The ancient Hindu symbol of circle with a dot in the middle known as bindu or bindhu, symbolizing the void and The negation of the self, was Probably instrumental in the use of a circle as a representation of the concept of zero.

The ancient Hindu symbol of circle with a dot in the middle known as bindu or bindhu, symbolizing the void and The negation of the self, was Probably instrumental in the use of a circle as a representation of the concept of zero.

Introduction:

Despite developing quite independently of Chinese (and probably also of Babylonian mathematics), some very advanced mathematical discoveries were made at a very early time in India.

On the basis of historical and archaeological conclusion, the history of the Indian subcontinent may be divided into seven major periods discussed in chapter I.

Mantras from the early Vedic period (before 1000 BC) invoke powers of ten from a hundred all the way up to a trillion, and provide evidence of the use of arithmetic operations such as addition, subtraction, multiplication, fractions, squares, cubes and roots. A 4th Century AD Sanskrit text reports Buddha enumerating numbers up to 1053, as well as describing six more numbering systems over and above these, leading to a number equivalent to 10421. Given that there are an estimated 1080 atoms in the whole universe, this is as close to infinity as any in the ancient world came. It also describes a series of iterations in decreasing size, in order to demonstrate the size of an atom, which comes remarkably close to the actual size of a carbon atom (about 70 trillionths of a metre).

As early as the 8th Century BC, long before Pythagoras, a text known as the “Sulba Sutras” (or "Sulva Sutras") listed several simple Pythagorean triples, as well as a statement of the simplified Pythagorean theorem for the sides of a square and for a rectangle (indeed, it seems quite likely that Pythagoras learned his basic geometry from the "Sulba Sutras"). The Sutras also contain geometric solutions of linear and quadratic equations in a single unknown, and give a remarkably accurate figure for the square root of 2, obtained by adding 1 + 1⁄3 + 1⁄(3 x 4) + 1⁄(3 x 4 x 34), which yields a value of 1.4142156, correct to 5 decimal places.

As early as the 3rd or 2nd Century BC, Jain mathematicians recognized five different types of infinities: infinite in one direction, in two directions, in area, infinite everywhere and perpetually infinite. Ancient Buddhist literature also demonstrates a prescient awareness of indeterminate and infinite numbers, with numbers deemed to be of three types: countable, uncountable and infinite.

Like the Chinese, the Indians early discovered the benefits of a decimal place value number system, and were certainly using it before about the 3rd Century AD. They refined and perfected the system, particularly the written representation of the numerals, creating the ancestors of the nine numerals that (thanks to its dissemination by medieval Arabic mathematicians) we use across the world today, sometimes considered one of the greatest intellectual innovations of all time.

The Indians were also responsible for another hugely important development in mathematics. The earliest recorded usage of a circle character for the number zero is usually attributed to a 9th Century engraving in a temple in Gwalior in central India. But the brilliant conceptual leap to include zero as a number in its own right is usually credited to the 7th Century Indian mathematicians Brahmagupta - or possibly another Indian, Bhaskara I - even though it may well have been in practical use for centuries before that. The use of zero as a number which could be used in calculations and mathematical investigations would revolutionize mathematics.

CONTRIBUTIONS OF INDIAN MATHEMATICIANS BEFORE 20 TH CENTURY

I. Aryabhata:

Aryabhata (475 A.D. -550 A.D.) is the first well known Indian mathematician. Born in Kerala, he completed his studies at the university of Nalanda. In the section Ganita (calculations) of his astronomical treatise Aryabhatiya (499 A.D.), he made the fundamental advance in finding the lengths of chords of circles, by using the half chord rather than the full chord method used by Greeks. He gave the value of as 3.1416, claiming, for the first time, that it was an approximation. (He gave it in the form that the approximate circumference of a circle of diameter 20000 is 62832.) He also gave methods for extracting square roots, summing arithmetic series, solving indeterminate equations of the type ax -by = c, and also gave what later came to be known as the table of Sines. He also wrote a text book for astronomical calculations, Aryabhatasiddhanta. Even today, this data is used in preparing Hindu calendars (Panchangs). In recognition to his contributions to astronomy and mathematics, India's first satellite was named Aryabhata. 

He was the first person to say that Earth is spherical and it revolves around the sun.

He gave the formula (a + b)2 = a2 + b2 + 2ab He taught the method of solving the following problems:

Aryabhata wrote the Aryabhatiya. He described the important fundamental principles of mathematics in 332 shlokas. The treatise contained:

Quadratic equations Trigonometry The value of π, correct to 4 decimal places.

Aryabhata also wrote the Arya Siddhanta, which is now lost. Aryabhata’s contributions include:Trigonometry:

Introduced the trigonometric functions. Defined the sine (jya) as the modern relationship between half an angle and half a

chord. Defined the cosine (kojya). Defined the versine (ukramajya). Defined the inverse sine (otkram jya). Gave methods of calculating their approximate numerical values. Contains the earliest tables of sine, cosine and versine values, in 3.75° intervals

from 0° to 90°, to 4 decimal places of accuracy. Contains the trigonometric formula sin (n + 1) x – sin nx = sin nx – sin (n – 1) x –

(1/225) sin nx. Spherical trigonometry.

Arithmetic: Continued fractions.

Algebra: Solutions of simultaneous quadratic equations. Whole number solutions of linear equations by a method equivalent to the modern

method. General solution of the indeterminate linear equation.

Mathematical astronomy: Proposed for the first time, a heliocentric solar system with the planets spinning on

their axes and following an elliptical orbit around the Sun. Accurate calculations for astronomical constants, such as the: Solar eclipse. Lunar eclipse. The formula for the sum of the cubes, which was an important step in the

development of integral calculus.

Calculus: Infinitesimals: In the course of developing a precise mapping of the lunar eclipse, Aryabhatta was

obliged to introduce the concept of infinitesimals (tatkalika gati) to designate the near instantaneous motion of the moon.

Differential equations: He expressed the near instantaneous motion of the moon in the form of a basic

differential equation.

Exponential function: He used the exponential function e in his differential equation of the near

instantaneous motion of the moon.

Varahamihira Varahamihira (505-587) produced the Pancha Siddhanta (The Five Astronomical Canons). He made important contributions to trigonometry, including sine and cosine tables to 4 decimal places of accuracy and the following formulas relating sine and cosinefunctions:

BrahmaguptaBrahmagupta (Sanskrit: ब्रह्मगुप्-597–668 CE) was an Indian mathematician and astronomer who wrote two important works in Mathematics and Astronomy: the Brāhmasphuṭasiddhānta (Extensive Treatise of Brahma) (628), a theoretical treatise, and the Khaṇḍakhādyaka, a more practical text. There are reasons to believe that Brahmagupta originated from Bhinmal.

Brahmagupta's Brahmasphuṭasiddhanta is the first book that mentions zero as a number[citation needed], hence Brahmagupta is considered the first to formulate the concept of zero. But his description of division by zero differs from our modern understanding.

He gave rules of using zero with negative and positive numbers. Zero plus a positive number is the positive number and negative number plus zero is a negative number etc. The Brahmasphuta siddhanta 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. In chapter eighteen of his Brahmasphutasiddhanta, Brahmagupta describes operations on negative numbers. He first describes addition and subtraction,

He discovered the position of nine planets and stated that these planets revolve around the sun. He also stated the correct number of days in a year that is 365.

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

Four fundamental operations (addition, subtraction, multiplication and division) were known to many cultures before Brahmagupta. This current system is based on the Hindu Arabic number system and first appeared in Brahmasphutasiddhanta. Brahmagupta describes the multiplication as thus “The multiplicand is repeated like a string for cattle, as often as there are integrant portions in the multiplier and is repeatedly multiplied by them and the products are added together. It is multiplication. Or the multiplicand is repeated as many times as there are component parts in the multiplier”. 

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

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.

In chapter twelve of his Brahmasphutasiddhanta, Brahmagupta provides a formula useful for generating Pythagorean triples:

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.

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 is where

.

In verse 40, he gives values of π.

Trigonometry:In Chapter 2 of his Brahmasphutasiddhanta, entitled Planetary True Longitudes, Brahmagupta presents a sine table.

Interpolation formula: In 665 Brahmagupta devised and used a special case of the Newton–Stirling interpolation formula of the second-order to interpolate new values of the sine function from other values already tabulated.[22] The formula gives an estimate for the value of a function   at a value a + xh of its argument (with h > 0 and −1 ≤ x ≤ 1) when its value is already known at a − h, a and a + h.

The formula for the estimate is:

.

where Δ is the first-order forward-difference operator, i.e.

Some of the important contributions made by Brahmagupta in astronomy are: methods for calculating the position of heavenly bodies over time (ephemerides), their rising and setting, conjunctions, and the calculation of solar and lunar eclipses.[27] Brahmagupta criticized the Puranic view that the Earth was flat or hollow. Instead, he observed that the Earth and heaven were spherical.

Brahma Gupta was born in 598A.D in Pakistan. He gave four methods of multiplication. He gave the following formula, used in G.P series a + ar + ar2 + ar3 +……….. + arn-1 = (arn-1) ÷ (r – 1) He gave the following formulae : Area of a cyclic quadrilateral with side a, b, c, d= √(s -a)(s- b)(s -c)(s- d) where 2s

= a + b + c + d Length of its diagonals = 

Mahavira : Mahavira was a 9th-century Indian mathematician from Gulbarga who asserted that the square root of a negative number did not exist. He gave the sum of a series whose terms are squares of an arithmetical progression and empirical rules for area and perimeter of an ellipse. He was patronised by the great Rashtrakuta king Amoghavarsha. Mahavira was the author of Ganit Saar Sangraha. He separated Astrology from Mathematics. He expounded on the same subjects on which Aryabhata and Brahmagupta contended, but he expressed them more clearly. He is highly respected among Indian Mathematicians, because of his establishment of terminology for concepts such as equilateral, and isosceles triangle; rhombus; circle and semicircle. Mahavira’s eminence spread in all

South India and his books proved inspirational to other Mathematicians in Southern India.

Higher Order Equations:Mahavira solved higher order equations of n degree of the forms: axn=q and

Formula For Cyclic Quadrilateral:Mahavira expressed characteristics of a cyclic quadrilateral, like Brahmagupta did previously. He also established equations for the sides and diagonal of Cyclic Quadrilateral.If sides of Cyclic Quadrilateral are a,b,c,d and its diagonals are x and y while

and . Then xy = ac+bd.

Bhāskara II Bhāskara (also known as Bhāskarāchārya ("Bhāskara the teacher") and as

Bhāskara II to avoid confusion with Bhāskara I) (1114–1185), was anIndian mathematician and astronomer. He was born near Vijjadavida (Bijapur in modern Karnataka). Bhāskara is said to have been the head of an astronomical observatory at Ujjain, the leading mathematical center of medieval India. He lived in the Sahyadri region (Patnadevi, in Jalgaon district, Maharashtra). 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 RenaissanceEuropean 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.[9]

• Solutions of Diophantine equations of the second order, such as 61x2 + 1 = y2. This very equation was posed as a problem in 1657 by the French mathematicianPierre 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.[citation needed]

• Preliminary concept of mathematical analysis.• Preliminary concept of infinitesimal calculus, along with notable contributions

towards integral calculus.[11]• 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. (See Trigonometry section below.)

Arithmetic:Bhaskara's arithmetic text Leelavati covers the topics of definitions, arithmetical terms, interest computation, arithmetical and geometrical progressions, plane geometry, solid geometry, the shadow of the gnomon, methods to solve indeterminate equations, and combinations.Lilavati is divided into 13 chapters and covers many branches of mathematics, arithmetic, algebra, geometry, and a little trigonometry and mensuration. More specifically the contents include:

1. Definitions.2. Properties of zero (including division, and rules of operations with zero).3. Further extensive numerical work, including use of negative numbers and surds.4. Estimation of π.5. Arithmetical terms, methods of multiplication, and squaring.6. Inverse rule of three, and rules of 3, 5, 7, 9, and 11.7. Problems involving interest and interest computation.8. Indeterminate equations (Kuttaka), integer solutions (first and second order). His

contributions to this topic are particularly important, since the rules he gives are (in effect) the same as those given by the renaissance European mathematicians of the 17th century, yet his work was of the 12th century. Bhaskara's method of solving was an improvement of the methods found in the work of Aryabhata and subsequent mathematicians.

His work is outstanding for its systemization, improved methods and the new topics that he has introduced. Furthermore the Lilavati contained excellent recreate problems.Algebra:His Bijaganita ("Algebra") was a work in twelve chapters. It was the first text to recognize that a positive number has two square roots (a positive and negative square root). His work Bijaganita is effectively a treatise on algebra and contains the following topics:

Positive and negative numbers. Zero .

The 'unknown' (includes determining unknown quantities).

Determining unknown quantities.

Surds  (includes evaluating surds).

Kuttaka (for solving indeterminate equations and Diophantine equations).

Simple equations (indeterminate of second, third and fourth degree).

Simple equations with more than one unknown.

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

Solutions of indeterminate equations of the second, third and fourth degree.

Quadratic equations.

Quadratic equations with more than one unknown.

Operations with products of several unknowns.

Bhaskara derived a cyclic, chakravala method for solving indeterminate quadratic equations of the form ax2 + bx + c = y.[13] Bhaskara's method for finding the solutions of the problem Nx2 + 1 = y2 (the so-called "Pell's equation") is of considerable importance.Trigonometry:The Siddhānta Shiromani (written in 1150) demonstrates Bhaskara's knowledge of trigonometry, including the sine table and relationships between different trigonometric functions. He also discovered spherical trigonometry, along with other interesting trigonometrical results. In particular Bhaskara seemed more interested in trigonometry for its own sake than his predecessors who saw it only as a tool for calculation. Among the many interesting results given by Bhaskara, discoveries first found in his works include computation of sines of angles of 18 and 36 degrees, and the now well known formulae for sin(a+b)  and sin(a-b).

Calculus:His work, the Siddhānta Shiromani, is an astronomical treatise and contains many theories not found in earlier works. Preliminary concepts of infinitesimal calculus and mathematical analysis, along with a number of results in trigonometry, differential calculus and integral calculus that are found in the work are of particular interest.

Evidence suggests Bhaskara was acquainted with some ideas of differential calculus.[14] It seems, however, that he did not understand the utility of his researches, and thus historians of mathematics generally neglect this achievement. Bhaskara also goes deeper into the 'differential calculus' and suggests the differential coefficient vanishes at an extremum value of the function, indicating knowledge of the concept of 'infinitesimals'.

There is evidence of an early form of Rolle's theorem in his work If  f(a)=d(b)=0 then  for some x with  He gave the result that if   then  , thereby finding

the derivative of sine, although he never developed the notion of derivatives. Bhaskara uses this result to work out the position angle of the ecliptic, a quantity

required for accurately predicting the time of an eclipse. In computing the instantaneous motion of a planet, the time interval between

successive positions of the planets was no greater than a truti, or a 1⁄33750 of a second, and his measure of velocity was expressed in this infinitesimal unit of time.

He was aware that when a variable attains the maximum value, its differentialvanishes.

He also showed that when a planet is at its farthest from the earth, or at its closest, the equation of the centre (measure of how far a planet is from the position in which it is predicted to be, by assuming it is to move uniformly) vanishes. He therefore concluded that for some intermediate position the differential of the equation of the centre is equal to zero. In this result, there are traces of the general mean value theorem, one of the most important theorems in analysis, which today is usually derived from Rolle's theorem. The mean value theorem was later found by Parameshvara in the 15th century in the Lilavati Bhasya, a commentary on Bhaskara's Lilavati.

Madhava (1340–1425) and the Kerala School mathematicians (including Parameshvara) from the 14th century to the 16th century expanded on Bhaskara's work and further advanced the development of calculus in India.

Astronomy:Using an astronomical model developed by Brahmagupta in the 7th century, Bhaskara accurately defined many astronomical quantities, including, for example, the length of the sidereal year, the time that is required for the Earth to orbit the Sun, as 365.2588 days which is the same as in Suryasiddhanta. The modern accepted measurement is 365.2563 days, a difference of just 3.5 minutes. His mathematical astronomy text Siddhanta Shiromani is written in two parts: the first part on mathematical astronomy and the second part on the sphere.The twelve chapters of the first part cover topics such as:

1. Mean longitudes of the planets.2. True longitudes of the planets.3. The three problems of diurnal rotation.4. Syzygies .

5. Lunar eclipses .6. Solar eclipses .7. Latitudes  of the planets.8. Sunrise equation 9. The Moon's crescent.10. Conjunctions  of the planets with each other.11. Conjunctions of the planets with the fixed stars.12. The paths of the Sun and Moon.

The second part contains thirteen chapters on the sphere. It covers topics such as:1. Praise of study of the sphere.2. Nature of the sphere.3. Cosmography  and geography.4. Planetary mean motion.5. Eccentric  epicyclic model of the planets.6. The armillary sphere.7. Spherical trigonometry .8. Ellipse  calculations. 9. First visibilities of the planets.10. Calculating the lunar crescent.11. Astronomical instruments.12. The seasons.13. Problems of astronomical calculations.

Engineering:The earliest reference to a perpetual motion machine date back to 1150, when Bhāskara II described a wheel that he claimed would run forever. Bhāskara II used a measuring device known as Yasti-yantra. This device could vary from a simple stick to V-shaped staffs designed specifically for determining angles with the help of a calibrated scale.

He was born in a village of Mysore district. He was the first to give that any number divided by 0 gives infinity (00). He has written a lot about zero, surds, permutation and combination. He wrote, “The hundredth part of the circumference of a circle seems to be

straight. Our earth is a big sphere and that’s why it appears to be flat.” He gave the formulae like sin(A ± B) = sinA.cosB ± cosA.sinB

Madhva : Madhava sometimes called the greatest mathematician-astronomer of medieval India. He came from the town of Sangamagrama in Kerala, near the southern tip of India, and founded the Kerala School of Astronomy and Mathematics in the late 14th Century.

Although almost all of Madhava's original work is lost, he is referred to in the work of later Kerala mathematicians as the source for several infinite series expansions (including the sine, cosine, tangent and arctangent functions and the value of π), representing the first steps from the traditional finite processes of algebra to considerations of the infinite, with its implications for the future development of calculus and mathematical analysis.

Unlike most previous cultures, which had been rather nervous about the concept of infinity, Madhava was more than happy to play around with infinity, particularly infinite series. He showed how, although one can be approximated by adding a half plus a quarter plus an eighth plus a sixteenth, etc, (as even the ancientEgyptians and Greeks had known), the exact total of one can only be achieved by adding up infinitely many fractions.

But Madhava went further and linked the idea of an infinite series with geometry and trigonometry. He realized that, by successively adding and subtracting different odd number fractions to infinity, he could home in on an exact formula for π (this was two centuries before Leibniz was to come to the same conclusion in Europe). Through his application of this series, Madhava obtained a value for π correct to an astonishing 13 decimal places.

He went on to use the same mathematics to obtain infinite series expressions for the sine formula, which could then be used to calculate the sine of any angle to any degree of accuracy, as well as for other trigonometric functions like cosine, tangent and arctangent. Perhaps even more remarkable, though, is that he also gave estimates of the error term or correction term, implying that he quite understood the limit nature of the infinite series.

Madhava’s use of infinite series to approximate a range of trigonometric functions, which were further developed by his successors at the Kerala School, effectively laid the foundations for the later development of calculus and analysis, and either he or his disciples developed an early form of integration for simple functions. Some historians have suggested that Madhava's work, through the writings of the Kerala School, may have been transmitted to Europe via Jesuit missionaries and traders who were active around the ancient port of Cochin (Kochi) at the time, and may have had an influence on later European developments in calculus.

Among his other contributions, Madhava discovered the solutions of some transcendental equations by a process of iteration, and found approximations for some transcendental numbers by continued fractions. In astronomy, he discovered a procedure to determine the positions of the Moon every 36 minutes, and methods to estimate the motions of the planets.

Srinivasa Aaiyangar Ramanujan : Srinivasa Aaiyangar Ramanujan is undoubtedly the most celebrated Indian

Mathematical genius. He was born During an illness in England, Hardy visited Ramanujan in the hospital. When Hardy remarked that he had taken taxi number 1729, a singularly unexceptional number, Ramanujan immediately responded that this number was actually quite remarkable: it is the smallest integer that can be represented in two ways by the sum of two cubes: 1729=1³+12³=9³+10³ since then the number 1729 is called Ramanujan’s number.

He was born at Erode in Tamil Nadu on December 22, 1887. He failed in English in Intermediate, so his formal studies were stopped but his self-study of mathematics continued.

He made contributions to the analytical theory of numbers, infinite series, Mathematical analysis and continued fractions.

An equation for him has no meaning unless it expresses a thought of God By 17, Ramanujan conducted his own mathematical research on Bernoulli

numbers and the Euler- Mascheroni Constant. Encouraged by Seshu Iyer, Ramanujan wrote a letter and he sent a set of 120

theorems to Professor Hardy of Cambridge, Fellow of Trinity College. Hardy received his letter on 16-02-1913 as a result he invited Ramanujan to England, which changed the lives of both Ramanujan and Hardy.

Ramanujan showed that any big number can be written as sum of not more than four prime numbers.

He showed that how to divide the number into two or more squares or cubes. Ramanujan sailed to England on 17-3-1914. The English men in no time found

that Ramanujan had extraordinary memory. Out of 32 papers, 7 were published in Collaboration with hardy.

He was the first Indian mathematician to get FRS (Fellow of the Royal Society-The Royal Society of London for Improving Natural Knowledge) at the age of 30 on 28-2-1918.

Both Hardy and Ramanujan Published a joint Paper in the Proceedings of Royal mathematical society on a formula P(n) for partitions, It is a growing achievement in the theory of partition.

P(n) =the no of ways a given integer n can be represented as a sum of other +ve integers for example,

P(4) = 5. (1+1+1+1=4, 3+1=4, 2+2=4, 1+2+1=4, and 4). On 100 scale Hardy rated as follows.

Hardy gave himself 25% To Little wood 30% To David Hilbert (German mathematician) 80 % To Srinivasa Ramanujan 100 % .

Ramanujan the mathematical genius stayed during last year of his life in chetput in the house of Diwan Bahadur T Namberumal Chetty who took up the responsibility of looking after Ramanujan till his death.

His last letter to Hardy on 12-1-1920 was on mock-theta functions. The mathematical giants heart stopped beating on 26-4-1920, who lived 32 years,

4 months and 4 days are all powers of 2. An English mathematician G. H. Hardy said that Ramanujan's talent was, by the

prominent to be in the same league as legendary mathematicians such as Euler, Gauss, Newton and Archimedes. 

It was in recognition of Srinivasa Ramanujan contribution to mathematics the Government of India decided to celebrate Ramanujan's birthday 'December 22' as the National Mathematics Day every year and to celebrate 2012 as the National Mathematical Year.

APPLICATIONS OF HIS THEORY His work on modular forms is exactly what physicists when they work on a 26-

dimensional mathematical models on super theory in Astronomy. His theory has wide application in Fast Algorithms, Molecular System in

Statistical Dynamics, Cryptology, Space research of which he knew nothing William GOSPER of USA developed an algorithm 175 lakhs digits to find the

value of π and the methods he developed are from Ramanujan results. Srinivasa Ramanujan I Hypothesis, as Hardy called it, or the TAU conjecture as it

more generally known was proved by DELIGNE a Belgian Mathematician in 1974, who was subsequently awarded the FIELDS MEDAL, the mathematical community’s counter part of NOBEL Prize, Using the tools of ALGEBRAIC GEOMETRY – this helped Ramanujan’s reputation all the more.

The Ramanujan Prize In 2004, the Abdus Salam International Centre for Theoretical Physics (ICTP)

announced the creation of a Ramanujan Prize of $ 15,000 to be awarded annually to a mathematician under 45 of the 3rd world. . Ramanujan Prize of the ICTP is also supported by Abel foundation.

Sastra Ramanujan Prize (Sastra -Shanmukha Arts, Science Technology Academy). Award carries $

10,000, age limit is 32, since Ramanujan achieved so much credit and brief life span of 32 years.

P.C.Mahalanobis ( 29 June 1893 – 28 June 1972)

He founded the Indian Statistical Research Institute in Calcutta. In 1958, he started the National Sample Surveys which gained international fame.. Professor P.C. Mahalanobis’ contribution to the subject of statistics is incredibly varied. Mahalanobis had held the position of Statistical Advisor to the Government of India and Member, Planning Commission, beside many other distinguished appointments. The formulation of the D²-Statistics, derivation of its properties and its application are some of his most profound contributions. He received the Weldon Medal from Oxford University in 1944 and was elected a Fellow of the Royal Society, London, in 1945 for his fundamental contributions to statistics, particularly in the area of large-scale sample surveys. As Chairman of the United Nations Sub-Commission on Statistical Sampling, a position to which he was appointed in 1947, he tirelessly advocated the use of sampling methods to be extended to all parts of the world. As the Chairman of the UN Statistical Commission, he also helped in the spread of robust and extremely useful statistical ideas to other countries. As a pioneer to government planning, his background to a number of initiatives in de-centralized planning is meritorious, one of them being the setting up of the National Sample Survey (NSS) entrusted with key responsibilities for the design, collection and analysis of sample survey data.

CONTRIBUTIONS OF INDIAN MATHEMATICIANS IN 20 TH CENTURY

D.R. Karakar :Dattaraya Ramchandra Kaprekar (1905–1986) was an Indian recreational mathematician who described several classes of natural numbers including the Kaprekar, Harshad and Self numbers and discovered the Kaprekar constant, named after him. Despite having no formal postgraduate training and working as a schoolteacher, he published extensively and became well known in recreational mathematics circles. Fond of numbers.

Well known for "Kaprekar Constant" 6174. Take any four digit number in which all digits are not alike. Arrange its digits in descending order and subtract from it the number formed by arranging the digits in ascending order. If this process is repeated with reminders, ultimately number 6174 is obtained, which then generates itself. Thus, starting with 1234, we have4321 − 1234 = 3087, then

8730 − 0378 = 8352, and8532 − 2358 = 6174.

Kaprekar numberAnother class of numbers Kaprekar described are the Kaprekar numbers.[ A Kaprekar number is a positive integer with the property that if it is squared, then its representation can be partitioned into two positive integer parts whose sum is equal to the original number (e.g. 45, since 452=2025, and 20+25=45, also 9, 55, 99 etc.) However, note the restriction that the two numbers are positive; for example, 100 is not a Kaprekar number even though 1002=10000, and 100+00 = 100. This operation, of taking the rightmost digits of a square, and adding it to the integer formed by the leftmost digits, is known as the Kaprekar operation.

Some examples of Kaprekar numbers in base 10, besides the numbers 9, 99, 999, …, are (sequence A006886 in OEIS):

Number Square Decomposition703 703² = 494209 494+209 = 7032728 2728² = 7441984 744+1984 = 27285292 5292² = 28005264 28+005264 = 5292857143 857143² = 734694122449 734694+122449 = 857143

Devlali or Self numberIn 1963, Kaprekar defined the property which has come to be known as self numbers, which are integers that cannot be generated by taking some other number and adding its own digits to it. For example, 21 is not a self number, since it can be generated from 15: 15 + 1 + 5 = 21. But 20 is a self number, since it cannot be generated from any other integer. He also gave a test for verifying this property in any number. These are sometimes referred to as Devlali numbers (after the town where he lived); though this appears to have been his preferred designation, the term self number is more widespread. Sometimes these are also designated Colombian numbers after a later designation.

Harshad numberKaprekar also described the Harshad numbers which he named harshad, meaning "giving joy" (Sanskrit harsha, joy +da taddhita pratyaya, causative); these are defined by the property that they are divisible by the sum of their digits. Thus 12, which is divisible by 1 + 2 = 3, is a Harshad number. These were later also called Niven numbers after a 1977 lecture on these by the Canadian mathematician Ivan M. Niven. Numbers which are Harshad in all bases (only 1, 2, 4, and 6) are called all-Harshad numbers. Much work has been done on Harshad numbers, and their distribution, frequency, etc. are a matter of considerable interest in number theory today.

Demlo number

Kaprekar also studied the Demlo numbers, named after a train station 30 miles from Bombay on the then G. I. P. Railway where he had the idea of studying them. These are the numbers 1, 121, 12321, …, which are the squares of the repunits 1, 11, 111, ….

Sri Lakkoju Sanjeevaraya Sharma: anjeevaraya Sharma born on 22nd Novemebr 1907 in Kadapa District Kallur

village Proddatur mandal, Andhra Pradesh, India. Renowned Mathematics Genius, who won 13 Gold Medals worldwide, and blind

by birth. In one Avadhaanam Sanjeevaraya Sharma has given answer to the question “2 to

the Power of 103″ within seconds, the resultant number has 32 digits. But he is not a educated person and even He is a blind by birth. He has conducted around 6000 Ganitha avadhaanas across the world.

SHe was Identified as a blind child when he is born ,so People around told just put a “Vadla ginja” in his mouth to kill him, he servived Even after doing that.

He gains all the knowledge by just listening. In his time there is no Braille system to which helps blind people for reading.He was listening to his sister whenever she reads the lessons taught in School. Without knowing how 1,2,3 digits looks he became Ganitha Brahma.He learned the Voileen also.

Anibisent,Nehru,Rajendra Prasad and Shakunthala Devi all saw his Avadhanams. Shakunthala Devi agreed that he is talented than her. Once British ViceRoy told If he would have Born in Our Country We could have worshipped him by placing his Idol in the center of the City.

He is also known as Anka VidyaSagara, Vishwa Sankhya Acharya. In his last days he spent his time in the Premises of Kala Hasti Eeshwara Temple

by Playing the Voileen. He died on Dec 2nd 1997. Throughout his life he lived in Poverty. He is one among the world’s top most 6 Mathematicians.

 

C.R.Rao: (born 10 September 1920)C R Rao was born in Hadagali, Karnataka, India into a Telugu Velama family. He received an M.Sc. in mathematics from Andhra University and an M.A. in statistics from Calcutta University in 1943. He was among the first few people in the world to hold a Master's degree in Statistics.

Rao worked at the Indian Statistical Institute and the Anthropological Museum in Cambridge before acquiring a Ph.D. degree at King's College in Cambridge University

under R.A. Fisher in 1948, to which he added a Sc.D. degree, also from Cambridge, in 1965.

He held several important positions, as the Director of the Indian Statistical Institute, Jawaharlal Nehru Professor and National Professor in India, University Professor at the University of Pittsburgh and Eberly Professor and Chair of Statistics and Director of the Center for Multivariate Analysis at the Pennsylvania State University. As Head and later Director of the Research and Training School at the Indian Statistical Institute for a period of over 40 years, Rao developed research and training programs and produced several leaders in the field of Mathematics. On the basis of Dr. Rao's recommendation, the ASI (The Asian Statistical Institute) now known as Statistical Institute for Asia and Pacific was established in Tokyo to provide training to statisticians working in government and industrial organizations.

Among his best-known discoveries are the Cramér–Rao bound and the Rao–Blackwell theorem both related to the quality of estimators. Other areas he worked in include multivariate analysis, estimation theory, and differential geometry. His other contributions include the Fisher–Rao Theorem, Rao distance, and orthogonal arrays. He is the author of 14 books and has published over 400 journal publications.

Rao has received over 37 honorary doctoral degrees from universities in 19 countries around the world and numerous awards and medals for his contributions to statistics and science. He is a member of eight National Academies in India, the United Kingdom, the United States, and Italy. Rao was awarded the United States National Medal of Science, that nation's highest award for lifetime achievement in fields of scientific research, in June 2002. The latest addition to his collection of awards is the India Science Award for 2010, the highest honour conferred by the government of India in scientific domain. He has most recently been honoured with his 38th honorary doctorate by the university at Buffalo in 2013.

He has been the President of the International Statistical Institute, Institute of Mathematical Statistics (USA), and the International Biometric Society. He was inducted into the Hall of Fame of India's National Institution for Quality and Reliability (Chennai Branch) for his contribution to industrial statistics and the promotion of quality control programs in industries.

Rao has been honoured by numerous colloquia, honorary degrees, and festschrifts and was awarded the US National Medal of Science in 2002. The American Statistical Association has described him as "a living legend whose work has influenced not just statistics, but has had far reaching implications for fields as varied as economics, genetics, anthropology, geology, national planning, demography, biometry, and medicine."The Times of India listed Rao as one of the top 10 Indian scientists of all time.

Harish Chandra:Harish-Chandra FRS (Harish Chandra Mehrotr ; 11 October 1923 – 16 October 1983) was anIndian American mathematician and physicist who did fundamental work in representation theory, especially harmonic analysis on semisimple Lie groups. He was influenced by the mathematicians Hermann Weyl and Claude Chevalley. From 1950 to 1963 he was at the Columbia University and worked 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 analogues of the Peter–Weyl theory in the non-compact case.

He is also known for work with Armand Borel on 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 member of the National Academy of Sciences of the U.S. and a Fellow of the Royal Society. 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. In 1981, he received an honorary degree from Yale University.

The Indian Government named the Harish-Chandra Research Institute, an institute dedicated to Theoretical Physics and Mathematics, after him.

Robert Langlands wrote in a biographical article of Harish-Chandra: “He was considered for the Fields Medal in 1958, but a forceful member of the selection committee in whose eyes Thom was a Bourbakist was determined not to have two. So Harish-Chandra, whom he also placed on the Bourbaki camp, was set aside”.

Shakuntala Devi :She was born in 1939. Devi traveled the world demonstrating her arithmetic talents, including a tour of Europe in 1950 and a performance in New York City in 1976.  In 1988, she traveled to the U.S. to have her abilities studied by Arthur Jensen, a professor of psychology at the University of California, Berkeley. Jensen tested her performance of several tasks, including the calculation of large numbers. Examples of the problems presented to Devi included calculating the cube root of 61,629,875, and the seventh root of 170,859,375. Jensen reported that Devi provided the solution to the aforementioned problems (395 and 15, respectively) before Jensen could copy them down in his notebook. Jensen published his findings in the academic journal Intelligence in 1990. In addition to her work as a mental calculator, Devi was an astrologer and an author of several books, including cookbooks and novels.

Achievements

In 1977, at Southern Methodist University, she was asked to give the 23rd root of a 201-digit number; she answered in 50 seconds. Her answer—546,372,891—was confirmed by calculations done at the U.S. Bureau of Standards by the UNIVAC 1101 computer, for which a special program had to be written to perform such a large calculation.

On June 18, 1980, she demonstrated the multiplication of two 13-digit numbers—7,686,369,774,870 × 2,465,099,745,779—picked at random by the Computer Department of Imperial College, London. She correctly answered 18,947,668,177,995,426,462,773,730 in 28 seconds. This event is mentioned in the 1982 Guinness Book of Records. Writer Steven Smith states that the result is "so far superior to anything previously reported that it can only be described as unbelievable".

In 1980, she gave the product of two, thirteen digit numbers within 28 seconds, many countries have invited her to demonstrate her extraordinary talent.

In Dallas she competed with a computer to see who give the cube root of 188138517 faster, she won. At university of USA she was asked to give the 23rd root of

91674867692003915809866092758538016248310668014430862240712651642793465704086709659 32792057674808067900227830163549248523803357453169351119035965775473400756818688305 620821016129132845564895780158806771.

She answered in 50seconds. The answer is 546372891. It took a UNIVAC 1108 computer, full one minute (10 seconds more) to confirm that she was right after it was fed with 13000 instructions.

Now she is known to be Human Computer. She died in the hospital on April 21, 2013. On November 4, 2013, Devi was honored with a Google Doodle for what would

have been her 84th birthday

Some of her books include: Astrology for You (New Delhi: Orient, 2005). ISBN 978-81-222-0067-6 Book of Numbers (New Delhi: Orient, 2006). ISBN 978-81-222-0006-5 Figuring: The Joy of Numbers (New York: Harper & Row, 1977), ISBN 978-0-

06-011069-7, OCLC 4228589 In the Wonderland of Numbers (New Delhi: Orient, 2006). ISBN 978-81-222-

0399-8 Mathability: Awaken the Math Genius in Your Child (New Delhi: Orient, 2005).

ISBN 978-81-222-0316-5 More Puzzles to Puzzle You (New Delhi: Orient, 2006). ISBN 978-81-222-0048-5 Perfect Murder (New Delhi: Orient, 1976), OCLC 3432320

Puzzles to Puzzle You (New Delhi: Orient, 2005). ISBN 978-81-222-0014-0 Super Memory: It Can Be Yours (New Delhi: Orient, 2011). ISBN 978-81-222-

0507-7; (Sydney: New Holland, 2012). ISBN 978-1-74257-240-6, OCLC 781171515

The World of Homosexuals (Vikas Publishing House, 1977), ISBN 978-0706904789

Narendra Karmaker :Narendra Karmarkar was born in Gwalior In 1957. After securing an All India Rank 1 in the Joint Entrance Examination conducted by the prestigious IITs (IIT-JEE), he took admission in the Indian Institute of Technology Bombay. 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.

He invented a polynomial algorithm for linear programming also known as the interior point method. The algorithm is a cornerstone in the field of Linear Programming. He published his famous result in 1984 while he was working for Bell Laboratories in New Jersey. Karmarkar was a professor at the Tata Institute of Fundamental Research in Mumbai. He is currently working on a new architecture for supercomputing. Some of the ideas are published at the IEEE website. Fab5 conference organized by MIT center for bits and atoms.

Karmarkar received a number of awards for his algorithm, among them:

Paris Kanellakis Award, 2000 given by The Association for Computing Machinery.

Srinivasa Ramanujan Birth Centenary Award for 1999, presented by the Prime Minister of India.

Distinguished Alumnus Award, Indian Institute of Technology, Bombay, 1996 Distinguished Alumnus Award, Computer Science and Engineering, University of

California, Berkeley (1993) Fulkerson Prize in Discrete Mathematics given jointly by the American

Mathematical Society & Mathematical Programming Society (1988) Fellow of Bell Laboratories (1987- ) Texas Instruments Founders’ Prize (1986) Marconi International Young Scientist Award (1985) Frederick W. Lanchester Prize of the Operations Research Society of America for

the Best Published Contributions to Operations Research (1984) President of India gold medal, I.I.T. Bombay (1978) National Science Talent Award in Mathematics, India (1972, India)

Karmarkar's algorithmKarmarkar'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.

Paris Kanellakis AwardThe Association for Computing Machinery awarded him the prestigious Paris Kanellakis Award in 2000 for his work on polynomial time interior point methods for linear programming.

Galois geometryAfter 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 investigationsCurrently, 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, Poland on 16 July 2008, and at MIT on 25 July 2008. He also delivered a lecture on his recent work at IIT Bombay in September 2013. Some of his recent work is published at ieeexplore.

Indian mathematics

I.Ancient MathematiciansApastambaBaudhayanaKatyayanaManava

PāṇiniPingala

Yajnavalkya

II. Classical MathematiciansĀryabhaṭa IĀryabhaṭa IIBhāskara IBhāskara IIMelpathur Narayana BhattathiriBrahmadevaBrahmaguptaBrihaddeshiGovindasvāmiHalayudhaJyeṣṭhadevaKamalakaraMādhava of SaṅgamagrāmaMahāvīraMahendra SūriMunishvaraNarayana PanditParameshvaraAchyuta PisharatiJagannatha SamratNilakantha SomayajiŚrīpatiSridharaGangesha UpadhyayaVarāhamihiraSankara VariarVirasena

III. Modern Mathematicians Shreeram Shankar AbhyankarRaj Chandra BoseSatyendra Nath BoseHarish-ChandraSubrahmanyan ChandrasekharDijen K. Ray-ChaudhuriSarvadaman ChowlaGopinath KallianpurNarendra KarmarkarPrasanta Chandra MahalanobisJayant NarlikarVijay Kumar Patodi

Ganesh PrasadC. P. RamanujamSrinivasa RamanujanC. R. RaoSamarendra Nath RoySharadchandra Shankar ShrikhandeS. R. Srinivasa Varadhan more