutta to England. He had been offered a position undertaking statistical work at the
Anthropological Museum in Cambridge and he also registered as a research student at
Cambridge University. His Ph.D. studies at King's College, Cambridge, were supervised by R
A Fisher. CR worked at the Anthropological Museum every day and every evening he spent a
few hours in Fisher's genetics laboratory mapping the chromosomes of thousands of live
mice. While at Cambridge CR wrote his influential paper Large sample tests of statistical
hypotheses concerning several parameters with applications to problems of estimation which
was published in the Proceedings of the Cambridge Philosophical Society in 1948. He was
awarded his doctorate in 1948 for his thesis Statistical Problems of Biological Classification
which was examined by John Wishart.

He returned to Calcutta in August to his post at the Indian Statistical Institute. Soon after
arriving, on 9 September 1948, he married SmtBhargavi, a girl he had known from
childhood; they had a daughter Tejaswini and a son Veerendra. CR was appointed as a
Superintending Statistician on his return but a major grant to the Institute from the Indian
government allowed the Institute to set up a Research and Training School and appoint
professors, assistant professors and other academic grades. Those senior to CR, R C Bose and
A Bhattacharya, left the Institute around this time. S N Roy was appointed to a professorship
and CR to an assistant professorship. However, Roy left for the United States soon after his
appointment and CR became a professor in July 1949 at the young age of 28. The Institute
was given authority to award its own degrees in 1959 and started up its own undergraduate
programme. Being understaffed CR found himself doing quite a bit of undergraduate
teaching but found that a rewarding experience. In 1964 he became the director of the
Research and Training School, and then in 1972 he was appointed Director-Secretary of the
Indian Statistical Institute. He was named Jawaharlal Nehru Professor in 1976.

Let us not look at some of the highly influential books CR has published. The first was
Advanced statistical methods in biometric research (1952) which he began writing while
working for his doctorate at Cambridge. His aim, stated in the Preface, is:-

... to present a number of statistical techniques, keeping in view the requirements of both the
student who questions the basis of a particular method employed and the practical worker
who seeks a recipe for the reduction of his data.

His next book Linear statistical inference and its applications (1965) was more
mathematical, and designed to be used for mathematical statistics courses in universities. He
published Computers and the Future of Human Society in 1970, and in the following year,
jointly with Sujit Kumar Mitra, he published Generalized inverse of matrices and its
applications. R J Plemmons explains that they:-
... present a general unified treatment of the concept of inversability of singular, and in
general rectangular, matrices over the complex field. ... the material should make a good text
for a one term course in matrix algebra or mathematical statistics.

Further books followed such as Characterization Problems of Mathematical Statistics

(Russian edition 1972, English translation 1973) with A Kagan and Yu V Linnik.

In 1979 CR left the Indian Statistical Institute, just before reaching the mandatory retirement
age of sixty, and went to the United States where he was appointed to a University
Professorship at the University of Pittsburgh. In 1988 he moved to Pennsylvania State
University in Pittsburgh where he was named Eberly Family Professor in Statistics. After
retiring he became Emeritus Eberly Professor.

Books he published after moving to the United States include: (with J Kleffe) Estimation of
variance components and applications (1988), Statistics and truth. Putting chance to work
(1989), (with D N Shanbhag) Choquet-Deny type functional equations with applications to
stochastic models (1994), (with H Toutenburg) Linear models. Least squares and alternatives
(1995), and (with M B Rao) Matrix algebra and its applications to statistics and
econometrics (1998).

C R Rao has received so many honours that it would be quite impossible to list them all. Here
is a selection to give an impression of the high esteem that he is held in throughout the world.
First we list some of the prizes and awards he has received: the S SBhatnagar Prize of
Council of Scientific and Industrial Research (1963), the Guy Medal in Silver of the Royal
Statistical Society (1965), awarded the title of Padma Bhushan by the Indian Government
(1968), received the MegnadhSaha Medal of the Indian National Science Academy (1969),
the Jagdish Chandra Bose Gold Medal of the Bose Institute (1979), the Silver Plate of the
Andhra Pradesh Academy of Sciences (1984), The Times of India listed Rao as one of the top
10 Indian scientists of all time (1988), awarded the Samuel S Wilks Memorial Award of the
American Statistical Association (1989):-

For major contributions to the theory of multivariate statistics and applications of that theory
to problems of biometry; for world wide activities as advisor to national and international
organizations; for long time conscientious as a teacher, editor, author and founder of
academic institutions; and for the great influence he has had on the applications of statistical
thinking in different scientific disciplines, embodying over a career of more than 40 years the
spirit and ideals of Samuel S Wilks.

Other awards include: the Mahalanobis Birth Centenary Gold Medal by the Indian Science
Congress (1996), the Distinguished Achievement Medal from the American Statistical
Association (1997):-

... for outstanding contributions to the development of methods, issues, concepts, and
applications of environmental statistics ...

the Padma Vibhushan by the Government of India (2001), the National Medal of Science
(2002), presented by President George W Bush on 12 June at a ceremony in the White
House:-
... for his pioneering contributions to the foundations of statistical theory and multivariate
statistical methodology and their applications, enriching the physical biological,
mathematical, economic and engineering sciences ...

the SrinivasaRamanujan Medal of the Indian National Science Academy (2003), the
International Mahalanobis Prize of the International Statistical Institute (2003), and the Guy
Medal in Gold of the Royal Statistical Society (2011):-

In the 115-year history of Royal Statistical Society, he is the 34th recipient of the award. He
is the first Asian, first non-European and first non-American to receive the award.

He has been elected to the Royal Society of London (1967), the National Academy of
Sciences, USA (1995), the American Academy of Arts and Science, the Indian National
Science Academy, the Lithuanian Academy of Sciences, and the Third World Academy of
Sciences. He was made an Honorary Member of the International Statistical Institute (1983),
the International Biometric Society (1986), the Royal Statistical Society (1969), the Finnish
Statistical Society (1990), the Portuguese Statistical Society, the Institute of Combinatorics
and Applications, and the World Innovation Foundation.

He has been awarded thirty-three honorary degrees by universities in eighteen countries, in

six continents, including: Andhra University, India (1967), Leningrad University, USSR
(1970), Delhi University, India (1973), University of Athens, Greece (1976), Osmania
University, Hyderabad, India (1977), Ohio State University, Columbus, USA (1979),
Universidad Nacional de San Marcos, Lima, Peru (1982), University of the Philippines,
Manila (1983), University of Tampere, Finland (1985), Indian Statistical Institute, India
(1989), Université de Neuchâtel, Switzerland (1989), Colorado State University, Fort Collins,
USA (1990), University of Hyderabad, India (1991), Agricultural University of Poznan,
Poland (1991), Slovak Academy of Sciences, Bratislava, Slovakia (1994), University of
Barcelona, Spain (1995), University of Munich, Germany (1995), Sri Venkateswara
University, Tirupati, India (1996), University of Guelph, Canada (1996), University of
Calcutta, India (2003), University of Pretoria, South Africa (2004), University of Rhode
Island, Kingston, USA (2007), and Jawaharlal Nehru Technological University, Kakinada,
India (2011).

The Journal of Quantitative Economics published a special issue in Rao's honour in 1991.
The preface gives the following tribute:-

Dr Rao is a very distinguished scientist and a highly eminent statistician of our time. His
contributions to statistical theory and applications are well known, and many of his results,
which bear his name, are included in the curriculum of courses in statistics at bachelor's and
master's level all over the world. He is an inspiring teacher and has guided the research
work of numerous students in all areas of statistics. His early work had greatly influenced the
course of statistical research during the last four decades. One of the purposes of this special
issue is to recognize Dr Rao's own contributions to econometrics and acknowledge his major
role in the development of econometric research in India.

C R Rao has served as president of five statistical societies: the Indian Econometric Society
(1971-1976); the International Biometric Society (1973-1975); the Institute of Mathematical
Statistics (1976-1977); the International Statistical Institute (1977-1979), and the Forum for
Interdisciplinary Mathematics (1982-1984).
Finally we record that CR has many hobbies such as gardening, photography, cooking, and
Indian classical dance. In Calcutta he played soccer and badminton with staff and students in
the evenings. In the United Sates his relaxation is walking.

Article by:J J O'Connor and E F Robertson

List of References (14 books/articles)

Harish-Chandra's mother was Satyagati Seth, who was the daughter of the lawyer Ram
Sanehi Seth, and his father was Chandrakishore who was a civil engineer. Chandrakishore
had been educated at the Thomason Engineering College at Roorkee, then entered the Indian
Service of Engineers and the first part of his career was [14]:-

... spent in the field, usually on horseback, inspecting and maintaining the dikes of the
extensive network of canals in the northern plains.

Since Harish-Chandra's father spent most of his time travelling around the country inspecting
canals, Harish-Chandra spent most of his childhood living in his maternal grandfather's home
in Kanpur. Ram Sanehi Seth was a wealthy man with a large house which was home to many
of his relatives. Harish-Chandra, despite being a rather weakly child, did sometimes spend
time travelling with his father. His education was treated by the family as of the utmost
importance [14]:-

A tutor was hired, and there were visits from a dancing master and a music master. At the
age of nine he was enrolled, younger than his schoolmates, in the seventh class. He
completed Christ Church High School at fourteen, and remained in Kanpur for intermediate
college, which he finished at sixteen ...

He then attended the University of Allahabad. Here he studied theoretical physics, this
direction being the result of reading Principles of Quantum Mechanics by Dirac which he
found himself in the university library. He was awarded a B.Sc. in 1941, then a master's
degree in 1943. Harish-Chandra worked as a postgraduate research fellow on problems in
theoretical physics under HomiBhabha, at the Indian Institute of Science at Bangalore in
Southern India. Bhabha had been a student of Dirac in the 1930s. Harish-Chandra began
publishing papers on theoretical physics while at Bangalore, and he published a couple of
joint papers with Bhabha extending some of Dirac's results. For the first six months he spent
in Bangalore he lived with Dr Kale, a botanist from Allahabad, and his wife who had taught
Harish-Chandra French when he was an undergraduate. The Kales had a daughter Lalitha
who at this time was a young girl, but about eight years later in 1952 he returned from the
United States to India and there married Lalitha.

Bhabha and Harish-Chandra's teacher, K S Krishnan at Allahabad University, recommended

him to Dirac for research work at Cambridge which would lead to Ph.D. degree. In 1945
Harish-Chandra went to Gonville and Caius College, Cambridge, where he studied for his
doctorate under Dirac's supervision. However Harish-Chandra saw comparatively little of his
supervisor, giving up attending Dirac's lecture course when he realised that Dirac was
essentially reading from one of his books. He wrote that Dirac was (see for example [14]):-

... very gentle and kind and yet rather aloof and distant. ... [I decided] I should not bother him
too much and went to see him about once each term.

During his time in Cambridge he began to move away from physics and became more
interested in mathematics attending the lecture courses of Littlewood and Hall. He also
attended a lecture by Pauli and pointed out a mistake in Pauli's work. The two were to
become life long friends. Harish-Chandra obtained his degree in 1947 for his thesis Infinite
irreducible representations of the Lorentz group in which he gives a complete classification
of the irreducible unitary representations of SL(2,C).
Dirac visited Princeton for the year 1947-48 and Harish-Chandra went to the United States
with him, working as his assistant during this time. However he was greatly influenced by the
leading mathematicians Weyl, Artin and Chevalley who were working there. It was during
this year in Princeton that he finally decided that he was a mathematician and not a physicist.
He later wrote (see for example [14]):-

Soon after coming to Princeton I became aware that my work on the Lorentz group was
based on somewhat shaky arguments. I had naively manipulated unbounded operators
without paying any attention to their domains of definition. I once complained to Dirac about
the fact that my proofs were not rigorous and he replied, "I am not interested in proofs but
only in what nature does." This remark confirmed my growing conviction that I did not have
the mysterious sixth sense which one needs in order to succeed in physics and I soon decided
to move over to mathematics.

After Dirac returned to Cambridge, Harish-Chandra remained at Princeton for a second year.
He spent 1949-50 at Harvard where he was influenced by Zariski. The period 1950 to 1963,
spent at Columbia University, New York, was his most productive. During this time he
worked on representations of semisimpleLie groups. Also during this period he had close
contact with Weil. When we say that he spent the years 1950-63 at Columbia University we
really mean that he was on the faculty there for this period, but it is fair to say that he spent
long periods in other institutions. He spent 1952-53 at the Tata Institute in Bombay and it was
during this year that he married Lalitha Kale (often known as Lily); they had two daughters
[14]:-

Lily ... with good spirits, generous affection, patience, and all-round competence was to
pamper him for thirty years.

He spent 1955-56 at the Institute for Advanced Study at Princeton, 1957-58 as a Guggenheim
Fellow in Paris where he was able to work with Weil and the two often went walking
together. He was a Sloan Fellow at the Institute for Advanced Study from 1961 to 1963.

In [14] Harish-Chandra is quoted as saying that he believed that his lack of background in
mathematics was in a way responsible for the novelty of his work:-

I have often pondered over the roles of knowledge or experience, on the one hand, and
imagination or intuition, on the other, in the process of discovery. I believe that there is a
certain fundamental conflict between the two, and knowledge, by advocating caution, tends to
inhibit the flight of imagination. Therefore, a certain naiveté, unburdened by conventional
wisdom, can sometimes be a positive asset.

Armand Borel describes some of Harish-Chandra's contributions in [6]. Schwermer reviews

that work as well as [25] and what follows is taken from these reviews:-

The impression one has of Harish-Chandra's work as a whole is one of surprising

cohesiveness and uniformity. His primary interests revolved about the representation theory
of reductive groups and harmonic analysis on these groups and their related homogeneous
spaces. However Harish-Chandra made several attempts to get into algebraic geometry or
number theory.
He has built a fundamental theory of representations of Lie groups and Lie algebras,
respectively of harmonic analysis on these groups and their homogeneous spaces. It was
Harish-Chandra who extended the concept of a character of finite-dimensional
representations of semisimple Lie groups to the case of infinite-dimensional representations;
he proved an analogue of Weyl's character formula.

Some major contributions by Harish-Chandra's work may be singled out: the explicit
determination of the Plancherel measure for semisimple groups, the determination of the
discrete series representations, his results on Eisenstein series and in the theory of
automorphic forms, his "philosophy of cusp forms", as he called it, as a guiding principle to
have a common view of certain phenomena in the representation theory of reductive groups
in a rather broad sense, including not only the real Lie groups, but p-adic groups or groups
over adele rings. His scientific work, being a synthesis of analysis, algebra and geometry, is
still of lasting influence.

Harish-Chandra worked at the Institute of Advanced Study at Princeton from 1963. He was
appointed IBM-von Neumann Professor in 1968.

He died of a heart attack at the end of a week long conference in Princeton, having earlier
suffered from three heart attacks, the first being in 1969 and the third in 1982. In October
1983 a conference in honour of Armand Borel was held in Princeton [14]:-

... for the week of the conference, [Harish-Chandra's] vigour and force reasserted
themselves. Princeton's warm, clear autumn weather prevailed and between lectures at the
conference, on a lawn or a terrace of the Institute, he was the centre of a lively crowd,
expressing his views on a variety of topics. On Sunday 16 October, the last day of the
conference he and Lily had many of the participants to their home. He was a sparkling host.
In the late afternoon, after the guests had departed, he went for his customary walk, and
never returned alive.

Harish-Chandra received many awards in his career. He was a Fellow of the Royal Society
(1973), a Fellow of the National Academy of Sciences (United States) (1981), the Indian
Academy of Sciences and the Indian National Science Academy (1975). He won the Cole
prize from the American Mathematical Society in 1954 for his papers on representations of
semisimple Lie algebras and groups, and particularly for his paper On some applications of
the universal enveloping algebra of a semisimple Lie algebra which he had published in the
Transactions of the American Mathematical Society in 1951. In 1974, he received the
Ramanujan Medal from Indian National Science Academy. He was awarded honorary
degrees by Delhi University (1973) and Yale University (1981).

Having been both a physicist and a mathematician it is interesting to see Harish-Chandra's

view of their relationship. This is described nicely in [14]:-

Although he was convinced that the mathematician's very mode of thought prevented him
from comprehending the essence of theoretical physics, where, he felt, deep intuition and not
logic prevailed, and sceptical of any mathematician who presumed to attempt to understand
it, he was even more impatient with those mathematicians in whom a sympathy for theoretical
physics was lacking, a failing he attributed in pacourtesy that did not conceal the depth of his
feelings and thought. In his early years he liked to paint and later expressed a fondness for
the French Impressionists. ... One of his colleagues suggested that