词条 | Katapayadi system | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
释义 |
Ka·ṭa·pa·yā·di (Devanagari: कटपयादि) system (also known as Paralppēru, Malayalam: പരല്പ്പേര്) of numerical notation is an ancient Indian system to depict letters to numerals for easy remembrance of numbers as words or verses. Assigning more than one letter to one numeral and nullifying certain other letters as valueless, this system provides the flexibility in forming meaningful words out of numbers which can be easily remembered. HistoryThe oldest available evidence of the use of Kaṭapayādi (Sanskrit: कटपयादि) system is from Grahacāraṇibandhana by Haridatta in 683 CE.[1] It has been used in Laghu·bhāskarīya·vivaraṇa written by Śaṅkara·nārāyaṇa in 869 CE.[2] Some argue that the system originated from Vararuci.[3] In some astronomical texts popular in Kerala planetary positions were encoded in the Kaṭapayādi system. The first such work is considered to be the Chandra-vakyani of Vararuci, who is traditionally assigned to the fourth century CE. Therefore, sometime in the early first millennium is a reasonable estimate for the origin of the Kaṭapayādi system.[4] Aryabhata, in his treatise Ārya·bhaṭīya, is known to have used a similar, more complex system to represent [[astronomical number]]s. There is no definitive evidence whether the Ka-ṭa-pa-yā-di system originated from Āryabhaṭa numeration.[5] Geographical spread of the useAlmost all evidences of the use of Ka-ṭa-pa-yā-di system is from south India, especially Kerala. Not much is known about its use in north India. However, on a Sanskrit astrolabe discovered in north India, the degrees of the altitude are marked in the Kaṭapayādi system. It is preserved in the Sarasvathy Bhavan Library of Sampurnanand Sanskrit University, Varanasi.[1] The Ka-ṭa-pa-yā-di system is not confined to India. Some Pali chronograms based on the Ka-ṭa-pa-yā-di system have been discovered in Burma.[6] Rules and practicesFollowing verse found in Śaṅkaravarman's Sadratnamāla explains the mechanism of the system.[7][8]
Transiliteration: nanyāvacaśca śūnyāni saṃkhyāḥ kaṭapayādayaḥ Translation: na (न), nya (ञ) and a (अ)-s, i.e., vowels represent zero. The nine integers are represented by consonant group beginning with ka, ṭa, pa, ya. In a conjunct consonant, the last of the consonants alone will count. A consonant without vowel is to be ignored. Explanation: The assignment of letters to the numerals are as per the following arrangement.
Variations
UsageMathematics and astronomy
അനൂനനൂന്നാനനനുന്നനിത്യൈ-സ്സമാഹതാശ്ചക്രകലാവിഭക്താഃചണ്ഡാംശുചന്ദ്രാധമകുംഭിപാലൈര്-വ്യാസസ്തദര്ദ്ധം ത്രിഭമൗര്വിക സ്യാത് Transliteration anūnanūnnānananunnanityaissmāhatāścakra kalāvibhaktoḥcaṇḍāṃśucandrādhamakuṃbhipālairvyāsastadarddhaṃ tribhamaurvika syāt It gives the circumference of a circle of diameter, anūnanūnnānananunnanityai (10,000,000,000) as caṇḍāṃśucandrādhamakuṃbhipālair (31415926536).
भद्राम्बुद्धिसिद्धजन्मगणितश्रद्धा स्म यद् भूपगी: Transliteration bhadrāṃbuddhisiddhajanmagaṇitaśraddhā sma yad bhūpagīḥ Splitting the consonants gives,
Reversing the digits to modern day usage of descending order of decimal places, we get 314159265358979324 which is the value of pi (π) to 17 decimal places, except the last digit might be rounded off to 4.
गोपीभाग्यमधुव्रात-श्रुग्ङिशोदधिसन्धिग ॥ खलजीवितखाताव गलहालारसंधर ॥ ಗೋಪೀಭಾಗ್ಯಮಧುವ್ರಾತ-ಶೃಂಗಿಶೋದಧಿಸಂಧಿಗ || ಖಲಜೀವಿತಖಾತಾವ ಗಲಹಾಲಾರಸಂಧರ || This verse directly yields the decimal equivalent of pi divided by 10: pi/10 = 0.31415926535897932384626433832792 గోపీభాగ్యమధువ్రాత-శృంగిశోదధిసంధిగ | ఖలజీవితఖాతావ గలహాలారసంధర || Carnatic music
Raga DheerasankarabharanamThe katapayadi scheme associates dha9 and ra2, hence the raga's melakarta number is 29 (92 reversed). Now 29 36, hence Dheerasankarabharanam has Ma1. Divide 28 (1 less than 29) by 6, the quotient is 4 and the remainder 4. Therefore, this raga has Ri2, Ga3 (quotient is 4) and Da2, Ni3 (remainder is 4). Therefore, this raga's scale is Sa Ri2 Ga3 Ma1 Pa Da2 Ni3 SA. Raga MechaKalyaniFrom the coding scheme Ma 5, Cha 6. Hence the raga's melakarta number is 65 (56 reversed). 65 is greater than 36. So MechaKalyani has Ma2. Since the raga's number is greater than 36 subtract 36 from it. 65-36=29. 28 (1 less than 29) divided by 6: quotient=4, remainder=4. Ri2 Ga3 occurs. Da2 Ni3 occurs. So MechaKalyani has the notes Sa Ri2 Ga3 Ma2 Pa Da2 Ni3 SA. Exception for SimhendramadhyamamAs per the above calculation, we should get Sa 7, Ha 8 giving the number 87 instead of 57 for Simhendramadhyamam. This should be ideally Sa 7, Ma 5 giving the number 57. So it is believed that the name should be written as Sihmendramadhyamam (as in the case of Brahmana in Sanskrit). Representation of datesImportant dates were remembered by converting them using Kaṭapayādi system. These dates are generally represented as number of days since the start of Kali Yuga. It is sometimes called kalidina sankhya.
This number is the time at which the work was completed represented as number of days since the start of Kali Yuga as per the Malayalam calendar. Others
പലഹാരേ പാലു നല്ലൂ, പുലര്ന്നാലോ കലക്കിലാംഇല്ലാ പാലെന്നു ഗോപാലന് - ആംഗ്ലമാസദിനം ക്രമാല് Transiliteration palahāre pālu nallū, pularnnālo kalakkilāṃillā pālennu gopālan - āṃgḷamāsadinaṃ kramāl Translation: Milk is best for breakfast, when it is morning, it should be stirred. But Gopālan says there is no milk - the number of days of English months in order. Converting pairs of letters using Kaṭapayādi yields - pala (പല) is 31, hāre (ഹാരേ) is 28, pālu പാലു = 31, nallū (നല്ലൂ) is 30, pular (പുലര്) is 31, nnālo (ന്നാലോ) is 30, kala (കല) is 31, kkilāṃ (ക്കിലാം) is 31, illā (ഇല്ലാ) is 30, pāle (പാലെ) is 31, nnu go (ന്നു ഗോ) is 30, pālan (പാലന്) is 31. See also
References1. ^1 Sreeramamula Rajeswara Sarma (1999), Kaṭapayādi Notation on a Sanskrit Astrolabe {{dead link|date=December 2017 |bot=InternetArchiveBot |fix-attempted=yes }} 2. ^{{cite web |url=http://www-history.mcs.st-and.ac.uk/Biographies/Sankara.html|title=Sankara Narayana|last=J J O'Connor |author2=E F Robertson|date=November 2000 |publisher=School of Mathematics and Statistics, University of St Andrews, Scotland |accessdate=1 January 2010}} 3. ^{{cite web|url=http://www-wireless.usenet-replayer.com/data/humanities/language/sanskrit/4154.html |archive-url=https://archive.is/20110717175501/http://www-wireless.usenet-replayer.com/data/humanities/language/sanskrit/4154.html |dead-url=yes |archive-date=17 July 2011 |title=Aryabhatta's numerical encoding |last=Usenet Discussion |accessdate=1 January 2010 |df= }} 4. ^{{cite book|last=Plofker|first=Kim |title=Mathematics in India|publisher=Princeton University Press|year=2008|pages=384|isbn= 978-0-691-12067-6}} 5. ^{{cite journal|last=J. F. Fleet|date=Apr 1912|title=The Ka-ta-pa-ya-di Notation of the Second Arya-Siddhanta |journal=The Journal of the Royal Asiatic Society of Great Britain and Ireland|publisher=Royal Asiatic Society of Great Britain and Ireland|pages=459–462 |jstor=25190035}} 6. ^{{cite journal|last=J.F. Fleet|date=Jul 1911|title=The Katapayadi System of Expressing Numbers|journal=The Journal of the Royal Asiatic Society of Great Britain and Ireland|publisher=Royal Asiatic Society of Great Britain and Ireland|pages=788–794 |jstor=25189917}} 7. ^Sarma, K.V. (2001). "Sadratnamala of Sankara Varman". Indian Journal of History of Science (Indian National Academy of Science, New Delhi) 36 (3–4 (Supplement)): 1–58. {{cite web|url=http://www.new.dli.ernet.in/rawdataupload/upload/insa/INSA_1/20005b67_s1.pdf |title=Archived copy |accessdate=17 December 2009 |deadurl=yes |archiveurl=https://web.archive.org/web/20150402140113/http://www.new.dli.ernet.in/rawdataupload/upload/insa/INSA_1/20005b67_s1.pdf |archivedate=2 April 2015 }} 8. ^{{cite journal|author=Anand Raman|title=The Ancient Katapayadi Formula and the Modern Hashing Method|url=http://www.speech.sri.com/people/anand/Papers/ieee-annals19-4.pdf|deadurl=yes|archiveurl=https://web.archive.org/web/20110616032135/http://www.speech.sri.com/people/anand/Papers/ieee-annals19-4.pdf|archivedate=16 June 2011|df=dmy-all}} 9. ^Aryabhatta's Numerical Encoding{{dead link|date=March 2017|bot=medic}}{{cbignore|bot=medic}} 10. ^Francis Zimmerman, 1989, Lilavati, gracious lady of arithmetic - India - A Mathematical Mystery Tour {{cite web|url=http://findarticles.com/p/articles/mi_m1310/is_1989_Nov/ai_8171045/ |title=Archived copy |accessdate=2010-01-03 |deadurl=yes |archiveurl=https://web.archive.org/web/20090906215627/http://findarticles.com/p/articles/mi_m1310/is_1989_Nov/ai_8171045/ |archivedate= 6 September 2009 |df= }} 11. ^Dr. C Krishnan Namboodiri, Chekrakal Illam, Calicut, Namboothiti.com {{cite web|url=http://www.namboothiri.com/articles/katapayaadi.htm|title="Katapayaadi" or "Paralpperu"|last= Dr. C Krishnan Namboodiri|publisher=Namboothiri Websites Trust|accessdate=1 January 2010}} 12. ^Visti Larsen, Choosing the auspicious name 13. ^ Further reading
4 : Numeral systems|Mnemonics|Indian mathematics|Kerala school |
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