Indian Mathematics In reading about these areas of mathematics that we often take as obvious. it often was, once someone had worked it out beforehand. This series is about some of the lesser-known of those people.

Place value system and zero It is speculated that Aryabhata used the term “approaching” to The placevalue system, first seen mean that not only is this an in the Bakhshali Manuscript, approximation but that the value which has been dated between is incommensurable (or irrational). 200BC and 300AD, was clearly in If this is correct, it is quite a place in his work. The author of sophisticated insight, because the the Bakhshali Manuscript is not irrationality of pi was proved in known. Europe only in 1761 by Lambert). While Aryabhata did not use a After Aryabhatiya was translated symbol for zero, the French into Arabic (ca. 820 AD) this mathematician Georges Ifrah approximation was mentioned in argues that knowledge of zero was AlKhwarizmi's book on algebra. implicit in Aryabhata's placevalue system as a place holder for the Algebra powers of ten with null coefficients. In Aryabhatiya, Aryabhata However, Aryabhata did not use provided elegant results for the the brahmi numerals. Continuing summation of series of squares the Sanskritic tradition from Vedic and cubes: times, he used letters of the alphabet to denote numbers, 12 + 2 2 + … + n 2 = (n + 1)(2n + 1) expressing quantities, such as the 6 table of sines in a mnemonic form. and

3 3 3 2 Pi as irrational 1 + 2 + … + n = (1+2+ … +n) Aryabhata’s work influenced the Aryabhata worked on the Indian astronomical tradition and approximation for Pi (π), and may several neighbouring cultures have come to the conclusion that π through translations. His definition is irrational. In the second part of of sine, cosine and several other the Aryabhatiyam he writes: trigonometrical functions "Add four to 100, multiply by eight, influenced the birth of and then add 62,000. By this rule trigonometry. He was also the first the circumference of a circle with a to specify sine and versine (1cos x) diameter of 20,000 can be tables in 3.75 o intervals to 90 o to approached." an accuracy of 4 decimal places. This implies that the ratio of the Aryabhata's astronomical circumference to the diameter is calculation methods were also very ((4+100)×8+62000)/20000 = influential. Along with the 3.1416, which is accurate to five trigonometric tables, they came to significant figures. be widely used in the Islamic world and used to compute many Arabic astronomical tables (zijes). In that uses zero in this way. In 628 particular, the astronomical tables AD, Brahmagupta wrote a book in the work of the Arabic Spain explaining how zero worked, with scientist AlZarqali (11 th century) rules like "The sum of zero and were translated into Latin as the zero is zero" and "The sum of a Tables of Toledo (12 th century) and positive and a negative is their remained the most accurate difference; or, if they are equal, ephemeris used in Europe for zero.". With the formation of the centuries. Islamic Empire a few years later, the use of zero spread quickly from Calendric calculations devised by India to West Asia and Africa (by Aryabhata and his followers have the 800's), and then more slowly to been in continuous use in India for Christian Europe (not until the the practical purposes of fixing the 1200's AD, and only specialists Panchangam (the Hindu used it until the 1500's). calendar). In the Islamic world, they formed the basis of the Jalali A later landmark in Indian calendar introduced in 1073 AD by mathematics was the development a group of astronomers including of the series expansions for Omar Khayyam, versions of which trigonometric functions (sine, (modified in 1925) are the national cosine, and arc tangent) by calendars in use in Iran and mathematicians of the Kerala Afghanistan today. The dates of School in the fifteenth century AD. the Jalali calendar are based on Their remarkable work, completed actual solar transit, as in two centuries before the invention Aryabhata and earlier Siddhanta of calculus in Europe, provided calendars. This type of calendar what is now considered the first requires an ephemeris for example of a power series (apart calculating dates. Although dates from the geometric series). were difficult to compute, seasonal However, they did not formulate a errors were less in the Jalali systematic theory of differentiation calendar than in the Gregorian and integration, nor is there any calendar. direct evidence of their results being transmitted outside Kerala. India's first satellite Aryabhata and the lunar crater Aryabhata are Baudhayana (c. 8th century BCE) named in his honour. An Institute composed the Baudhayana Sulba for conducting research in Sutra, the bestknown Sulba astronomy, astrophysics and Sutra, which contains examples of atmospheric sciences is the simple Pythagorean triples, such Aryabhatta Research Institute of as: (3,4,5), (5,12,13), (8,15,17), observational sciences (ARIES) (7,24,25), and (12,35,37) as well near Nainital, India. The inter as a statement of the Pythagorean school Aryabhata Maths theorem for the sides of a square. Competition is also named after Baudhayana also gave a formula him, as is Bacillus aryabhata, a for the square root of two giving species of bacteria discovered by 1.4142156…. ISRO scientists in 2009. The formula is accurate up to five In 458 AD, Indian mathematicians decimal places, the true value wrote a book, the Lokavibhaaga, being 1.41421356…