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

François Patte (1)

1 MAP5 - UMR 8145 - Mathématiques Appliquées Paris 5

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.

Mots clés: mathematics; history; India; Sanskrit

Domaines: Histoire, Philosophie et Sociologie des sciences

François Patte. The karaṇī: How to use integers to make accurate calculations on square roots. Rajendra Bhatia, Indian Statistical Institute, New Delhi. Contributions to the History of Indian Mathematics, Hindustan Book Agency, pp.115, 2005, Culture and History of Mathematics. ⟨hal-00382536⟩