History of Numbers in India-1

Evolution of numbers

The ingenious method of expressing every possible number using a set of ten symbols (each symbol having a place value and an absolute value) emerged in India. The idea seems so simple nowadays that its significance and profound importance is no longer appreciated. Its simplicity lies in the way it facilitated calculation and placed arithmetic foremost amongst useful inventions. the importance of this invention is more readily appreciated when one considers that it was beyond the two greatest men of Antiquity, Archimedes and Apollonius.- Laplace

History is not just about kings, battles and empires, it is also about the evolution of sciences, art and human thinking. History is not just the wars fought, it is also the ideas that have generated from time to time, and played a major role in influencing mankind. The evolution of mathematics in India, always fascinated me, how we have made significant contributions to the world of numbers. And no these are not some random achievements, number system, zero, concepts in geometry, trignometry, calculus, almost every field of maths has seen significant contribution. Unfortunately as in so many cases, the contribution of India to the world of numbers is not really known to many Indians. No it’s not just lack of knowledge, there is a reluctance of many Indians to recognize their contribution. Quite ironic, that a renowned mathematician like Laplace could understand the significance of the numbers system that came from India, but many of us fail to do so. Anyway that is another topic altogether, the purpose of my series is to share my knowledge and insights on the evolution of Mathematics in India, through a series of blog posts.

As are the crests of a peacock, as are the gem-stones of a snake, placed on the highest place of the body, the forehead, so is mathematics (Ganita) the head of all Vedah and shastras.What is the use of much speaking. Whatever object exists in this moving and nonmoving world, cannot be understood without the base of Ganita (Mathematics).- Vedanga Jyotisha

India confronts Egypt and Babylonia by the 3rd millennium with a thoroughly individual and independent civilisation of her own, technically the peer of the rest. And plainly it is deeply rooted in Indian soil. The Indus civilisation represents a very perfect adjustment of human life to a specific environment. And it has endured; it is already specifically Indian and forms the basis of modern Indian culture.-V.G.Childe

If we go back into ancient Indian history, much before the Vedic era, the first mathematical concepts were used in the Indus Valley civiliazation, that admittedly was more practical in nature. Though technically speaking, both Mohenjodaro and Harappa are now in Pakistan, they have been considered as part of the Indian sub continent from long, and the civilization there, has given a foundation of sorts to later Indian culture. Studies have shown that people in the Indus Valley, were among the first to develop a system of uniform weights and measures. An Ivory scale found in Lothal, had the smallest division around 1.704 mm, which was the smallest for the Bronze Age. For all practical purposes people in Harappa followed the decimal division in measurement, including using hexadron weights in mass. These weights made out of chert, were in a ratio of 5:2:1 with weights of 0.05, 0.1, 0.2, 0.5, 1, 2, 5, 10, 20, 50, 100, 200, and 500 units, with each unit weighing approximately 28 grams, somewhat similiar to the English ounce. There were other scales too discovered in the Indus Valley, one a decimal scale based on a unit of measurement of 1.32 inches now called the “Indus Inch”, and another a bronze rod marked in lengths of 0.367 inches, 100 units of those make around 36.7 inches, which is a stride. Excavations in the Indus Valley have also shown mass produced weights in shapes of hexahedra, barrels, cones and cylinders. One more interesting feature was use of a Mohenjo Daro ruler, whose unit of length(1.32 inches or 3.4 cm), was divided into 10 equal parts. What is apparent that the inhabitants of Indus Valley, had a good knowledge of basic geometry, and measures.  What exactly caused the decline of the Indus Valley civilization is not known, and is also beyond the scope of this article.

Hindu units of time on a logarithmic scale.

Vedic Period

The Vedic period that roughly stretched from 1500 BC to 800 BC, saw the flourishing of mathematics in India, through the sacred texts, and the usage of numbers. The Vedic texts mostly contained hymns, ritual procedures, observations, that were transmitted orally first and later in written form. It is interesting to note that you had Sanskrit terms for every power of 10, right up to 10⁶²
. For example both Lakh and Crore, derived from Sanskrit words Laksha and Prakrit word Krodi  to denote 10⁵    and 10⁷ respectively. The Yajurveda Samhita also had numbers as high as 10¹²  being mentioned in the texts, a very interesting example is the mantra at the end of annahoma during the Aswamedha Yagna, uttered just after sunrise, that invokes numbers going right up to a trillion.

Hail to śata (“hundred,” 10²), hail to sahasra (“thousand,” 10³), hail to ayuta (“ten thousand,” 10⁴), hail to niyuta (“hundred thousand,” 10⁵), hail to prayuta (“million,” 10⁶), hail to arbuda (“ten million,” 10⁷), hail to nyarbuda (“hundred million,” 10⁸), hail to samudra (“billion,” 10⁹, literally “ocean”), hail to madhya (“ten billion,” 10¹⁰, literally “middle”), hail to anta (“hundred billion,” 10¹¹,lit., “end”), hail to parārdha (“one trillion,” 10¹² lit., “beyond parts”), hail to the dawn (us’as), hail to the twilight (vyuṣṭi), hail to the one which is going to rise (udeṣyat), hail to the one which is rising (udyat), hail to the one which has just risen (udita), hail to svarga (the heaven), hail to martya (the world), hail to all.- Source from Wiki.

Though the Sulabasutras are considered the definitive Indian text on Geometry, research has shown that some of the basic geometrical concepts were discussed in Shatapatha Brahmana and the Taittiriya Samhita. The mathematics here though was for more practical purposes, Vedic rites, often went on for one full year, and were believed to mark the passage of time. A large part of the Stapatha Brahmana is dedicated to altar construction for the agnicayana rite, which occurs for 12 days.  This 12 day ritual often take place in a mahavedi,  that was shaped in the form of a trapezoid, and there was a smaller rectangular area close by called the pragvamsa. Agni in this case refers to a year, and the construction of the agnicayana altar, contained the basic geometrical concepts too. It was not just the construction of the altars, or the layers, even the bricks used in construction had specified dimensions. You basically had the ordinary bricks named lokamprna, and the special ones named yajusmati, each having it’s own specific measurements.  What is apparent, that if one has an indepth look at the procedures laid down for the construction of the altars in Shatapatha Brahmana, it is clear, that many scientific and mathematical principles were in use here. However, most of the mathematic, geometric principles enunciated in Shatapatha Brahmana and the Sulbha Sastras later, were primarily for application, more practical in nature. We shall be looking at these two texts in detail, in the following posts.
Sources

1) Gonitsora http://gonitsora.com/a-brief-history-of-ancient-indian-mathematics-part-i/

2) Wikipedia-http://en.wikipedia.org/wiki/Indian_mathematics

3) Overview of Indian Mathematics by McTutor- http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Indian_mathematics.html

4) Astronomy in Stapatha Brahmana by Subash Kak-  http://www.new.dli.ernet.in/rawdataupload/upload/insa/INSA_1/20005b5c_15.pdf