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around 300–1000 CE

Bakhshali Manuscript

One of the earliest known uses of a positional zero symbol

This ancient Indian manuscript is a collection of mathematical problems of various types, including several types of arithmetic. It contains one of the earliest known uses of symbol for zero used in a place-value system, and is perhaps the oldest known manuscript in Indian mathematics.

Bakhshali Manuscript

The Bakhshali manuscript is an ancient Indian mathematical text written on birch bark. Its dating is controversial, with proposed dates ranging from 300 to 1000 CE. The manuscript is a compendium of rules and illustrative examples in which each example is stated as a problem, the solution is described and the solution of the problem is verified. Its problems involve fractions, square roots, progressions and linear and quadratic equations. It also contains one of the earliest known uses of symbol for zero used in a place-value system.

Artifact format

70 leaves; too fragile for examination

Artifact origin

Mardan, Pakistan

Current artifact location

Bodleian Library

Catalog number

MS. Sansk. d. 14


Arithmetic timeline Rhind Papyrus Egyptian Mathematical Problem Papyrus Salamis Tablet Bakhshali Manuscript Pacioli's Summa Incan Yupana Napier's Rabdologiæ

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Computational Explanation

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Additional Reading

  • Datta, B. "The Bakhshali mathematics." The Bulletin of the Calcutta Mathematical Society, Vol. 21, pp. 1–60, 1929.
  • Hoernle, A. On the Bakshali Manuscript. Vienna: Alfred Hölde, 1887.
  • Joseph, G. G. The Crest of the Peacock: Non-European Roots of Mathematics, 3rd ed. Princeton, NJ: Princeton University Press, pp. 357–359, 364–367 and 380–382, 2011.
  • Kaye, G. R. The Bakhshali Manuscript: A Study in Medieval Mathematics. Calcutta: The Archaeological Survey of India, 1927.
  • Plofker, K. "The Bakhshāli Manuscript." In The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook (Ed. V. J. Katz). Princeton, NJ: Princeton University Press, pp. 398 and 434–445, 2007.
  • Plofker, K. "The Bakhshāli Manuscript." § 5.2 in Mathematics in India. Princeton, NJ: Princeton University Press, pp. 157–162, 2009.
  • Pickover, C. The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics. New York: Sterling, pp. 76–77, 2012.
  • Sarasvati, S. S. P.; and Jyotishmati, U. The Bakhshali Manuscript: An Ancient Treatise of Indian Arithmetic. Allahabad: Dr. Ratna Kumari Svadhyaya Sansthan, 1979.
  • Takao, H. The Bakhshālī Manuscript: an Ancient Indian Mathematical Treatise. Groningen, Netherlands: Egbert Forsten, 1995.

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