# Difference between revisions of "Talk:A brief history of indian mathematics"

By Prof. Vijaya Kumar Murty

Mathematics has played a significant role in the development of Bhartiya culture for millennia. Basic principles like counting 1, 2, 3, etc. to zero, are based on Sanskrit figures. Algebra is based on these mathematical developments[1].

Aryabhatta

In ancient times, mathematics was mainly used in an applied role. Thus, mathematical methods were used to solve problems in architecture and construction (as in the public works of the Harappa), in astronomy and astrology (as in the works of the Jain mathematicians), and in the construction of Vedic altars (as in the case of the Shula Sutras of Baudhayana and his successors). By the sixth or fifth century BCE, mathematics was being studied for its own sake. In modern times, mathematics is studied both for its own sake, as well as for its applications in other fields of knowledge.

## Mathematics in Ancient Times (3000 to 600 BCE)

The Indus valley civilization is considered to have existed around 3000 BCE. Two of its most famous cities, Harappa and Mohenjo-Daro, provide evidence that construction of buildings followed a standardized measurement which was decimal in nature. Here, we see mathematical ideas developed for the purpose of construction. This civilization had an advanced brick-making technology (having invented the kiln). Bricks were used in the construction of buildings and embankments for food control.

The study of astronomy is considered to be even older, and there must have been mathematical theories on which it was based. Even in later times, we find that astronomy motivated considerable mathematical development, especially in the field of trigonometry.

Much has been written about the mathematical constructions that are to be found in Vedic literature. In particular, the Shatapatha Brahmana, which is a part of the Shukla Yajur Veda, contains detailed descriptions of the geometric construction of altars for yajnas. Here, the brick-making technology of the Indus valley civilization was put to a new use. As usual, there are different interpretations of the dates of Vedic texts, and in the case of this Brahman, the range is from 1800 to about 800 BCE. Perhaps it is even older.

Supplementary to the Vedas are the Shula Sutras. These texts are considered to date from 800 to 200 BCE. Four in number, they are named after their authors: Baudhayana (800 BCE), Manava (750 BCE), Apastamba (600 BCE), and Katyayana (200 BCE). The sutras contain the famous theorem commonly attributed to Pythagoras. Some scholars (such as Seidenberg) feel that this theorem is already present in the Shatapatha Brahmana. Much later, Bhaskaracharya gave an algebraic proof of this theorem, as opposed to the geometric proof that the Greeks, and possibly the Chinese, were aware of.[2]

The Shulba Sutras introduce the concept of irrational numbers, numbers that are not the ratio of two whole numbers. For example, the square root of 2 is one such number. The sutras give a way of approximating the square root of a number using rational numbers through a recursive procedure which in modern language would be a ‘series expansion’. This predates, by far, the European use of Taylor series.

It is interesting that the mathematics of this period seems to have been developed for solving practical geometric problems, especially the construction of religious altars. However, the study of the series expansions for certain functions already hints at the development of an algebraic perspective. In later times, we find a shift towards algebra, with simplification of algebraic formulae and summation of series acting as catalysts for mathematical discovery.

## Brahmi Numerals, the Place-value System, and Zero

No account of historical mathematics would be complete without a discussion of numerals, the place-value system, and the concept of zero. The numerals that we use even today can be traced to the Brahmi numerals that seem to have made their appearance in 300 BCE. But Brahma numerals were not part of a place-value system. They evolved into the Gupta numerals around 400 CE and subsequently into the Devanagari numerals, which developed slowly between 600 and 1000 CE.

By 600 CE, a place-value decimal system was well in use. This means that when a number is written down, each symbol that is used has an absolute value, but also a value relative to its position. For example, the numbers 1 and 5 have a value on their own, but also have a value relative to their position in the number 15. The importance of a place-value system need hardly be emphasized. It would suffice to cite an oft-quoted remark by La-place: ‘It is India that gave us the ingenious method of expressing all numbers by means of ten symbols, each symbol receiving a value of position as well as an absolute value; a profound and important idea which appears so simple to us now that we ignore its true merit. But its very simplicity and the great ease which it has lent to computations put our arithmetic in the first rank of useful inventions; and we shall appreciate the grandeur of the achievement the more when we remember that it escaped the genius of Archimedes and Apollonius, two of the greatest men produced by antiquity.’[3]

A place-value system of numerals was apparently known in other cultures; for example, the Babylonians used a sexagesimal place-value system as early as 1700 BCE, but the Indian system was the first decimal system. Moreover, until 400 BCE, the Babylonian system had an inherent ambiguity as there was no symbol for zero. Thus, it was not a complete place-value system in the way we think of it today.

The elevation of zero to the same status as other numbers involved difficulties that many brilliant mathematicians struggled with. The main problem was that the rules of arithmetic had to be formulated so as to include zero. While addition, subtraction, and multiplication with zero were mastered, division was a more subtle question. Today, we know that division by zero is not well-defined and so has to be excluded from the rules of arithmetic. But this understanding did not come all at once, and took the combined eforts of many minds. It is interesting to note that it was not until the seventeenth century that zero was being used in Europe, and the path of mathematics from India to Europe is the subject of much historical research.

## The Classical Era of Indian Mathematics (500 to 1200 CE)

The most famous names of mathematics belong to what is known as the classical era. This includes Aryabhata I (500 CE), Brahmagupta (700 CE), Bhaskara I (900 CE), Mahavira (900 CE), Aryabhata II (1000 CE), and Bhaskaracharya or Bhaskara II (1200 CE).

During this period, two centers of mathematical research emerged, one at Kusumapura near Pataliputra and the other at Ujjain. Aryabhata I was the dominant figure at Kusumapura and may even have been the founder of the local school. His fundamental work, the Aryabhatiya, set the agenda for research in mathematics and astronomy in India for many centuries.

One of Aryabhata’s discoveries was a method for solving linear equations of the form ax + by = c. Here a, b, and c are whole numbers, and we are seeking values of x and y in whole numbers satisfying the above equation. For example, if a = 5, b = 2, and c = 8, then x = 8 and y = −16 is a solution. In fact, there are infinitely many solutions:

x = 8 − 2m

y = 5m − 16

where m is any whole number, as can easily be verified. Aryabhata devised a general method for solving such equations, and he called it the kuttaka (or pulverizer) method. He called it the pulverizer because it proceeded by a series of steps, each of which required the solution of a similar problem, but with smaller numbers. Thus, a, b, and c were ‘pulverized’ into smaller numbers.

The Euclidean algorithm, which occurs in the Elements of Euclid, gives a method to compute the greatest common divisor of two numbers by a sequence of reductions to smaller numbers. As far as I am aware, Euclid does not suggest that this method can be used to solve linear equations of the above sort. Today, it is known that if the algorithm in Euclid is applied in reverse order, then in fact it will yield Aryabhata’s method. Unfortunately, the mathematical literature still refers to this as the extended Euclidean algorithm, mainly out of ignorance of Aryabhata’s work.

It should be noted that Aryabhata studied the above linear equations because of his interest in astronomy. In modern times, these equations are of interest in computational number theory and are of fundamental importance in cryptography.

Amongst other important contributions of Aryabhata is his approximation of π to four decimal places (3.1416)[4]. Also of importance is Aryabhata’s work on trigonometry, including his tables of values of the sine function, as well as algebraic formulae for computing the sine of multiples of an angle.

The other major center of mathematical learning during this period was Ujjain, which was home to Varahamihira, Brahmagupta, and Bhaskaracharya. The text Brahma-sphuta-siddhanta by Brahmagupta, published in 628 CE, dealt with arithmetic involving zero and negative numbers.

As with Aryabhata, Brahmagupta was an astronomer, and much of his work was motivated by problems that arose in astronomy. He gave the famous formula for a solution to the quadratic equation ax² + bx + c = 0, namely x = (−b + √(b² − 4ac))/2a

It is not clear whether Brahmagupta gave just this solution or both solutions to this equation. Brahmagupta also studied quadratic equations in two variables and sought solutions in whole numbers. Such equations were studied only much later in Europe. We shall discuss this topic in more detail in the next section.

This period closes with Bhaskaracharya (1200 CE). In his fundamental work on arithmetic (titled Lilavati) he refined the kuttaka method of Aryabhata and Brahmagupta. The Lilavati is impressive for its originality and diversity of topics. Until recently, it was a popularly held view that there was no original Indian mathematics before Bhaskaracharya. However, the above discussion shows that his work was the culmination of a series of distinguished mathematicians who came before him. Also, after Bhaskaracharya, there seems to have been a gap of two hundred years before the next recorded work. Perhaps this is another time period about which more research is needed.

## The Solution of Pell’s equation

In Brahmagupta’s work, Pell’s equation had already made an appearance. This is the equation that, for a given whole number D, asks for whole numbers x and y satisfying the equation

x² − Dy² = 1.

In modern times, it arises in the study of units of quadratic fields and is a topic in the field of algebraic number theory. If D is a whole square (such as 1, 4, 9, and so on), the equation is easy to solve, as it factors into the product(x − my)(x + my) = 1where D = m². This implies that each factor is +1 or −1, and the values of x and y can be determined from that. However, if D is not a square, then it is not even clear that there is a solution. Moreover, if there is a solution, it is not clear how one can determine all solutions. For example, consider the case D = 2. Here, x = 3 and y = 2 gives a solution. But if D = 61, then even the smallest solutions are huge.

Brahmagupta discovered a method, which he called samāsa, by which, given two solutions of the equation, a third solution could be found. That is, he discovered a composition law on the set of solutions. Brahmagupta’s lemma was known one thou-sand years before it was rediscovered in Europe by Fermat, Legendre, and others.

This method appears now in most standard text-books and courses in number theory. The name of the equation is a historical accident. The Swiss mathematician Leonhard Euler mistakenly assumed that the English mathematician John Pell was the first to formulate the equation, and began referring to it by this name.

The work of Bhaskaracharya gives an algorithmic approach—which he called the cakravāla (cyclic) method—to finding all solutions of this equation. The method depends on computing the continued fraction expansion of the square root of D and using the convergents to give values of x and y. Again, this method can be found in most modern books on number theory, though the contributions of Bhaskaracharya do not seem to be well-known.

## Mathematics in South India

We described above the centers at Kusumapara and Ujjain. Both of these cities are in North India. There was also a flourishing tradition of mathematics in South India.

Mohair is a mathematician belonging to the ninth century who was most likely from modern-day Karnataka. He studied the problem of cubic and quadratic equations and solved them for some families of equations. His work had a significant impact on the development of mathematics in South India. His book Ganita-sara-sangraha amplifies the work of Brahmagupta and provides a very useful reference for the state of mathematics in his day. It is not clear what other works he may have published; further research into the extent of his contributions would probably be very fruitful.

Another notable mathematician of South India was Madhava, from Kerala. Madhava belongs to the fourteenth century. He discovered series expansions for some trigonometric functions such as the sine, cosine, and arctangent that were not known in Europe until after Newton. In modern terminology, these expansions are the Taylor series of the functions in question.

Madhava gave an approximation to π of 3.14159265359, which goes far beyond the four decimal places computed by Aryabhata. Madhava deduced his approximation from an infinite series expansion for π /4 that became known in Europe only several centuries after Madhava (due to the work of Leibniz).

Madhava’s work with series expansions suggests that he either discovered elements of the differential calculus or nearly did so. This is worth further analysis. In a work in 1835, Charles Whish suggested that the Kerala school had ‘laid the foundation for a complete system of fluxions’. The theory of fluxions is the name given by Newton to what we today call the differential calculus. On the other hand, some scholars have been very dismissive of the contributions of the Kerala school, claiming that it never progressed beyond a few series expansions. In particular, the theory was not developed into a powerful tool as was done by Newton. We note that it was around 1498 that Vasco da Gama arrived in Kerala and the Portuguese occupation began. Judging by evidence at other sites, it is not likely that the Portuguese were interested in either encouraging or preserving the sciences of the region. No doubt, more research is needed to discover where the truth lies.

Madhava spawned a school of mathematics in Kerala, and among his followers may be noted Nilakantha and Jyesthadeva. It is due to the writings of these mathematicians that we know about the work of Madhava, as all of Madhava’s own writings seem to be lost.

On 22 May the King of Norway presented the Abel Prize for 2007 to the distinguished probabilist S R Srinivasa Varadhan. Considered as the Nobel Prize for mathematics, the Abel Prize is awarded by the Norwegian Academy of Science and Letters for outstanding work in mathematics, work of extraordinary depth and influence in the mathematical sciences. Varadhan received the prize for ‘his fundamental contributions to probability theory and in particular for creating a unified theory of large deviations’. Kristian Seip, chairman of the Abel Committee, says: ‘Varadhan’s work has great conceptual strength and ageless beauty. His ideas have been hugely influential and will continue to stimulate further research for a long time.’

## Mathematics in the Modern Age:

In more recent times, there have been many important discoveries made by mathematicians of Indian origin. We shall mention the work of three of them: Shrinivas Ramanujan, Harish-Chandra, and Manjul Bhargava.

Ramanujan

Ramanujan (1887–1920) is perhaps the most famous of modern Indian mathematicians. Though he produced significant and beautiful results in many aspects of number theory, his most lasting discovery may be the arithmetic theory of modular forms. In an important paper published in 1916, he initiated the study of the τ function. The values of this function are the Fourier coefficients of the unique normalized cusp form of weight 12 for the modular group SL2(Z). Ramanujan proved some properties of the τ function and conjectured many more. As a result of his work, the modern arithmetic theory of modular forms, which occupies a central place in number theory and algebraic geometry, was developed by Hecke.[5]

Harish Chandra

Harish-Chandra (1923–83) is perhaps the least known Indian mathematician outside of mathematical circles. He began his career as a physicist, working under Dirac. In his thesis, he worked on the representation theory of the group SL2(C). This work convinced him that he was really a mathematician, and he spent the remainder of his academic life working on the representation theory of semi-simple groups. For most of that period, he was a professor at the Institute for Advanced Study in Princeton, New Jersey. His Collected Papers published in four volumes contain more than 2,000 pages.[6] His style is known as meticulous and thorough and his published work tends to treat the most general case at the very outset. This is in contrast to many other mathematicians, whose published work tends to evolve through special cases. Interestingly, the work of Harish-Chandra formed the basis of Langlands’s theory of automorphic forms, which are a vast generalization of the modular forms considered by Ramanujan.[7]

Manjul Bhargava (b. 1974) discovered a composition law for ternary quadratic forms. In our discussion of Pell’s equation, we indicated that Brahmagupta discovered a composition law for the solutions. Identifying a set of importance and discovering an algebraic structure such as a composition law is an important theme in mathematics. Karl Gauss, one of the greatest mathematicians of all time, showed that binary quadratic forms, that is, functions of the form

ax² + bxy + cy²

where a, b, and c are integers, have such a structure. More precisely, the set of primitive SL2(Z) orbits of binary quadratic forms of a given discriminant D has the structure of an abelian group. In fact, this is the ideal class group. After this fundamental work of Gauss, there had been no progress for several centuries on discovering such structures in other classes of forms. Manjul Bhargava’s stunning work in his doctoral thesis, published as several papers in the Annals of Mathematics, shows how to address this question for cubic (and other higher degree) binary and ternary forms.[8] The work of Bhargava, who is currently Professor of Mathematics at Princeton University, is deep, beautiful, and largely unexpected. It has many important ramifications and will likely form a theme of mathematical study at least for the coming decades. It is also sure to be a topic of discussion at the 2010 International Congress of Mathematicians in Hyderabad.

## Select Bibliography

1. André Weil, Number Theory: An Approach through History from Hammurapi to Legendre (Boston: Birkhäuser, 1984).
2. George G Joseph, The Crest of the Peacock: Non-European Roots of Mathematics (London: Penguin, 1991).
3. Georges Ifrah, The Universal History of Numbers: From Prehistory to the Invention of the Computer (London: Wiley , 1998).
4. A Seidenberg, ‘The Geometry of the Vedic Rituals’, in Agni: The Vedic Ritual of the Fire Altar, ed. Frits Staal, (Berkeley: Asian Humanities, 1983).
5. J J O’Connor and E F Robertson, ‘An overview of Indian mathematics’ [1] accessed 2 July 2007
6. I G Pearce, ‘Indian Mathematics: Redressing the Balance’ [2] accessed 2 July 2007.

## References

1. The Complete Works of Swami Vivekananda, 9 vols (Calcutta: Advaita Ashrama, 1–8, 1989; 9, 1997), 2.20
2. See V Kumar Murty, ‘Contributions of the Indian Subcontinent to Civilization’, Prabuddha Bharata, 100/1 (1995), 134–5.
3. Pierre Simon Laplace (1749–1827) was a French mathematician famous for, among other things, his contributions to the theory of probability and the differential equation named after him.
4. By comparison, the Greeks were using the weaker approximation 3.1429
5. Ramanujan’s works are available in Collected Papers of Srinivasa Ramanujan, eds G H Hardy, P V Seshu Aiyar, and B M Wilson (Urbana: University of Illinois, 1927).
6. Harish-Chandra, Collected Papers, 4 volumes, ed. V S Varadarajan (New York: Springer-Verlag, 1981).
7. For a discussion, see V Kumar Murty, ‘Ramanujan and Harish-Chandra’, Mathematical Intelligencer, 15 (1993), 33–39.
8. M Bhargava, ‘Higher Composition Laws’, Annals of Mathematics, 159/1 (2004), 217–50; 159/2 (2004), 865-86.

• Originally published as "A Brief History of Indian Mathematics" by Prabhuddha Bharata September 2007 edition. Reprinted with permission.