View of MATHEMATICAL HERITAGE OF INDIA

(1)

Vol. 05,Special Issue 02, (IC-IRSHEM-2020) February 2020, Available Online: www.ajeee.co.in/index.php/AJEEE

1

MATHEMATICAL HERITAGE OF INDIA SONAM CHHABRA

Assistant Professor, Department of Mathematics, Surendera Group of Institution, Sri Ganganagar, Rajasthan

Abstract:- This essay will testify the History of Mathematics in India and also features a brief overview of some of the Indian Mathematicians of ancient time as well as of modern time. The origin of the term mathematics is in the Greek word `mathemata‟, which was used to the utmost in early writings to specify any subject of instruction or study. As learning advanced, it was found appropriate to limit the scope of this term to particular elds of knowledge. Between 1000 B.C. and 1800 A.D., numerous study on mathematics were inspired by Indian mathematicians in which were set forth for the first time, the theory of zero, numeral system, techniques of algebra and algorithm, square root and cube root. The fact is that marvellous contributions made by Indian mathematicians over many hundreds of years are owned now a immense debt by today mathematics.

Keywords: Indian, mathematics, mathematicians, history.

1. INTRODUCTION

Despite developing quite separately of Chinese, some very well known mathematical discoveries were made by Indians at very early stage of time. It is well spoken that the history of mathematics is the history of civilization. The preliminary works obtainable , the Vedas (c. 3,000 B.C. or perhaps much earlier), even though including mainly of hymns of praise and poems of worship, manifest a high state of civilisation and engaged in the development of Indian culture by the mathematicians. A great contribution has been made by Ancient India to the world’s mathematical heritage such as:

 The early Vedic period (before 1000 BCE) indicates the application of arithmetic operations such as addition, subtraction, multiplication, fractions, squares, cubes and roots.

 A 4th Century CE Sanskrit text describes Buddha listing numbers up to 1053, as in addition with explaining six more numbering systems over and above these, directing to a number identical to 10421

 As long ago the 8th Century BCE, pretty soon Pythagoras, a text known as the

“Sulba Sutras” (or “Sulva Sutras”) catalogued different simple Pythagorean triples, moreover for the sides of a square and for a rectangle a statement of the simplified Pythagorean Theorem is enumerated.

 As soon as the 3rd or 2nd Century BCE, five different types of infinities: infinite in one direction, in two directions, in area, infinite everywhere and perpetually infinite were acknowledged by Jain mathematicians.

 A prophetic consciousness of indeterminate and infinite numbers, with numbers supposed to be of three types: countable, uncountable and infinite etc. were illustrated by Ancient Buddhist literature.

Now, there is an increasing awareness around the globe about the great contribution of Indian mathematicians in the field of mathematical astronomy. Their great contribution lead to the developments of arithmetic, algebra, trigonometry and secondarily geometry as in addition with combinatorics., By giving up many exciting mathematical ideas, the country has witnessed steady mathematical evolution in the world over last 3,000 years, though at times they lagged behind, paticularly in the recent centuries.

1.1 Overview

Indian mathematics history can be divided into 5 different parts as follows:

1) Ancient Period

a. Vedic Period (around 3000 B.C. – 1000 B.C.) b. Post Vedic Period (1000 B.C. – 500 B.C.) 2) Pre Middle period (500 B.C. – 400 A.D.)

3) Middle period or classic period (400 A.D. –1200 A.D.) 4) End of classic period (1200 A.D. – 1800 A.D.)

5) Current period (After 1800 A.D.)

(2)

2 2. ANCIENT PERIOD

2.1 Vedic Period (around 3000 B.C. – 1000 B.C.)

In the Vedic period, list of mathematical activity are mostly to be found in Vedic texts linked with ritual activities. However, the study of arithmetic and geometry were forced by secular considerations in early agricultural civilizations, as the system of land grants, agriculture tax evaluation, and problems of mensuration arise the required solution for this. As the plots or land could not all be of the same shape, so just to ensure that everyone will get same amount of plot or irrigated and non-irrigated lands with equivalent fertility respectively. This meant that an understanding of geometry and arithmetic was required by the local administrators or cultivators. By the help of this, conversion of plots into rectangular plots or triangular plots to squares of equivalent sizes and so on and the tax evaluation based on their fixed proportions of annual or seasonal crop incomes made easy by the local administrators and cultivators respectively. Mathematics was thus brought into the service by the both religious as well as non-religious domains. Numerals and decimals are also described in the Vedas. In this age, great input by India in the arena of mathematics is the disclosure of „Zero‟ and the ‟10 place value method‟ such as ,10 to the power n’ i.e. .

In Vedic works, many rules and evolution of geometry are found as mentioned below:

 Use of geometric shapes, including triangles, rectangles, squares, trapezium and circles.

 Equivalence through numbers and area

 Equivalence led to the problem of squaring the circle and vice versa

 Early problems of Pythagoras theorem

 Estimation forπ– Three values forπare found in Shatapata Brahmana.

 All the four arithmetical operators (addition, subtraction, multiplication and division)

This whole discussion verify that many mathematical methods which were not in concept stage still they were used in methodical as well as in expanded manner.

2.2 Post-Vedic Period (1000 B.C. – 500 B.C.)

To execute rituals, altars were to be formulated. The altar had to conform to very precise measurements, if this ritual sacrifice was to be successful. Geometrical mathematics was developed just to make those precise measurements. The rules were applicable in the form of Shulv sutras (also Sulbasutras) which means „rope‟ and this helped in measuring geometry while making altars.

a) Sulbasutras: It is believed that the Sulbasutras have been composed around 800 BC but according to the mathematical knowledge recorded in these sutras (aphorisms) are much more ancient. The Sulba authors prominence that they were entirely stating facts was already known to the composers of the Brahmanas and Samhitas of the early Vedic age.

The oldest known mathematics texts in actuality are the SuZba-sutras of Baudhayana, Apastamba and Katyayana which contours part of the literature of the Sutra period of the later Vedic age. The sulbasutras give a collection of consequences in mathematics that had been used for the designing and constructions of the numerous dignified Vedic fire-altars right from the dawn of civilization such as designs of several of these brick-altars and geometric knowledge. For example:

 constructions describing a falcon in flight with curved wings

 a chariot wheel complete with spokes or a tortoise with extended head and legs

 techniques for the construction of ritual altars in use during the Vedic era

 displays an understanding of basic geometric shapes and techniques of converting one geometric shape (such as a rectangle) to another of equivalent (or multiple, or fractional) area (such as a square) etc.

While some of the formulations are estimsted, others are exact which disclose a certain degree of practical creativity as well as some theoretical understanding of basic geometric principles. Modern methods of multiplication and addition perhaps emerged from the techniques explained in the Sulva-Sutras.

(3)

3

b) Panini: In the history of Indian science specially in the field of Sanskrit, grammar and linguistics, an important progress has been made a great impact on all mathematical work that was followed by the pioneering work of Panini (6th C BC). Panini provided formal production rules and definitions describing Sanskrit grammar in his treatise called Asthadhyayi, apart from explaining a comprehensive and scientific theory of phonetics, phonology and morphology. Elements like vowels and consonants and those which are the parts of speech such as nouns and verbs were placed in classes. The construction of compound words and sentences was developed through ordered rules operating on underlying structures in a manner related to formal language theory. Today, Panini's constructions can also be seen as proportional to modern definitions of a mathematical function. Thus Panini's work gave an example of a scientific notational model that could have moved later mathematicians to utilize abstract notations in characterizing algebraic equations and presenting algebraic theorems and results in a scientific format.

c) Jain Mathematics: The Jain tradition has also been very important in the mathematical development in the country. As similar to the Vedic people, Jain scholars were not motivated from the rituals but from the thought of cosmos. In the Jaina cosmography the universe is assumed to be a flat plane with concentric annular regions encircle an innermost circular region with a diameter of 100000 yojanas studied as the Jambudvipa (island of Jambu), and the annular regions rather consist of water and land, and their widths increasing two fold with each successive ring, it may be stated that this cosmography is also established in the Puranas.

One of the remarkable features of the Jaina tradition is the deviation from old belief of 3 as the ratio of the circumference to the diameter, Suryaprajnapti recalls the then traditional value 3 for it, and discards it in favour of √10. From the early times the Jainas were aware that the ratio of the area of the circle to the square of its radius is the equal as the ratio of the circumference to the diameter. They had also interesting approximate formulae for the lengths of circular arcs and the areas attained by them together with the corresponding chord. How they arrived at these formula is not understood. A elaborate discussion is found in Jaina literature on Permutations and combinations, sequences, categorisation of infinities are some of the other mathematical topics.

d) Bakshali Manuscript: The Bakshali manuscript, which was unearthed in the 19th Century, does not appear to belong to any specific period. A few historians class it as a work of the early classical period and others suggest it may be a work of Jaina mathematics. However there is still a debate surrounding the date of the Bakshali manuscript.

The Bakshali manuscript was written on leaves of Birch in Sarada characters and in the Gatha dialect, which is a combination of Sanskrit and Prakrit. The Bakshali manuscript highlights developments in arithmetic and algebra. The arithmetic contained within the work is of high quality. There are eight principal topics in the Bakshali manuscript:

 Examples of the rules of three(and profit and loss and interest)

 Solution to linear equations with as many as five unknowns

 Solutions of quadratic equations

 Arithmetic and Geometric Progressions

 Compound Series

 Quadratic Indeterminate equations (origin of type y= )

 Simultaneous Equations

3. PRE MIDDLE PERIOD (500 B.C. – 400 A.D.)

Almost all the writings of this time are lost, except for a few books and few pages of Vaychali Ganit, Surya Siddhanta and Ganita Anuyog. During this period too, mathematics underwent sufficient development.

Sathnanga sutra, bhagvathi sutra and Anuyogdwar Sutra are famous books of this time. Apart from these, the books titled Tatvarthaadigyam Sutra Bhashya of Jain philosopher Omaswati (135 B.C.) and Tiloyapannati of Acharya Yativrisham (176B.C.) are famous writings of this time.

Vaychali Ganit discusses in detail the following:

 Basic calculations of mathematics

(4)

4

 Numbers based on 10

 Fractions

 Squares and Cubes of numbers

 Rule of false position

 Interest methods

Sathananga Sutra has mentioned five types of infinite and Anuyogdwar Sutra has mentioned four types of measure and also describes permutation and combination and some rule of exponents. Roots of modern trigonometry lie in the book Surya Siddhanta. By around 3rd century B.C., Brahmi numerals began to appear. Here is one style of Brahmi numerals.

4. THE CLASSICAL PERIOD OR MIDDLE PERIOD (400 A.D. – 1200 A.D.)

4.1 Aryabhatta: Aryabhata, who is occasionally known as Aryabhata I, or Aryabhata the elder to distinguish him from a 10th Century astronomer of the same name, stands a pioneer of the revival of Indian mathematics, and the so called „classical period‟ or „golden era‟ of Indian mathematics

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.

In mathematical verses of Aryabhatiya, the following topics are covered:

A. Approximation of Pi: Aryabhata worked on the approximation for pi (π), and may have come to the conclusion that π is irrational. He wrote that if 4 is added to 100, then multiplied by 8 then added to 62,000 the answer will be equal to circumference of a circle of diameter 20000. Aryabhata was aware that it was an irrational number and that his value was an approximation, which shows his incredible insight.

B. Trigonometry: Aryabhata discussed the concept of sine in his work by the name of ardha-jya, which literally 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.

C. Algebra: In Aryabhatiya, Aryabhata provided elegant results for the summation of series of squares and cubes.

D. Astronomy: Aryabhata's system of astronomy was called the audAyaka system.

Aryabhata correctly insisted that the earth rotates about its axis daily; Aryabhata’s work on astronomy was also pioneering, and was far less tinged with a mythological flavour. He even computed the circumference of the earth as 25835 miles which is close to modern day calculation of 24900 miles. The work of Aryabhata was also extremely influential in India. Aryabhata died in 550 A.D. This remarkable man was a genius and continues to baffle many mathematicians of today.

4.2 Bhaskara I: Bhāskara (c. 600 – c. 680) (commonly called Bhaskara I to avoid confusion with the 12th-century mathematician Bhāskara II) was a 7th-century mathematician, who was the first to write numbers in the Hindu decimal system with a circle for the zero, and who gave a unique and remarkable rational approximation of the sine function in his commentary on Aryabhata's work. This commentary, Āryabhaṭīyabhāṣya, written in 629 CE, is among the oldest known prose works in Sanskrit on mathematics and astronomy.

Bhaskara wrote three astronomical contributions. In 629 he annotated the Aryabhatiya, written in verses, about mathematical astronomy. The comments referred exactly to the 33 verses dealing with mathematics. There he considered variable equations and trigonometric formulae.His work Mahabhaskariya divides into eight chapters about mathematical astronomy. In chapter 7, he gives a remarkable approximation formula for sin x i.e.

which he assigns to Aryabhata. It reveals a relative error of less than 1.9% (the greatest deviation ( -1 ≈ 1.859 at x=0).Moreover, relations between sine and cosine, as well as between the sine of an angle >90°>180° or >270° to the sine of an angle <90° are given.

Parts of Mahabhaskariya were later translated into Arabic.

(5)

5

4.3 Vahamishira: Varāhamihira (c. early 6th-century), also called Vārāha or Mihira, was a Hindu polymath and himself an astronomer.

His Contributions during his life follow as:

A. Trigonometry: Varahamihira improved the accuracy of the sine tables of Aryabhata.

B. Combinatorics: He was among the first mathematicians to discover a version of what is now known as the Pascal's triangle. He used it to calculate the binomial coefficients. He also records the first known 4×4 magic square.

4.4 Brahmagupta: Brahmagupta (born c. 598CE, died c. 668CE) was an Indian mathematician and astronomer. He is the author of three early works on mathematics and astronomy: The Brāhmasphuṭasiddhānta, a theoretical treatise, and the Khaṇḍakhādyaka, a more practical text. Brahmagupta was the first to give rules to work out with zero. In the Brahmasphutasiddhant among the major developments are those in the areas of:

A. Algebra: Brahmagupta gave the solution of the general linear equation in chapter eighteen of Brahmasphutasiddhanta. He further gave two equivalent solutions to the general quadratic equation which are respectively:

He went on to solve systems of simultaneous indeterminate equations stating that the desired variable must first be isolated, and then the equation must be divided by the desired variable's coefficient.

B. Airthmetic: The 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. In Brahmasphutasiddhanta, multiplication was named Gomutrika. In the starting of chapter twelve of his Brahmasphutasiddhanta, named 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:

C. Zero: Brahmagupta's Brahmasphuṭasiddhanta is the first book that provides rules for arithmetic manipulations that apply to zero and to negative numbers. In chapter eighteen of his Brahmasphutasiddhanta, Brahmagupta describes operations on negative numbers.

Here Brahmagupta states that = 0 and as for the question of where a/0 ≠ 0 he did not commit himself. His rules for arithmetic on negative numbers and zero are quite close to the modern understanding, except that in modern mathematics division by zero is left undefined.

D. Geometry: Brahmagupta's formula

(6)

6 Diagram for reference

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,

The approximate area is the product of the halves of the sums of the sides and opposite sides of a triangle and a quadrilateral. The accurate [area] is the square root from the product of the halves of the sums of the side’s diminished by [each] side of the quadrilateral.

So given the lengths p, q, r and s of a cyclic quadrilateral, the approximate area is

while, letting the exact area is

Heron's formula is a special case of this formula and it can be derived by setting one of the sides equal to zero.

E. Astronomy: Some of the important contributions made by Brahmagupta in astronomy are his methods for calculating the position of heavenly bodies over time, their rising and setting, conjunctions, and the calculation of solar and lunar eclipses.

In chapter seven of his Brahmasphutasiddhanta, entitled Lunar Crescent, Brahmagupta rebuts the idea that the Moon is farther from the Earth than the Sun. He does this by explaining the illumination of the Moon by the Sun.

Further work exploring the longitudes of the planets, diurnal rotation, lunar and solar eclipses, risings and settings, the moon's crescent and conjunctions of the planets, are discussed in his treatise Khandakhadyaka.

4.5 Mahaviracharya: Mahāvīra (or Mahaviracharya, "Mahavira the Teacher") was a 9th- century Jain mathematician as well as the author of Gaṇitasārasan graha. He presented on the similar subjects on which Aryabhata and Brahmagupta contended, but he expressed them more clearly. His work is a highly syncopated approach to algebra and the emphasis in much of his text is on developing the techniques necessary to solve algebraic problems.

He discovered algebraic identities like He also found out the formula for nCr as

. He asserted that the square root of a negative number does not exist.

His major contributions to mathematics include:

A. Arithmetic

 Detailed operations with fractions(and unit fractions)

 Geometric progressions – He gave almost all the required formulas

 General formula for permutation and combination.

B. Algebra

 Work on quadratic equations

 Indeterminate equations

 Simultaneous equations C. Geometry

 Definitions for most of the geometric shapes

 Repeated Brahmagupta’s construction for cyclic quadrilateral

 He referred to the ellipse and gave its perimeter.

4.6 Sridhara: Sridharacharya (c. 870 CE – c. 930 CE) a great Indian mathematician as well as Sanskrit pandit and philosopher born in the 8th Century AD. However, it’s a fact that he wrote Patganita Sara, a work on arithmetic and mensuration.

Notable Work are as follows:

(7)

7

 Gave an explaination on the zero and also wrote that "If zero is added or subtracted to any number, the number remains unchanged and if it is multiplied by any number, the product is zero".

 In the case of dividing a fraction he gave the method of multiplying the fraction by the reciprocal of the divisor.

 gave the practical applications of algebra

 separated algebra from arithmetic

 The first person to give a formula for solving quadratic equations.

4.7 Aryabhatta II: Aryabhata II (920 A.D. – 1000 A.D.) a mathematical-astronomer who gives results of algebra in this work such as Mahasiddhanta i.e. 20 verses of detailed rules for solving by = ax + c

4.8 Sripati: Sripati (1019 A.D. – 1066 A.D.) the most important Indian mathematician of the 11th century and a follower of the teaching of Lalla. In his work, he gave the identity:

4.9 Bhaskaracharya I I: Bhāskara (1114–1185) also known as Bhāskarāchārya and as Bhaskara II to avoid uncertainity with Bhāskara I, was an Indian mathematician and astronomer.

Some of Bhaskara's work done during his life time are as follows:

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

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

Bhaskara II gave the first general method for finding the solutions of the problem

= 1 (so-called "Pell's equation").

A. Airthmetic: Bhaskara's arithmetic text Leelavati enclosed 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 measurement. More clearly the contents contain:

 Definitions

 Properties of zero (including division, and rules of operations with zero),

 Further extensive numerical work, including use of negative numbers and surds,

 Estimation of π.,

 Arithmetical terms, methods of multiplication, and squaring, inverse rule of three, and rules of 3, 5, 7, 9, and 11,

 Problems involving interest and interest computation,

 Indeterminate equations (Kuṭṭaka), integer solutions (first and second order).

B. Algebra: His Bijaganita ("Algebra") was a work in twelve chapters. A positive number has two square rootsIt was the first text to recognize. His work Bijaganita is sucessively a study on algebra and includes the following topics:

 Positive and negative numbers.

 The 'unknown' (includes determining unknown quantities)

 Determining unknown quantities.

 Surds (includes evaluating surds).

 Kuṭṭaka (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).

(8)

8

 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.

C. Astronomy: Using an astronomical model intiated by Brahmagupta in the 7th century, Bhāskara correctly 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 approximately 365.2588 days which is the similar as in Suryasiddhanta. The modern accepted measurement is 365.25636 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:

 Mean longitudes of the planets.

 True longitudes of the planets.

 The three problems of diurnal rotation. (Diurnal motion is an astronomical term referring to the apparent daily motion of stars around the Earth, or more precisely around the two celestial poles. It is caused by the Earth's rotation on its axis, so every star apparently moves on a circle, that is called the diurnal circle.)

 Syzygies.

 Lunar eclipses.

 Solar eclipses.

 Latitudes of the planets.

 Sunrise equation

 The Moon's crescent.

 Conjunctions of the planets with each other.

 Conjunctions of the planets with the fixed stars.

 The paths of the Sun and Moon.

 The second part contains thirteen chapters on the sphere.

5. END OF CLASSICAL PERIOD OR POST MIDDLE PERIOD (1200 A.D – 1800 A.D.) 5.1 The Kerala School of Mathematics

The Kerala School of Astronomy and Mathematics was a school of mathematics and astronomy established by Madhava of Sangamagrama in Kerala, India, founded in the late 14th Century by Madhava of Sangamagrama, sometimes called the greatest mathematician- astronomer of medieval India including members: Parameshvara, Neelakanta Somayaji, Jyeshtadeva, Achyuta Pisharati, Melpathur Narayana Bhattathiri and Achyuta Panikkar.

He developed infinite series approximations for a range of trigonometric functions, including π, sine, etc. Some of his contributions to geometry and algebra and his early forms of differentiation and integration for simple functions may have been transmitted to Europe via Jesuit missionaries, and it is possible that the later European development of calculus was influenced by his work to some extent. Their work involve a series expansion for and the arc-tangent series, and the series for sine and cosine functions that were attained in Europe by Gregory, Leibnitz and Newton, respectively, over two hundred years later. Some numerical values for that are accurate to 11 decimals are also a highlight of the work. In many ways the work of the Kerala mathematicians predicted the Calculus as it developed in Europe later, and in particular involves manipulations with indefinitely small quantities (in the determination of circumference of the circle, etc.) analogous to the infinitesimals in Calculus; it has also been argued by some authors that the work is indeed Calculus already.

6. CURRENT PERIOD (1800 A.D. ONWARDS)

6.1 Srinivasa Ramanujan (1887-1920): Srinivasa Ramanujan (22 December 1887 – 26 April 1920) was an Indian mathematician who lived through the British Rule in India. He was the second Indian who was the youngest individual member of the Royal Society and became the first Indian who elected an individual of Trinity College, Cambridge, Though.

Although he had no formal training in pure mathematics, still he was capable to give markable contributions to mathematical analysis, number theory, infinite series, and

(9)

9

continued fractions, including solutions to mathematical problems then considered unsolvable. He also proved that any big number can be written as sum of not more than four prime numbers and 1729 is the smallest number which can be written in the form of sum of cubes of two numbers in two ways, i.e. 1729 = + = + , later on this number 1729 is called Ramanujan‟s Number.

In 1913 he started a postal partnership with the English mathematician G. H. Hardy at the University of Cambridge, England. In his notes, Ramanujan had introduced revolutionary new theorems, During his wholew life, he was able to assembled nearly 3,900 results (mostly identities and equations) in which many were completely novel, his original and highly unconventional results, like the Ramanujan prime, the Ramanujan theta function, partition formulae and mock theta functions, have provided entire new areas of work and inspired a huge amount of further research.

6.2 Swami Bharti Krisnateerthaji (1884 A.D. – 1960 A.D.): Swami Bharti Krisnateerthaji author of the book „Vedic Ganit‟ as well as the originator and father of Vedic Ganit. Bharati Krishnaji got the key to Ganita Sutra coded in the Atharva Veda and with the help of lexicographs he rediscovered Vedic Mathematics. He also found “Sixteen Sutras” or word formulas which cover all the branches of Mathematics Arithmetic, Algebra, Geometry, Trigonometry, Physics, plan and spherical geometry, conics, calculus- both differential and integral, applied mathematics of all various kinds, dynamics, hydrostatics and all.

6.3 Shakuntala Devi (1939A.D. – 2013 A.D.): Shakuntala Devi travelled all around the world exhibiting her arithmetic talents, including a tour of Europe in 1950 and a performance in New York City in 1976 where her abilities were studied by Arthur Jensen.

Jensen examined her performance of various tasks, including the calculation of large numbers such as the problems presented to Devi contained calculating the cube root of 61,629,875 and the seventh root of 170,859,375. In 1977, at Southern Methodist University, she provided the 23rd root of a 201-digit number in 50 seconds. On 18 June 1980, she illustrated the multiplication of two 13-digit numbers 7,686,369,774,870 × 2,465,099,745,779 selected 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, later her performance was recorded in the 1982 Guinness Book of Records.

REFERENCES

1. B Datta and A N Singh, History of Hindu Mathematics: A source book, Parts 1 and 2 (single volume), Asia Publishing House, Bombay, 1962.

2. On the History of Indian Mathematics, (IJITR) INTERNATIONAL JOURNAL OF INNOVATIVE TECHNOLOGY AND RESEARCH Volume No.3, Issue No.2, February – March 2015, 1915 – 1924.

3. Ancient Indian Mathematics –AConspectus* by S G Dani

4. M. C. Apaṭe. The Laghubhāskarīya, with the commentary of Parameśvara. Anandāśrama, Sanskrit series no. 128, Poona, 1946.

5. Bhau Daji (1865). "Brief Notes on the Age and Authenticity of the Works of Aryabhata, Varahamihira, Brahmagupta, Bhattotpala, and Bhaskaracharya". Journal of the Royal Asiatic Society of Great Britain and Ireland. pp. 392–406.

6. Bhattacharyya, R. K. (2011), "Brahmagupta: The Ancient Indian Mathematician", in B. S. Yadav; Man Mohan (eds.), Ancient Indian Leaps into Mathematics, Springer Science & Business Media, pp. 185–192, ISBN 978-0-8176-4695-0

7. https://www.storyofmathematics.com/indian.html 8. http://veda.wikidot.com/history-of-mathematics-in-india 9. https://en.wikipedia.org/wiki/List_of_Indian_mathematicians

10. https://en.wikipedia.org/wiki/Kerala_School_of_Astronomy_and_Mathematics 11. https://en.wikipedia.org/wiki/Srinivasa_Ramanujan

12. https://en.wikipedia.org/wiki/Shakuntala_Devi 13. https://en.wikipedia.org/wiki/Bh%C4%81skara_I 14. https://en.wikipedia.org/wiki/Aryabhata

15. https://en.wikipedia.org/wiki/Var%C4%81hamihira 16. https://en.wikipedia.org/wiki/Brahmagupta

17. https://en.wikipedia.org/wiki/Mah%C4%81v%C4%ABra_(mathematician) 18. https://en.wikipedia.org/wiki/Sridhara

19. https://en.wikipedia.org/wiki/Bh%C4%81skara_II