799:, pages 54–56: "In the Chandah-sutra of Pingala, dating perhaps the third or second century BC, Pingala's use of a zero symbol 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) = 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."
472:
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
1667:
1632:
1819:
1369:
1644:
1065:
1799:
1691:
1586:
1761:
1814:
1024:
1865:
1850:
1824:
571:
235:
style, not fully comprehensible without a commentary. It has been dated to the last few centuries BCE. In the 10th century CE,
468:
to explicitly refer to the number. Pingala's binary representation increases towards the right, and not to the left as modern
1855:
1766:
865:
823:
659:
554:
1339:
1036:
1860:
1809:
1319:
1058:
1559:
1794:
1409:
1163:
1203:
1082:
946:
291:
1784:
1029:
838:
B. van Nooten, "Binary
Numbers in Indian Antiquity", Journal of Indian Studies, Volume 21, 1993, pp. 31–50
1804:
1789:
1051:
540:
1845:
1299:
971:
1309:
1599:
1465:
1314:
199:
144:
1725:
1607:
1198:
857:
1746:
1673:
260:
1486:
651:
Sanskrit
Prosody and Important Literary and Geographical Names in the Ancient History of India
649:
1603:
1595:
1097:
936:
546:
307:
1685:
1396:
1364:
941:
579:
110:
8:
1751:
1720:
1528:
1514:
1451:
1344:
1289:
1284:
1238:
1188:
734:
211:
180:
815:
The mathematics of harmony: from Euclid to contemporary mathematics and computer science
1253:
1074:
1005:
931:
850:
780:
754:
746:
159:
71:
901:
699:
1756:
1638:
1458:
1233:
1228:
1102:
1009:
861:
819:
655:
596:
550:
478:
190:
171:
135:
115:
600:
1472:
1354:
1324:
1294:
997:
926:
758:
742:
738:
592:
315:
217:
98:
75:
67:
1437:
1208:
884:
256:
1679:
1626:
1554:
1500:
1349:
1218:
813:
690:(1983:68) considers the Chandas-shāstra as "very late" within the Vedānga corpus.
1258:
1178:
464:
1650:
1570:
1507:
1430:
1416:
1334:
1329:
1263:
1107:
809:
588:
498:
469:
287:
1158:
1839:
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1662:
1591:
1493:
311:
56:
1153:
1117:
252:
1697:
1535:
1479:
1423:
1243:
1223:
1148:
977:
687:
483:
454:
Because of this, Pingala is sometimes also credited with the first use of
306:
syllables, using a recursive formula, that results in a partially ordered
280:
241:
225:
216:'Pingala's Threads of Knowledge'), the earliest known treatise on
164:
87:
1703:
1656:
1444:
1390:
1173:
1127:
536:
474:
20:
750:
722:
624:
310:
representation. Pingala is credited with being the first to express the
1402:
1359:
1213:
1193:
1168:
1001:
155:
1521:
1183:
1143:
1043:
267:
248:
236:
1304:
1268:
1248:
988:
Van Nooten, B. (1993-03-01). "Binary numbers in Indian antiquity".
459:
232:
921:
128:
1370:
Shanti Swarup
Bhatnagar Prize recipients in Mathematical Science
1112:
572:"The So-called Fibonacci Numbers in Ancient and Medieval India"
1571:
Infinite series expansions for the trigonometric functions
286:
presents a formula to generate systematic enumerations of
959:
Amulya Kumar Bag, 'Binomial theorem in ancient India',
477:. Pingala's work also includes material related to the
270:, the 2nd century CE scholar who authored Mahabhashya.
204:
149:
1820:
Ramanujan
Institute for Advanced Study in Mathematics
1565:
455:
19:
For the subtle energy channel described in yoga, see
847:
647:
849:
185:'A Treatise on Prosody'), also called the
1837:
16:3rd-2nd century BC Indian mathematician and poet
515:Nirnayasagar Press, Chand Shastra, Bombay, 1938
1059:
899:
808:
531:
529:
1587:Kerala school of astronomy and mathematics
1066:
1052:
987:
341:contains only words of the desired length
654:. Motilal Banarsidass. pp. 648–649.
1815:Homi Bhabha Centre for Science Education
693:
526:
231:is a work of eight chapters in the late
796:
545:. Princeton University Press. pp.
535:
508:Janakinath Kabyatittha & brothers,
1838:
1073:
781:"A HISTORY OF PIṄGALA'S COMBINATORICS"
239:wrote a commentary elaborating on the
1047:
982:Geschichte der altindischen Literatur
569:
266:Another think tank identifies him as
856:. Indiana University Press. p.
721:Hall, Rachel Wells (February 2008).
720:
700:François & Ponsonnet (2013: 184)
1340:Subbayya Sivasankaranarayana Pillai
625:"Pingala – Timeline of Mathematics"
13:
1800:Institute of Mathematical Sciences
966:George Gheverghese Joseph (2000).
290:, of all possible combinations of
97:), the earliest known treatise on
14:
1877:
1810:Harish-Chandra Research Institute
1320:K. R. Parthasarathy (probabilist)
1017:
158:) was an ancient Indian poet and
778:
448:GGG LGG GLG LLG GGL LGL GLL LLL
273:
942:History of the binomial theorem
893:
877:
841:
832:
812:; Olsen, Scott Anthony (2009).
802:
790:
772:
247:. According to some historians
101:. Creator of Pingala's formula.
39:unclear, 3rd or 2nd century BCE
1795:Chennai Mathematical Institute
1164:Melpathur Narayana Bhattathiri
743:10.1080/10724117.2008.11974752
714:
705:
681:
668:
641:
617:
563:
409:syllables in a word of length
1:
1866:2nd-century BC mathematicians
1851:Ancient Indian mathematicians
953:
947:List of Indian mathematicians
848:Susantha Goonatilake (1998).
723:"Math for Poets and Drummers"
1856:Ancient Sanskrit grammarians
1785:Indian Statistical Institute
990:Journal of Indian Philosophy
648:Vaman Shivaram Apte (1970).
597:10.1016/0315-0860(85)90021-7
7:
1805:Indian Institute of Science
1790:Bhaskaracharya Pratishthana
1668:A. A. Krishnaswamy Ayyangar
1633:Shankar Balakrishna Dikshit
1560:Hindu–Arabic numeral system
1025:Math for Poets and Drummers
914:
492:
484:
281:
251:Pingala was the brother of
242:
226:
205:
165:
150:
88:
10:
1882:
1300:Subrahmanyan Chandrasekhar
972:Princeton University Press
18:
1775:
1739:
1713:
1617:
1579:
1545:
1382:
1310:Veeravalli S. Varadarajan
1277:
1136:
1090:
1081:
1030:Saint Joseph's University
570:Singh, Parmanand (1985).
401:Possible combinations of
194:
175:
139:
122:
105:
81:
63:
52:
47:
43:
35:
28:
1861:Indian Sanskrit scholars
1726:Henry Thomas Colebrooke
1315:S. R. Srinivasa Varadhan
968:The Crest of the Peacock
519:
375:to each element of list
365:to each element of list
162:, and the author of the
1199:Mādhava of Saṅgamagrāma
886:Chhanda Sutra - Pingala
852:Toward a Global Science
322:Create a syllable list
259:, considered the first
963:1 (1966), 68–74.
900:Pingalacharya (1938).
424:Possible combinations
326:comprising one light (
1410:Brāhmasphuṭasiddhānta
1037:Mathematics of Poetry
937:Indian mathematicians
676:Mathematics of Poetry
154:; c. 3rd–2nd century
116:binary numeral system
1686:T. A. Saraswati Amma
1397:Bakhshali manuscript
1365:Kannan Soundararajan
970:, p. 254, 355.
961:Indian J. Hist. Sci.
872:Virahanka Fibonacci.
818:. World Scientific.
735:Taylor & Francis
580:Historia Mathematica
542:Mathematics in India
510:ChhandaSutra-Pingala
261:descriptive linguist
1762:Islamic mathematics
1721:Walter Eugene Clark
1515:Vasishtha Siddhanta
1487:Siddhānta Shiromani
1466:Paitamaha Siddhanta
1345:Tilak Raj Prabhakar
1290:Satyendra Nath Bose
1285:Srinivasa Ramanujan
1239:Nilakantha Somayaji
1028:, Rachel W. Hall,
984:, Wiesbaden (1983).
413:
391:and rename as list
257:Sanskrit grammarian
1254:Gangesha Upadhyaya
1075:Indian mathematics
1002:10.1007/BF01092744
932:Indian mathematics
768:– via JSTOR.
400:
72:Indian mathematics
1846:Fibonacci numbers
1833:
1832:
1639:Sudhakara Dvivedi
1459:Paulisa Siddhanta
1378:
1377:
1234:Jagannatha Samrat
1229:Achyuta Pisharati
1040:, Rachel W. Hall
867:978-0-253-33388-9
825:978-981-277-582-5
711:Van Nooten (1993)
678:, has "c. 200 BC"
661:978-81-208-0045-8
556:978-0-691-12067-6
512:, Calcutta, 1931.
505:8, Leipzig, 1863.
479:Fibonacci numbers
458:, as he used the
452:
451:
337:Repeat till list
215:
203:
184:
148:
126:
125:
1873:
1473:Romaka Siddhanta
1355:Akshay Venkatesh
1325:M. S. Narasimhan
1295:P.C. Mahalanobis
1088:
1087:
1068:
1061:
1054:
1045:
1044:
1013:
927:Sanskrit prosody
908:
907:
897:
891:
890:
881:
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855:
845:
839:
836:
830:
829:
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800:
794:
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712:
709:
703:
697:
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685:
679:
672:
666:
665:
645:
639:
638:
636:
635:
621:
615:
614:
612:
611:
605:
599:. Archived from
576:
567:
561:
560:
533:
503:Indische Studien
487:
414:
399:
371:Append syllable
361:Append syllable
302:, for a word of
284:
245:
229:
218:Sanskrit prosody
210:
208:
198:
196:
179:
177:
168:
153:
143:
141:
99:Sanskrit prosody
91:
85:Author of the "
76:Sanskrit grammar
68:Sanskrit prosody
26:
25:
1881:
1880:
1876:
1875:
1874:
1872:
1871:
1870:
1836:
1835:
1834:
1829:
1777:
1771:
1735:
1709:
1680:C. T. Rajagopal
1627:Bapudeva Sastri
1619:
1613:
1575:
1555:Brahmi numerals
1547:
1541:
1501:Surya Siddhanta
1374:
1350:Manjul Bhargava
1273:
1132:
1077:
1072:
1020:
956:
951:
917:
912:
911:
898:
894:
883:
882:
878:
868:
846:
842:
837:
833:
826:
810:Stakhov, Alexey
807:
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783:
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719:
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698:
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574:
568:
564:
557:
534:
527:
522:
495:
347:Replicate list
276:
93:" (also called
31:
24:
17:
12:
11:
5:
1879:
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1701:
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1665:
1660:
1654:
1651:P. C. Sengupta
1648:
1642:
1636:
1630:
1623:
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1615:
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1611:
1589:
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1562:
1557:
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1549:
1543:
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1525:
1518:
1511:
1508:Tantrasamgraha
1504:
1497:
1490:
1483:
1476:
1469:
1462:
1455:
1448:
1441:
1434:
1431:Karanapaddhati
1427:
1420:
1417:Ganita Kaumudi
1413:
1406:
1399:
1394:
1386:
1384:
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1379:
1376:
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1373:
1372:
1367:
1362:
1357:
1352:
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1342:
1337:
1335:Harish-Chandra
1332:
1330:C. S. Seshadri
1327:
1322:
1317:
1312:
1307:
1302:
1297:
1292:
1287:
1281:
1279:
1275:
1274:
1272:
1271:
1266:
1264:Sankara Variar
1261:
1256:
1251:
1246:
1241:
1236:
1231:
1226:
1221:
1216:
1211:
1206:
1201:
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1171:
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1131:
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1125:
1120:
1115:
1110:
1105:
1100:
1094:
1092:
1085:
1083:Mathematicians
1079:
1078:
1071:
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1063:
1056:
1048:
1042:
1041:
1033:
1019:
1018:External links
1016:
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797:Plofker (2009)
789:
779:Shah, Jayant.
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589:Academic Press
562:
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470:binary numbers
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316:Sanskrit metre
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187:Pingala-sutras
166:Chhandaḥśāstra
124:
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120:
119:
107:
103:
102:
95:Pingala-sutras
83:
79:
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65:
64:Main interests
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59:or post-Maurya
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1209:Mahendra Sūri
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940:
938:
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903:Chand Shastra
896:
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821:
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727:Math Horizons
724:
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708:
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689:
684:
677:
671:
663:
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653:
652:
644:
630:
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606:on 2019-07-24
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1670:(1892– 1953)
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1179:Govindasvāmi
1149:Āryabhaṭa II
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1714:Translators
1706:(1919–2005)
1704:K. V. Sarma
1700:(1918–2007)
1694:(1918–1992)
1688:(1918–2000)
1682:(1903–1978)
1676:(1901–1954)
1674:A. N. Singh
1659:(1888–1958)
1657:B. B. Datta
1653:(1876–1962)
1647:(1861–1916)
1641:(1855–1910)
1635:(1853–1898)
1629:(1821–1900)
1620:mathematics
1564:Symbol for
1548:innovations
1445:Lokavibhaga
1391:Aryabhatiya
1189:Jyeṣṭhadeva
1174:Brahmagupta
1159:Bhāskara II
1144:Āryabhaṭa I
1128:Yajnavalkya
421:characters)
334:) syllable:
300:) syllables
21:Nadi (yoga)
1840:Categories
1663:T. Hayashi
1546:Pioneering
1529:Yuktibhāṣā
1452:Pātīgaṇita
1403:Bijaganita
1360:Ravi Vakil
1214:Munishvara
1194:Kamalakara
1169:Brahmadeva
1154:Bhāskara I
1103:Baudhayana
954:References
634:2021-08-21
610:2018-11-29
1692:S. N. Sen
1600:New Delhi
1522:Veṇvāroha
1383:Treatises
1184:Halayudha
1137:Classical
1108:Katyayana
1098:Apastamba
1010:171039636
737:: 10–12.
674:R. Hall,
485:mātrāmeru
481:, called
351:as lists
268:Patanjali
237:Halayudha
200:romanized
145:romanized
111:mātrāmeru
1608:Varanasi
1566:zero (0)
1438:Līlāvatī
1305:C.R. Rao
1269:Virasena
1249:Sridhara
1219:Narayana
1204:Mahāvīra
915:See also
751:25678735
629:Mathigon
539:(2009).
499:A. Weber
493:Editions
460:Sanskrit
387:to list
249:Maharshi
191:Sanskrit
172:Sanskrit
136:Sanskrit
1747:Babylon
1580:Centres
1244:Śrīpati
1123:Pingala
1091:Ancient
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922:Chandas
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318:, eg.
308:binary
288:metres
253:Pāṇini
140:पिङ्गल
57:Maurya
1752:China
1006:S2CID
784:(PDF)
755:S2CID
747:JSTOR
733:(3).
604:(PDF)
587:(3).
575:(PDF)
547:55–56
520:Notes
465:śūnya
462:word
407:Laghu
294:laghu
233:Sūtra
1825:TIFR
862:ISBN
820:ISBN
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