Ancient Indian Square Roots

  • David H. BaileyEmail author
  • Jonathan M. Borwein
Reference work entry


Our modern system of positional decimal notation with zero, together with efficient algorithms for computation, which were discovered in India some time prior to 500 CE, certainly must rank among the most significant achievements of all time. And it was not easy. As Pierre-Simon Laplace (1923) explained, “the difficulty of inventing it will be better appreciated if we consider that it escaped the genius of Archimedes and Apollonius, two of the greatest men of antiquity” (pp. 222–223).

The Mayans came close, with a system that featured positional notation with zero. However, in their system successive positions represented the mixed sequence (1, 20, 360, 7,200, 144,000, …), rather than the purely base-20 sequence (1, 20, 400, 8,000, 160,000, …), which precluded any possibility that their numerals could be used as part of a highly efficient arithmetic system (Ifrah, 2000, p. 311).

What is more, mathematicians in ancient India developed remarkably advanced schemes, at a very...

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Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.University of California, Davis, Department of Computer ScienceDavisUSA
  2. 2.CARMA Priority Research CentreUniversity of NewcastleCallaghanAustralia