The Karaṅī: How to Use Integers to Make Accurate Calculations on Square Roots

Abstract

This paper presents the karaṅī, a mathematical construction to use integers to make calculations with square roots. Indian mathematicians invented new operations for this purpose (e.g. ${\left( {\sqrt 2 + \sqrt 8 } \right)^2} = 2 + 8 + 2\sqrt {2 \times 8} = {\left( {\sqrt {18} } \right)^2}$ for the sum of what they call the karaṅī 2 and 8, the sum of which is the karaṅī 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 karaṅī, and 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.