The karaṇī: How to use integers to make accurate calculations on square roots

20 pages Published after the first Joint India-AMS meeting in Mathematics held in Bangalore in December 2003.This paper presents the karaṇī, a mathematical construction to use integers to make calculations with square roots. Indian mathematicians invented new operations for this purpose (e.g. $(\sqrt 2 + \sqrt 8)^2 = 2 + 8 + 2\,\sqrt{2\times8} = (\sqrt{18})^2$ for the sum of what they call the karaṇī 2 and 8, the sum of which is the karaṇī 18). This construction seems to be sophisticated, even useless, but we can find an explanation in a commentary (17th century): if all the calculations on square roots are made with karaṇī, and that the approximate value is taken only at the end, the result is more accurate than if approximate values are taken at the beginning of the calculation