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.
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References
See, for instance, Bījagaṅita of Bhāskara with the commentary Sūryaprakāśa of Sūryadāsa, vol. I, ed. Pushpa Kumari Jain, Gaekwad’s Oriental Series, No. 182, Vadodara, 2001.
See, for instance, Bījagaṅita of Bhäskara with the commentary Bījapallavam of Krṣṅa Daivajña, ed. T.V. Radhakrishna Sastri, Tanjore Saraswathi Mahal Series No.78, Tanjore, 1958.
See, for instance, Siddhāntaśiromani of Bhäskara with Vāsanābhāsya and Vāsanāvārttika, ed. Muralidhara Chaturveda, Varanasi, 1981.
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© 2005 Hindustan Book Agency
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Patte, F. (2005). The Karaṅī: How to Use Integers to Make Accurate Calculations on Square Roots. In: Emch, G.G., Sridharan, R., Srinivas, M.D. (eds) Contributions to the History of Indian Mathematics. Culture and History of Mathematics, vol 3. Hindustan Book Agency, Gurgaon. https://doi.org/10.1007/978-93-86279-25-5_5
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DOI: https://doi.org/10.1007/978-93-86279-25-5_5
Publisher Name: Hindustan Book Agency, Gurgaon
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