FAMOUS INDIAN

MATHEMATICIANS

-- SHIVANITA ROY

X-B

Roll No. 37

ACKNOWLEDGEMENT

SRINIVASA RAMANUJAN

Born On : 22 December 1887

Place : Erode, Tamil Nadu, India.

Education

Ramanujan first encountered formal mathematics at age 10. He demonstrated a natural ability, and was given books on advanced trigonometry written by S. L. Loney. He had mastered them by age 12, and even discovered theorems of his own. He demonstrated unusual mathematical skills at school, winning accolades and awards. By 17, Ramanujan conducted his own mathematical research on Bernoulli numbers and the Euler–Mascheroni constant. He received a scholarship to study at Government College in Kumbakonam, but lost it when he failed his non-mathematical coursework. He joined another college to pursue independent

mathematical research, working as a clerk in the Accountant-General's office at the Madras Port Trust Office to support himself. In 1912–1913, he sent samples of his theorems to three academics at the University of Cambridge. Only G. H. Hardy recognized the brilliance of his work, subsequently inviting Ramanujan to visit and work with him at Cambridge. He became a Fellow of the Royal Society and a Fellow of Trinity College, Cambridge.

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. It is said that Ramanujan's discoveries are unusually rich and that there is often more in it than what initially meets the eye. As a by-product, new directions of research were opened up. Examples of the most interesting of these formulae include the intriguing infinite series for π, one of which is given below

This result is based on the negative fundamental discriminant d = −4×58 with class number h(d) = 2 (note that 5×7×13×58 = 26390 and that 9801=99×99; 396=4×99) and is related to the fact that

Compare to Heegner numbers, which have class number 1 and yield similar formulae. Ramanujan's series for π converges extraordinarily rapidly (exponentially) and forms the basis of some of the fastest algorithms currently used to calculate π. Truncating the sum to the first term also gives the approximation for π, which is correct to six decimal places.

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. The unusual part was that it was the solution to the whole class of problems. Mahalanobis was astounded and asked how he did it. "It is simple. The minute I heard the problem, I knew that the answer was a continued fraction. Which continued fraction, I asked myself. Then the answer came to my mind", Ramanujan replied.[83][84]

His intuition also led him to derive some previously unknown identities, such as

for all θ, where Γ(z) is the gamma function. Equating coefficients of θ0, θ4, and θ8 gives some deep identities for the hyperbolic secant.

In 1918, Hardy and Ramanujan studied the partition function P(n) extensively and gave a non-convergent asymptotic series that permits exact computation of the number of partitions of an integer. Hans Rademacher, in 1937, was able to refine their formula to find an exact convergent series solution to this problem. Ramanujan and Hardy's work in this area gave rise to a powerful new method for finding asymptotic formulae, called the circle method.

He discovered mock theta functions in the last year of his life. For many years these functions were a mystery, but they are now known to be the holomorphic parts of harmonic weak Maass forms.

The Ramanujan 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, which has as generating function the discriminant modular form Δ(q), a typical cusp form in the theory of modular forms. It was finally proven 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's notebooks

While still in India, Ramanujan recorded the bulk of his results in four notebooks of loose leaf paper. These results were mostly written up without any derivations. This is probably the origin of the misperception that Ramanujan was unable to prove his results and simply thought up the final result directly. Mathematician Bruce C. Berndt, in his review of these notebooks and Ramanujan's work, says that Ramanujan most certainly was able to make the proofs of most of his results, but chose not to.

This style of working may have been for several reasons. Since paper was very expensive, Ramanujan would do most of his work and perhaps his proofs on slate, and then transfer just the results to paper. Using a slate was common for mathematics students in India at the time. He was also quite likely to have been influenced by the style of G. S. Carr's book, which stated results without proofs. Finally, it is possible that Ramanujan considered his workings to be for his personal interest alone; and therefore only recorded the results.

The first notebook has 351 pages with 16 somewhat organized chapters and some unorganized material. The second notebook has 256 pages in 21 chapters and 100 unorganized pages, with the third notebook containing 33 unorganized pages. The results in his notebooks inspired numerous papers by later mathematicians trying to prove what he had found. Hardy himself created papers exploring material from Ramanujan's work as did G. N. Watson, B. M. Wilson, and Bruce Berndt. A fourth notebook with 87 unorganized pages, the so-called "lost notebook", was rediscovered in 1976 by George Andrews.

ARYABHATA

Born On: 476 A.D.

Place : Kusumpur, India

Mathematics

Place value system and zero

The place-value system, first seen in the 3rd century Bakhshali Manuscript, was clearly in place in his work. While he did not use a symbol for zero, the French mathematician Georges Ifrah argues that knowledge of zero was implicit in Aryabhata's place-value system as a place holder for the powers of ten with null coefficients.

However, Aryabhata did not use the Brahmi numerals. Continuing the Sanskritic tradition from Vedic times, he used letters of the alphabet to denote numbers, expressing quantities, such as the table of sines in a mnemonic form.

Approximation of pi

Aryabhata worked on the approximation for pi (π), and may have come to the conclusion that π is irrational. In the second part of the Aryabhatiyam (gaṇitapāda 10), he writes:

caturadhikam śatamaṣṭaguṇam dvāṣaṣṭistathā sahasrāṇāmayutadvayaviṣkambhasyāsanno vṛttapariṇāhaḥ."Add four to 100, multiply by eight, and then add 62,000. By this rule the circumference of a circle with a diameter of 20,000 can be approached."

This implies that the ratio of the circumference to the diameter is ((4+100)×8+62000)/20000 = 62832/20000 = 3.1416, which is accurate to five significant figures.

It is speculated that Aryabhata used the word āsanna (approaching), to mean that not only is this an approximation but that the value is incommensurable (or irrational). If this is correct, it is quite a sophisticated insight, because the irrationality of pi was proved in Europe only in 1761 by Lambert.

After Aryabhatiya was translated into Arabic (c. 820 CE) this approximation was mentioned in Al-Khwarizmi's book on algebra.

Mensuration and trigonometry

In Ganitapada 6, Aryabhata gives the area of a triangle as

tribhujasya phalashariram samadalakoti bhujardhasamvargah

that translates to: "for a triangle, the result of a perpendicular with the half-side is the area."

Aryabhata discussed the concept of sine in his work by the name of ardha-jya. Literally, it means "half-chord". For simplicity, people started calling it jya. When Arabic writers translated his works from Sanskrit into Arabic, they referred it as

jiba. However, in Arabic writings, vowels are omitted, and it was abbreviated as jb. Later writers substituted it with jiab, meaning "cove" or "bay." (In Arabic, jiba is a meaningless word.) Later in the 12th century, when Gherardo of Cremona translated these writings from Arabic into Latin, he replaced the Arabic jiab with its Latin counterpart, sinus, which means "cove" or "bay". And after that, the sinus became sine in English.

Indeterminate equations

A problem of great interest to Indian mathematicians since ancient times has been to find integer solutions to equations that have the form ax + by = c, a topic that has come to be known as diophantine equations. This is an example from Bhāskara's commentary on Aryabhatiya:

Find the number which gives 5 as the remainder when divided by 8, 4 as the remainder when divided by 9, and 1 as the remainder when divided by 7

That is, find N = 8x+5 = 9y+4 = 7z+1. It turns out that the smallest value for N is 85. In general, diophantine equations, such as this, can be notoriously difficult. They were discussed extensively in ancient Vedic text Sulba Sutras, whose more ancient parts might date to 800 BCE. Aryabhata's method of solving such problems is called the kuṭṭaka (कु� ट्टकु) method. Kuttaka means "pulverizing" or "breaking into small pieces", and the method involves a recursive algorithm for writing the original factors in smaller numbers. Today this algorithm, elaborated by Bhaskara in 621 CE, is the standard method for solving first-order diophantine equations and is often referred to as the Aryabhata algorithm.[14] The diophantine equations are of interest in cryptology, and the RSA Conference, 2006, focused on the kuttaka method and earlier work in the Sulvasutras.

Algebra

In Aryabhatiya Aryabhata provided elegant results for the summation of series of squares and cubes:

and

BRAHMAGUPTA

Born On: 598 CE

Place: Bhinmal, Rajasthan, India.

Mathematics

Brahmagupta was the first to use zero as a number. He gave rules to compute with zero. Contrary to popular opinion, negative numbers did not appear first in Brahmasputa siddhanta. Negative numbers appear for the first time in history in the Nine Chapters on the Mathematical Art (Jiu zhang suan-shu) around 200 BC. Brahmagupta's most famous work is his Brahmasphutasiddhanta. It is composed in elliptic verse, as was common practice in Indian mathematics, and consequently has a poetic ring to it. As no proofs are given, it is not known how Brahmagupta's mathematics was derived.

Algebra

Brahmagupta gave the solution of the general linear equation in chapter eighteen of Brahmasphutasiddhanta.

Arithmetic

Contrary to popular opinion, the four fundamental operations (addition, subtraction, multiplication and division) did not appear first in BrahmasputhaSiddhanta, but they were already known by the Sumerians at least 2500 BC. In BrahmasputhaSiddhanta, Multiplication was named Gomutrika. In the beginning of chapter twelve of his Brahmasphutasiddhanta, entitled Calculation, Brahmagupta details operations on fractions. The reader is expected to know the basic arithmetic operations as far as taking the square root, although he explains how to find the cube and cube-root of an integer and later gives rules facilitating the computation of squares and square roots. He then gives rules for dealing with

five types of combinations of fractions, , , , ,

and .

Series

Brahmagupta then goes on to give the sum of the squares and cubes of the first n integers.

The sum of the squares is that [sum] multiplied by twice the [number of] step[s] increased by one [and] divided by three. The sum of the cubes is the square of that [sum] Piles of these with identical balls [can also be computed].

It is important to note here Brahmagupta found the result in terms of the sum of the first n integers, rather than in terms of n as is the modern practice.

He gives the sum of the squares of the first n natural numbers as n(n+1)(2n+1)/6 and the sum of the cubes of the first n natural numbers as (n(n+1)/2)².

Zero

Brahmagupta's Brahmasphuṭasiddhanta is the very first book that mentions zero as a number, hence Brahmagupta is considered as the man who found zero. He gave rules of using zero with other numbers. Zero plus a positive number is the positive number etc. Brahmagupta made use of an important concept in

mathematics, the number zero. The Brahmasphutasiddhanta is the earliest known text to treat zero as a number in its own right, rather than as simply a placeholder digit in representing another number as was done by the Babylonians or as a symbol for a lack of quantity as was done by Ptolemy and the Romans.

Diophantine analysis

Pythagorean triples

In chapter twelve of his Brahmasphutasiddhanta, Brahmagupta finds Pythagorean triples,

The height of a mountain multiplied by a given multiplier is the distance to a city; it is not erased. When it is divided by the multiplier increased by two it is the leap of one of the two who make the same journey.

or in other words, for a given length m and an arbitrary multiplier x, let a = mx and b = m + mx/(x + 2). Then m, a, and b form a Pythagorean triple.

Pell's equation

Brahmagupta went on to give a recurrence relation for generating solutions to certain instances of Diophantine equations of the second degree such as Nx2 + 1 = y2 (called Pell's equation) by using the Euclidean algorithm. The Euclidean algorithm was known to him as the "pulverizer" since it breaks numbers down into ever smaller pieces.

Geometry

Brahmagupta's formula

Brahmagupta's most famous result in geometry is his formula for cyclic quadrilaterals. Given the lengths of the sides of any cyclic quadrilateral, Brahmagupta gave an approximate and an exact formula for the figure's area,

Although Brahmagupta does not explicitly state that these quadrilaterals are cyclic, it is apparent from his rules that this is the case. Heron's formula is a special case of this formula and it can be derived by setting one of the sides equal to zero.

Triangles

Brahmagupta dedicated a substantial portion of his work to geometry. One theorem states that the two lengths of a triangle's base when divided by its altitude then follows,

The base decreased and increased by the difference between the squares of the sides divided by the base; when divided by two they are the true segments. The perpendicular [altitude] is the square-root from the square of a side diminished by the square of its segment.

Thus the lengths of the two segments are .

He further gives a theorem on rational triangles. A triangle with rational sides a, b, c and rational area is of the form:

for some rational numbers u, v, and w.

Brahmagupta's theorem

Brahmagupta's theorem states that AF = FD.

Brahmagupta continues,

The square-root of the sum of the two products of the sides and opposite sides of a non-unequal quadrilateral is the diagonal. The square of the diagonal is diminished by the square of half the sum of the base and the top; the square-root is the perpendicular [altitudes].

So, in a "non-unequal" cyclic quadrilateral (that is, an isosceles trapezoid), the length of each diagonal is .

Pi

In verse 40, he gives values of π,

The diameter and the square of the radius [each] multiplied by 3 are [respectively] the practical circumference and the area [of a circle]. The accurate [values] are the square-roots from the squares of those two multiplied by ten.

So Brahmagupta uses 3 as a "practical" value of π, and as an "accurate" value of π.

SHAKUNTALA DEVI

Born On:  November 4, 1939

Place:  Bangalore, India

Education

Her calculating gifts first demonstrated themselves while she was doing card tricks with her father when she was three. They report she "beat" them by memorization of cards rather than by sleight of hand. By age six she demonstrated her calculation and memorization abilities at the University of Mysore. At the age of eight she had success at Annamalai University by doing the same.

Mathematics

In 1977 she extracted the 23rd root of a 201-digit number mentally. On June 18, 1980 she demonstrated the multiplication of two 13-digit numbers 7,686,369,774,870 x 2,465,099,745,779 picked at random by the Computer Department of Imperial College, London. She answered the question in 28 seconds. However, this time is more likely the time for dictating the answer (a 26-digit number) than the time for the mental calculation (the time of 28 seconds was quoted on her own website). Her correct answer was

18,947,668,177,995,426,462,773,730. This event is mentioned on page 26 of the 1995 Guinness Book of Records.

In 2006 she has released a new book called In the Wonderland of Numbers with Orient Paperbacks which talks about a girl Neha and her fascination for numbers.

Books

Some of her books include

Puzzles to Puzzle You More Puzzles to Puzzle You Book of Numbers Figuring: The Joy of Numbers In the Wonderland of Numbers Mathability: Awaken the Math Genius in Your Child Astrology for You

BHASKARACHARYA

Born On: In 1114

Place: Bijjada Bida which is in present day Bijapur district,Karnataka, India.

Mathematics

Some of Bhaskara's contributions to mathematics include the following:

A proof of the Pythagorean theorem by calculating the same area in two different ways and then canceling out terms to get a² + b² = c².

In Lilavati, solutions of quadratic, cubic and quartic indeterminate equations are explained.

Solutions of indeterminate quadratic equations (of the type ax² + b = y²).

Integer solutions of linear and quadratic indeterminate equations (Kuttaka). The rules he gives are (in effect) the same as those given by the Renaissance European mathematicians of the 17th century

A cyclic Chakravala method for solving indeterminate equations of the form ax² + bx + c = y. The solution to this equation was traditionally attributed to William Brouncker in 1657, though his method was more difficult than the chakravala method.

The first general method for finding the solutions of the problem x² − ny² = 1 (so-called "Pell's equation") was given by Bhaskara II.

Solutions of Diophantine equations of the second order, such as 61x² + 1 = y². This very equation was posed as a problem in 1657 by theFrench mathematician Pierre de Fermat, but its solution was unknown in Europe until the time of Euler in the 18th century.

Solved quadratic equations with more than one unknown, and found negative and irrational solutions.

Preliminary concept of mathematical analysis.

Preliminary concept of infinitesimal calculus, along with notable contributions towards integral calculus.

Conceived differential calculus, after discovering the derivative and differential coefficient.

Stated Rolle's theorem, a special case of one of the most important theorems in analysis, the mean value theorem. Traces of the general mean value theorem are also found in his works.

Calculated the derivatives of trigonometric functions and formulae. (See Calculus section below.)

In Siddhanta Shiromani, Bhaskara developed spherical trigonometry along with a number of other trigonometric results.

HARISH-CHANDRA

Born On: 11 October 1923–16 

Place: Kanpur (then Cawnpore), British India

EducationHe was educated at Scindia School, Gwalior, and at the University of Allahabad. After receiving his master’s degree in 1943, he moved to Bangalore for further studies. In 1945, he moved to University of Cambridge as a research student of Paul Dirac. While at Cambridge, he attended lectures by Wolfgang Pauli, and during one of them pointed out a mistake in Pauli's work. The two were to become lifelong friends. During this time he became increasingly interested in mathematics. He obtained his PhD in 1947 and during the same year he moved to the USA.

When Dirac visited Princeton, Harish-Chandra worked as his assistant.

MathematicsHe was influenced by the mathematicians Hermann Weyl and Claude Chevalley. From 1950 to 1963 he was at the Columbia University and carried out some of his best research, especially on representations of semisimple Lie groups. During this period he established as his special area the study of the discrete series representations of semisimple Lie groups — which are the closest analogue of the Peter–Weyl theory in the non-compact case. The methods were formidable and inductive, using Lie group decompositions. See also: Harish-Chandra homomorphism.

He is also known for work with Armand Borel founding the theory of arithmetic groups; and for papers on finite group analogues. He enunciated a philosophy of cusp forms, a precursor of the Langlands philosophy.

He was a faculty member at the Institute for Advanced Study in New Jersey from 1963. He was appointed IBM von Neumann Professor in 1968.

AchievementsHarish-Chandra was a Fellow of the Royal Society and Fellow of National Academy of Sciences. He was the recipient of the Cole Prize of the American Mathematical Society, in 1954. The Indian National Science Academy honoured him with the Srinivasa Ramanujan Medal in 1974.

CALYAMPUDI RADHAKRISHNA RAO

Born On: September 10, 1920

Place: Hadagali, Karnataka, India

Mathematics

Areas of research contributions

Estimation theory Statistical inference and linear models Multivariate analysis Combinatorial design Biometry Statistical genetics Generalized matrix inverses Functional equations

Achievements

International Mahalanobis Prize (2003) of the International Statistical Institute Srinivasa Ramanujan Medal (2003) of the Indian National Science Academy President George W. Bush, on June 12, 2002, honored him with the National

Medal of Science as a "prophet of new age" with the citation, "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."

Padma Vibhushan (2001) by the Government of India Mahalanobis Centenary Gold Medal (1993?) of the Indian Science Congress Wilks Memorial Award (1989) of the American Statistical Association Megnadh Saha Medal (1969) of the Indian National Science Academy Guy Medal in Silver (1965) of the Royal Statistical Society S. S. Bhatnagar Award (1963) of Council of Scientific and Industrial Research JC Bose Gold Medal of the Bose Institute Gold Medal of the University of Calcutta He was also awarded an honorary Doctor of Science by the University of

Calcutta in 2003.

M.S. NARASIMHAN

Born On:  1932

Place: NIL

Education

Narasimhan did his undergraduate studies at Loyola College, Chennai, where he was taught by Fr Racine. Fr Racine had studied with the famous French mathematicians Elie Cartan and Jacques Hadamard, and connected his students with the latest developments in modern mathematics. Among Racine's other students who achieved eminence, we may count Minakshisundaram, K. G. Ramanathan, C S Seshadri,Raghavan Narasimhan, and C. P. Ramanujam.

Narasimhan went to the Tata Institute of Fundamental Research (TIFR), Bombay, for his graduate studies. He obtained his Ph.D. fromUniversity of Mumbai in 1960; his advisor was K. Chandrasekharan. Among Narasimhan's distinguished students is M. S. Raghunathan who followed in this footstep to bag the Shanti Swarup Bhatnagar Prize as well as become FRS. Two other students who made a mark as top-notch mathematicians are S. Ramanan and V. K. Patodi.

Mathematics

He is well known along with C S Seshadri for their work entitled "Stable and unitary vector bundles on a compact Riemann surface".

Achievements

Shanti Swarup Bhatnagar Prize (1975) Third World Academy Award for Mathematics (1987) Padma Bhushan (1990) King Faisal International Prize for Science , 2006 (jointly with Simon Donaldson,

Imperial College)

MANJUL BHARGAVA

Born On:

Place: Hamilton, Ontario

Education

He grew up on Long Island, NY and attended Plainedge High School, graduating in 1992. He obtained his B.A. from Harvard University in 1996 and received his doctorate from Princeton in 2001, working under Andrew Wiles.

He is an accomplished tabla player, having studied under Zakir Hussain. He is also interested in Sanskrit. His grandfather Purushottam Lal Bhargava is a scholar of Sanskrit and ancient Indian history.

MathematicsHis Ph.D. thesis generalized the classical Gauss composition law for quadratic forms to many other situations. One major use of his results is the parametrization of quartic and quintic orders in number fields, thus allowing the study of asymptotic behaviour of arithmetic properties of these rings.

His research includes fundamental contributions to the representation theory of quadratic forms, to interpolation problems and p-adic analysis, and to the study of ideal class groups of algebraic number fields. A short list of his specific mathematical contributions are:

14 new Gauss composition laws, including the quartic and quintic degree cases.

Proof of the 15 theorem, including an extension of the theorem to other number sets such as prime numbers.

Proof of the 290 theorem. A novel generalization of the factorial function, resolving a decades-old

conjecture by George Pólya.

AchievementsPrinceton hired him at the rank of full professor with tenure just two years after he finished graduate school, making him the second youngest full professor at Princeton, after Charles Fefferman (professor at Princeton at age 24).

Bhargava has won several awards for his research, including the Morgan Prize, the Merten M. Hasse Prize from the MAA in 2003, a ClayResearch Fellowship, the Clay Research Award in 2005, and the Leonard M. and Eleanor B. Blumenthal Award for the Advancement of Research in Pure Mathematics. He was named one of Popular Science Magazine’s “Brilliant 10” in November 2002. He recently won theAmerican Mathematical Society's Cole Prize in number theory and the $10,000 SASTRA Ramanujan Prize, shared with Kannan Soundararajan, awarded by SASTRA in Tanjavur, India, for his outstanding contributions to number theory.

NARENDRA KARMARKAR

Born On: In 1957

Place:  Gwalior

Education

Karmarkar received his B.Tech in Electrical Engineering from IIT Bombay in 1978, M.S. from the California Institute of Technology and Ph.D. in Computer Science from the University of California, Berkeley.

Mathematics

Karmarkar's algorithm

Karmarkar's algorithm solves linear programming problems in polynomial time. These problems are represented by "n" variables and "m" constraints. The previous method of solving these problems consisted of problem representation by an "x" sided solid with "y" vertices, where the solution was approached by traversing from vertex to vertex. Karmarkar's novel method approaches the solution by cutting through the above solid in its traversal. Consequently, complex optimization problems are solved much faster using the Karmarkar algorithm. A practical example of this efficiency is the solution to a complex problem in communications network optimization where the solution time was reduced from weeks to days. His algorithm thus enables faster business and policy decisions. Karmarkar's algorithm has stimulated the development of several other interior point methods, some of which are used in current codes for solving linear programs.

Galois geometry

After working on the Interior Point Method, Karmarkar worked on a new architecture for supercomputing, based on concepts from finite geometry , especially projective geometry over finite fields.

Current investigations

Currently, he is synthesizing these concepts with some new ideas he calls sculpturing free space (a non-linear analogue of what has popularly been described as folding the perfect corner). This approach allows him to extend this work to the physical design of machines. He is now publishing updates on his recent work, including an extended abstract. This new paradigm was presented at IVNC, Polandon 16 July 2008, and at MIT on 25 July 2008. Some of the recent work is published at and Fab5 conference organised by MIT center for bits and atoms

Achievements

The Association for Computing Machinery awarded him the prestigious Paris Kanellakis Award in 2000 for his work.