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.

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, mathematical ideas were developed for the purpose of construction. This civilization had an advanced brick-making technology (having invented the kiln), used in the construction of buildings and embankments for food control.

The study of astronomy, based on mathematical theories, is considered to be even older. Even in later times, we find that astronomy motivated considerable mathematical development like trigonometry.

Vedic literature also contain mathematical constructions. In particular, the Shatapatha Brahmana, 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. Different interpretations of the dates of Vedic texts assign this Brahman date from 1800 to about 800 BCE or older.

Supplementary Vedas like the Shula Sutras are considered to date from 800 to 200 BCE. There are four, named after their authors: Baudhayana (800 BCE), Manava (750 BCE), Apastamba (600 BCE), and Katyayana (200 BCE). The sutras contain the famous Pythagoras theorem . 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 existed.

The Shulba Sutras introduce the concept of irrational numbers, numbers that are not the ratio of two whole numbers like the square root of 2. 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.

Mathematics of this period seems to have been developed for solving practical geometric problems like 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, it shifted 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 used even today can be traced to the Brahmi numerals that appeared 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 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.
—Pierre Simon Laplace

Babylonians had 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 used presently.

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, 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 efforts 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.

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

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