Pingala
Pingala | |
|---|---|
| Born | unclear, 3rd / 2nd century BCE[1] |
| Residence | Indian subcontinent |
| Academic work | |
| Era | Maurya or post-Maurya |
| Main interests | Indian mathematics, Sanskrit grammar |
| Notable works | Author of the Chandaḥśāstra (also called Pingala-sutras), the earliest known treatise on Sanskrit prosody |
| Notable ideas | mātrāmeru, binary numeral system, arithmetical triangle |
Acharya Pingala[2] (Devanagari: पिङ्गल piṅgala) (c. 3rd/2nd century BCE)[1] was an ancient Indian mathematician who authored the Chandaḥśāstra (also called Pingala-sutras), the earliest known treatise on Sanskrit prosody.[3]
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.[4][5] The 10th century mathematician Halayudha wrote a commentary on the Chandaḥśāstra and expanded it.
Contents
1 Combinatorics
2 Editions
3 Notes
4 See also
5 References
6 External links
Combinatorics
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.[6] 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.[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 śūnya explicitly to refer to zero.[9]
Editions
A. Weber, Indische Studien 8, Leipzig, 1863.
Notes
^ ab Kim Plofker (2009). Mathematics in India. Princeton University Press. pp. 55–56. ISBN 0-691-12067-6..mw-parser-output cite.citation{font-style:inherit}.mw-parser-output .citation q{quotes:"""""""'""'"}.mw-parser-output .citation .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/12px-Wikisource-logo.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-maint{display:none;color:#33aa33;margin-left:0.3em}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}
^ Singh, Parmanand (1985). "The So-called Fibonacci Numbers in Ancient and Medieval India" (PDF). Historia Mathematica. Academic Press. 12: 232.
^ 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.
^ 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 [śūnya] 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.
See also
- Chandas
- Sanskrit prosody
- Indian mathematics
- Indian mathematicians
- History of the binomial theorem
- List of Indian mathematicians
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 2010-05-06.
External links
Math for Poets and Drummers, Rachel W. Hall, Saint Joseph's University, 2005.
Mathematics of Poetry, Rachel W. Hall
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