Pingala
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Pingala
Pingala
Bornunclear, 3rd / 2nd century BCE[1]
ResidenceIndian subcontinent
EraMaurya or post-Maurya
Main interestsIndian mathematics, Sanskrit grammar
Notable worksAuthor of the Chandastra (also called Pingala-sutras), the earliest known treatise on Sanskrit prosody
Notable ideasm?tr?meru, binary numeral system, arithmetical triangle

Acharya Pingala[2] (Devanagari: pi?gala) (c. 3rd/2nd century BCE)[1] was an ancient Indian mathematician who authored the Chandastra (also called Pingala-sutras), the earliest known treatise on Sanskrit prosody.[3]

The Chandastra is a work of eight chapters in the late S?tra style, not fully comprehensible without a commentary. It has been dated to the last few centuries BCE.[4][5] The 10th century mathematician Halayudha wrote a commentary on the Chandastra and expanded it.

## Combinatorics

The Chandastra 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.[6] The discussion of the combinatorics of meter corresponds to the binomial theorem. Hal?yudha's commentary includes a presentation of Pascal's triangle (called meruprast?ra). Pingala's work also includes material related to the Fibonacci numbers, called m?tr?meru.[7]

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.[8] Pingala used the Sanskrit word nya explicitly to refer to zero.[9]

## Editions

• A. Weber, Indische Studien 8, Leipzig, 1863.

## Notes

1. ^ a b Kim Plofker (2009). Mathematics in India. Princeton University Press. pp. 55-56. ISBN 0-691-12067-6.
2. ^ Singh, Parmanand (1985). "The So-called Fibonacci Numbers in Ancient and Medieval India" (PDF). Historia Mathematica. Academic Press. 12: 232.
3. ^ 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.
4. ^ R. Hall, Mathematics of Poetry, has "c. 200 BC"
5. ^ Mylius (1983:68) considers the Chandas-sh?stra as "very late" within the Ved?nga corpus.
6. ^ Van Nooten (1993)
7. ^ Susantha Goonatilake (1998). Toward a Global Science. Indiana University Press. p. 126. ISBN 978-0-253-33388-9.
8. ^ "Math for Poets and Drummers" (pdf). people.sju.edu.
9. ^ Kim Plofker (2009), Mathematics in India, Princeton University Press, ISBN 978-0691120676, page 54-56. Quote - "In the Chandah-sutra of Pingala, dating perhaps the third or second century BC, [...] Pingala's use of a zero symbol [nya] as a marker seems to be the first known explicit reference to zero." Kim Plofker (2009), Mathematics in India, Princeton University Press, ISBN 978-0691120676, 55-56. "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.

## References

• 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. Retrieved .