# Mathematical Treasure: Indian Zeros

Author(s):
Frank J. Swetz (The Pennsylvania State University) and Shaharir bin Mohamad Zain (National University of Malaysia)

As noted elsewhere in Convergence (see The Cambodian Zero and Indonesian Zeros), for centuries there have been scholars searching not only for the emergence of symbols denoting the concept of zero but specifically the origin of “$0$” as the standard way of denoting zero in place-value numeral systems. In this article, we look at some of the appearances of $0$ represented as a large hollow dot that have been found in India.

A frequently-cited example points to a set of engravings on the wall of a Hindu temple in the Gwalior region of India. The temple had fallen into ruin long before modern archaeologists began studying it in the late 19th century, so the inscription over the main door remained unnoticed after initial excavations. However, it was copied and translated into English in 1883 by the renowned German indologist and philologist, Eugen Julius Theodor Hultzsch (1857–1927). From the Sanskrit notation in the inscription, Hultzsch deciphered the numeral $270$ as expressed in our modern decimal system, where clearly the last symbol in the string of numerals is an empty placeholder. A photo of the relevant portion of the inscription is provided below. The numeral refers to a piece of land of size 270 x 187 hastas, where hasta is a unit of length in ancient Indian measure (in modern terms, 1 hasta = approximately 18 inches or 45 centimeters). The construction of this temple has been dated to 876 CE, so this is the date typically given for this early appearance of zero.

The Gwalior inscription in Sanskrit with the number $270$ shown near the center of the image. Photograph by Flickr user ccarlstead, available via the NonCommercial-NoDerivs 2.0 Generic (CC BY-NC-ND 2.0) license.

In 2017, the Bodleian Library of Oxford University commissioned radiocarbon dating on a mathematical text written on birch bark that is known as the Bakhshālī Manuscript for the location where it was discovered in 1881, which today is near Peshawar, Pakistan. Since this manuscript also contains zero symbols—and appears to use them in arithmetical computations—determining the date the manuscript was written is valuable to the history of the zero concept. Fragments were tested from three of the manuscript’s 70 folios, returning dates of CE 224–383, 680–779, and 885–993. Although initial reporting seized on the very early 3rd–4th century date range, specialists in the history of Indian mathematics have since argued that the manuscript was probably written and assembled all at one time, which means the 9th–10th century date range is most likely when it was created. These historians have also thought it likely that its dot symbol was used more as a placeholder than as a full-fledged abstract number.

Detail from the Bakhshālī Manuscript showing one of its zero symbols. Wikimedia Commons.

##### References

Aczel, Amir. 2015. Finding Zero: A Mathematician’s Odyssey to Uncover the Origins of Numbers. New York: Palgrave Macmillan.

Cœdès, George. 1918. Le royaume de Çrīvijaya. Bulletin de l’École française d'Extrême-Orient 18(6): 1–36.

Cœdès, George. 1931. A propos de l'origine des chiffres arabes. Bulletin of the School of Oriental Studies 6(2): 323–328.

Diller, Anthony. 1995.  Sriwijaya and the First Zeros. Journal of the Malaysian Branch of the Royal Asiatic Society 68(1): 53–66.

Plofker, Kim, Agathe Keller, Takao Hayashi, Clemency Montelle and Dominik Wujastyk. 2017. The Bakhshālī Manuscript: A Response to the Bodleian Library’s Radiocarbon Dating. History of Science in South Asia 5(1): 134–150.

Swetz, Frank J., and Shaharir bin Mohamad Zain. 2022, July 28. The Elusive Origin of Zero. Scientific American Online.

Zain, Shaharir bin Mohamad. 2001. A Note on the Decimal Numeral System. The Humanistic Mathematics Network Journal 24: 36-42.

Index to Mathematical Treasures

Frank J. Swetz (The Pennsylvania State University) and Shaharir bin Mohamad Zain (National University of Malaysia), "Mathematical Treasure: Indian Zeros," Convergence (September 2022)