Pingala

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title: "Pingala" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["fibonacci-numbers", "ancient-indian-mathematicians", "ancient-sanskrit-grammarians", "indian-sanskrit-scholars", "2nd-century-bc-mathematicians"] description: "3rd–2nd century BC Indian mathematician and poet" topic_path: "geography/india" source: "https://en.wikipedia.org/wiki/Pingala" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary 3rd–2nd century BC Indian mathematician and poet ::

::data[format=table title="Infobox scholar"]

Field	Value
name	Pingala
birth_date	c. 3rd or 2nd century BCE
era	Maurya or post-Maurya
main_interests	Sanskrit prosody, Indian mathematics, Sanskrit grammar
notable_ideas	Mātrāmeru, Binary numeral system.
major_works	Author of the "**" (also called Pingala Sutras), the earliest known treatise on Sanskrit prosody, Creator of Pingala's formula.
::

| image = | caption = | name = Pingala | birth_date = c. 3rd or 2nd century BCE | era = Maurya or post-Maurya | main_interests = Sanskrit prosody, Indian mathematics, Sanskrit grammar | notable_ideas = Mātrāmeru, Binary numeral system. | major_works = Author of the "**" (also called Pingala Sutras), the earliest known treatise on Sanskrit prosody, Creator of Pingala's formula. | influences = | influenced = Acharya Pingala (; c. 3rd2nd century BCE) was an ancient Indian poet and mathematician, and the author of the ** (), also called the Pingala Sutras (), the earliest known treatise on Sanskrit prosody.

The ** 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 CE, Halayudha wrote a commentary elaborating on the **. According to some historians Maharshi Pingala was the brother of Pāṇini, the famous Sanskrit grammarian, considered the first descriptive linguist. Others identify him as Patanjali, the 2nd century CE scholar who authored Mahabhashya.

Combinatorics

The ** presents a formula to generate systematic enumerations of metres, of all possible combinations of light (laghu) and heavy (guru) syllables, for a word of n syllables, using a recursive formula, that results in a partially ordered binary representation. Pingala is credited with being the first to express the combinatorics of Sanskrit metre, e.g.:

• Create a syllable list x comprising one light (L) and heavy (G) syllable
• Repeat till list x contains only words of the desired length n
  • Replicate list x as lists a and b
    • Append syllable L to each element of list a
    • Append syllable G to each element of list b
  • Append lists b to list a and rename as list x

::data[format=table title="Possible combinations of ''Guru'' and ''Laghu'' syllables in a word of length ''n''{{Cite web |last=Shah |first=Jayant |title=A History of Pingala's Combinatorics |url=https://web.northeastern.edu/shah/papers/Pingala.pdf}}"]

Word length (n characters)	Possible combinations
1	G L
2	GG LG GL LL
3	GGG LGG GLG LLG GGL LGL GLL LLL
::

Because of this, Pingala is sometimes also credited with the first use of zero, as he used the Sanskrit word śūnya to explicitly refer to the number. Pingala's binary representation increases towards the right, and not to the left as modern binary numbers usually do. In Pingala's system, the numbers start from number one, and not zero. Four short syllables "0000" is the first pattern and corresponds to the value one. The numerical value is obtained by adding one to the sum of place values. Pingala's work also includes material related to the Fibonacci numbers, called **.

Editions

• A. Weber, Indische Studien 8, Leipzig, 1863.
• Janakinath Kabyatittha & Brothers, Pingala Chhanda Sutram, Calcutta, 1931.
• Nirnayasagar Press, Chand Shastra, Bombay, 1938.

Notes

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).
• {{Cite journal | doi = 10.1007/BF01092744 | volume = 21 | issue = 1 | pages = 31–50 | last = Van Nooten | first = B. | title = Binary numbers in Indian antiquity | journal = Journal of Indian Philosophy | date = 1993-03-01 | s2cid = 171039636

References

1. Singh, Parmanand. (1985). "The So-called Fibonacci Numbers in Ancient and Medieval India". [[Academic Press]].
2. Plofker, Kim. (2009). "Mathematics in India". Princeton University Press.
3. "Pingala – Timeline of Mathematics".
4. Vaman Shivaram Apte. (1970). "Sanskrit Prosody and Important Literary and Geographical Names in the Ancient History of India". Motilal Banarsidass.
5. R. Hall, ''Mathematics of Poetry'', has "c. 200 BC"
6. [[Klaus Mylius. Mylius]] (1983:68) considers the Chandas-shāstra as "very late" within the Vedānga corpus.
7. [[Pāṇini#FPencyclo. François & Ponsonnet (2013: 184)]].
8. Van Nooten (1993)
9. Hall, Rachel Wells. (February 2008). "Math for Poets and Drummers". [[Taylor & Francis]].
10. Shah, Jayant. "A History of Pingala's Combinatorics".
11. {{harvtxt. Plofker
12. (2009). "The mathematics of harmony: from Euclid to contemporary mathematics and computer science". World Scientific.
13. B. van Nooten, "Binary Numbers in Indian Antiquity", Journal of Indian Studies, Volume 21, 1993, pp. 31–50
14. Susantha Goonatilake. (1998). "Toward a Global Science". Indiana University Press.
15. "Chhanda Sutra – Pingala".
16. Pingalacharya. (1938). "Chand Shastra".

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