Ancient Indian Mathematics

10 October 2016

There are opposing views prevalent regarding the prominence of mathematics in ancient India. One, there are those who make excessive claims for the antiquity of Indian mathematics with the motive of emphasizing the uniqueness of Indian mathematical achievements. Whereas, the other conflicting views denies the existence of any ‘real’ Indian mathematics before A. D. 500. This view is the result of deeply entrenched Euro centrism that does not negotiate with the idea of independent developments in early Indian mathematics.

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Whereas mathematics grew out of philosophy in ancient Greece, it was an outcome of linguistic developments in India. In fact the algebraic character of ancient Indian mathematics is but a byproduct of the well-established linguistic tradition of representing numbers by words. ? Around 800 B. C. Vedic mathematics declined and Jains School of mathematics gradually which was to do notable work in the field. ? From about 200 B. C. was period of instability and fragmentation due to foreign invasions but also of useful cross cultural contacts.

Probably the only piece of existing mathematical evidence from this period is the Bakhshali manuscript. ? This period ranges from 3rd to 12th centuries and is referred to as the classical period of Indian civilization. Mathematical activities reached a climax with the appearance of the famous quartet: Brahmagupta, Mahvika and Bhaskracharya. Indian work on astronomy and mathematics spread westward, reaching the Islamic world where it was absorbed, refined and augmented before being transmitted to Europe. This last period described as the medieval period of Indian history, saw the migration of astronomy and mathematics from the north to south. Particularly in present day state of Kerala, this was a period marked by remarkable studies of infinite series and mathematical analysis that predated similar works in Europe by about three hundred years. Harappan society was a highly organized society. There is every possibility that the town dwellers were skilled in mensuration and practical arithmetic of a bid similar to what was practiced in Egypt and Mesopotamia.

Archaeological findings from that period provide the following indications of the numerate culture of that society: ? It shows uniformity of weights over such a wide area and time which is quite unusual in the history of metrology. Taking 27. 584 grams as a standard, representing 1, the other weights form a series of 0. 05, 0. 1, 0. 2, 0. 5, 2, 5, 10, 20, 50, 100, 200 and 500. Such standardization and durability is a strong indication of a numerate culture with wellestablished, centralized system of weights and measures. Scales and instruments for measuring length have also been discovered with remarkably high accuracy. A notable feature of Harappan culture was its extensive use of kiln-fired bricks and the advanced level of its brick-making technology. These bricks are exceptionally well baked and of excellent quality and may still be used over and over again provided some care is taken in removing them in the first place. Fifteen different sizes of Harappan bricks have been identified with standard ratio of the three dimensions as 4:2:1.

It was thought until recently that from them evolved first the Bakhshali Number system and then the Gwalior system which is recognizably close to our present day number system. In both Bakshati and Gwalior number systems, ten symbols were used to represent 1 to 9 and zero. With them it became possible to express any number, irrespective of its largeness, by a decimal place value system. Long lists of number- names for powers of 10 are found in various early sources. In the Ramayana, it is reported that Rama had an army of 1010 +1014 +1020 +1024 +1030 +1034 +1040 +1044 +1052 +1057 +1062 +5 men.

The very existence of names for powers of ten up to sixty two indicates that the Vedic Indians were quite at home with very large numbers. This is to be compared with ancient Greeks, who had no words for numbers above the myriad (104). The Jains who came after the Vedic Indians were particularly fascinated by even larger numbers which were intimately tied up with their philosophy of time and space. For units of measuring time, the Jains suggested following relationship: 1 purvis = 756 * 1011 days 1 shirsa prahelika = (8,400,000)28 purvis The last number contains 194 digits!

The word numeral system was the logical outcome of proceeding by the multiples of 10. Such a system presupposes a scientifically based vocabulary of number names in which the principles of addition, subtraction and multiplication are used.  Due to oral mode of preserving and disseminating knowledge, the wordnumeral system persisted in India. As a replacement to this, a new concrete system was devised to help versification and memory, known as bhutasamkhya, wherein numbers were indicated by well-known objects or ideas.

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