|Born||unclear, 3rd or 2nd century BCE|
|Era||Maurya or post-Maurya|
|Main interests||Sanskrit prosody, Indian mathematics, Sanskrit grammar|
|Notable works||Author of the Chandaḥśāstra (also called Pingala-sutras), the earliest known treatise on Sanskrit prosody. Creator of Pingala's formula.|
|Notable ideas||mātrāmeru, binary numeral system, arithmetical triangle|
Acharya Pingala (piṅgala; c. 3rd/2nd century BCE) was an ancient Indian poet and mathematician, and the author of the Chandaḥśāstra (also called Pingala-sutras), the earliest known treatise on Sanskrit prosody.
The Chandaḥśāstra is a work of eight chapters in the late Sūtra style, not fully comprehensible without a commentary. It has been dated to the last few centuries BCE. In the 10th-century, Halayudha wrote a commentary elaborating on the Chandaḥśāstra.
The Chandaḥśāstra presents the first known description of a binary numeral system in connection with the systematic enumeration of meters with fixed patterns of short and long syllables. The discussion of the combinatorics of meter corresponds to the binomial theorem. Halāyudha's commentary includes a presentation of Pascal's triangle (called meruprastāra). Pingala's work also includes material related to the Fibonacci numbers, called mātrāmeru.
Use of zero is sometimes ascribed to Pingala due to his discussion of binary numbers, usually represented using 0 and 1 in modern discussion, but Pingala used light (laghu) and heavy (guru) rather than 0 and 1 to describe syllables. As Pingala's system ranks binary patterns starting at one (four short syllables—binary "0000"—is the first pattern), the nth pattern corresponds to the binary representation of n−1 (with increasing positional values).
Pingala is credited with using binary numbers in the form of short and long syllables (the latter equal in length to two short syllables), a notation similar to Morse code. Pingala used the Sanskrit word śūnya explicitly to refer to zero.
- A. Weber, Indische Studien 8, Leipzig, 1863.
- Plofker, Kim (2009). Mathematics in India. Princeton University Press. pp. 55–56. ISBN 0-691-12067-6.
- Singh, Parmanand (1985). "The So-called Fibonacci Numbers in Ancient and Medieval India" (PDF). Historia Mathematica. Academic Press. 12: 232.
- "Pingala – Timeline of Mathematics". Mathigon. Retrieved 2021-08-21.
- Vaman Shivaram Apte (1970). Sanskrit Prosody and Important Literary and Geographical Names in the Ancient History of India. Motilal Banarsidass. pp. 648–649. ISBN 978-81-208-0045-8.
- R. Hall, Mathematics of Poetry, has "c. 200 BC"
- Mylius (1983:68) considers the Chandas-shāstra as "very late" within the Vedānga corpus.
- Van Nooten (1993)
- Susantha Goonatilake (1998). Toward a Global Science. Indiana University Press. p. 126. ISBN 978-0-253-33388-9.
- "Math for Poets and Drummers" (pdf). people.sju.edu.
- Plofker (2009), pages 54–56: "In the Chandah-sutra of Pingala, dating perhaps the third or second century BC, [...] Pingala's use of a zero symbol [śūnya] as a marker seems to be the first known explicit reference to zero. ... In the Chandah-sutra of Pingala, dating perhaps the third or second century BC, there are five questions concerning the possible meters for any value “n”. [...] The answer is (2)7 = 128, as expected, but instead of seven doublings, the process (explained by the sutra) required only three doublings and two squarings – a handy time saver where “n” is large. Pingala’s use of a zero symbol as a marker seems to be the first known explicit reference to zero."
- Amulya Kumar Bag, 'Binomial theorem in ancient India', Indian J. Hist. Sci. 1 (1966), 68–74.
- George Gheverghese Joseph (2000). The Crest of the Peacock, p. 254, 355. Princeton University Press.
- Klaus Mylius, Geschichte der altindischen Literatur, Wiesbaden (1983).
- Van Nooten, B. (1993-03-01). "Binary numbers in Indian antiquity". Journal of Indian Philosophy. 21 (1): 31–50. doi:10.1007/BF01092744.