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{{Short description|Ancient Indian alphasyllabic numeral system}}
{{Use dmy dates|date=December 2019}}
{{Cleanup lang|date=August 2022}}
{{Contains special characters|Indic}}
[[File:Katapayadi-eng.svg|thumb|250px|KaTaPaYadi System – Values]]
{{Numeral systems|expand=Sign-value notation}}

'''''Kaṭapayādi'' system''' ([[Devanagari]]: कटपयादि, also known as ''Paralppēru'', Malayalam: [[:ml:പരല്‍പ്പേര്|പരല്‍പ്പേര്]]) of numerical notation is an [[ancient]] [[India]]n [[alphasyllabic numeral system]] to depict [[letter (alphabet)|letters]] to [[number|numerals]] for easy remembrance of [[number]]s as [[words]] or [[Verse (poetry)|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.

== History ==
The oldest available evidence of the use of ''Kaṭapayādi'' (Sanskrit: कटपयादि) system is from ''Grahacāraṇibandhana'' by [[Haridatta]] in 683 [[Common Era|CE]].<ref name="astro">Sreeramamula Rajeswara Sarma, THE ''KATAPAYADI'' SYSTEM OF NUMERICAL NOTATION AND ITS SPREAD OUTSIDE KERALA, ''Rev. d'Histoire de Mathmatique'' 18 (2012)
[https://srsarma.in/pdf/articles/2012_Katapayadi_System.pdf]</ref> It has been used in ''Laghu·bhāskarīya·vivaraṇa'' written by ''[[Sankara Narayana|Śaṅkara·nārāyaṇa]]'' in 869 [[Common Era|CE]].<ref>{{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 |access-date=1 January 2010}}</ref>

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 [[Common Era|CE]]. Therefore, sometime in the early first millennium is a reasonable estimate for the origin of the ''Kaṭapayādi'' system.<ref>{{cite book|last=Plofker|first=Kim |title=Mathematics in India|title-link=Mathematics in India (book)|publisher=[[Princeton University Press]]|year=2009|pages=384|isbn= 978-0-691-12067-6}}</ref>

[[Aryabhata]], in his treatise ''[[Ārya·bhaṭīya]]'', is known to have used a similar, more complex system to represent [[Large numbers#Astronomical|astronomical number]]s. There is no definitive evidence whether the ''Ka-ṭa-pa-yā-di'' system originated from [[Āryabhaṭa numeration]].<ref>{{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 | volume=44|  doi=10.1017/S0035869X00043197|jstor=25190035|s2cid=163907655 |url=https://zenodo.org/record/1970444}}</ref>

== Geographical spread of the use ==
Almost 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 Sarasvati Bhavan Library of [[Sampurnanand Sanskrit University]], [[Varanasi]].
<ref>Sreeramamula Rajeswara Sarma (1999), Kaṭapayādi Notation on a Sanskrit Astrolabe. Ind. J. Hist. Sc.34(4) (1999)[https://www.insa.nic.in/writereaddata/UpLoadedFiles/IJHS/Vol34_4_2_SRSarma.pdf]</ref>

The ''Ka-ṭa-pa-yā-di'' system is not confined to India. Some [[Pali]] [[chronogram]]s based on the ''Ka-ṭa-pa-yā-di'' system have been discovered in [[Burma]].<ref>{{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|volume=43|issue=3|publisher=[[Royal Asiatic Society of Great Britain and Ireland]]|pages=788–794 |doi=10.1017/S0035869X00041952|jstor=25189917|s2cid=163597699 |url=https://zenodo.org/record/1512928}}</ref>

== Rules and practices ==
Following verse found in [[Sankara Varman|Śaṅkaravarman's]] ''[[Sadratnamala|Sadratnamāla]]'' explains the mechanism of the system.<ref>[[K. V. Sarma|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 |access-date=17 December 2009 |url-status=dead |archive-url=https://web.archive.org/web/20150402140113/http://www.new.dli.ernet.in/rawdataupload/upload/insa/INSA_1/20005b67_s1.pdf |archive-date=2 April 2015 }}</ref><ref>{{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|url-status=dead|archive-url=https://web.archive.org/web/20110616032135/http://www.speech.sri.com/people/anand/Papers/ieee-annals19-4.pdf|archive-date=16 June 2011}}</ref>
<blockquote>
नञावचश्च शून्यानि संख्या: कटपयादय:।<BR>
मिश्रे तूपान्त्यहल् संख्या न च चिन्त्यो हलस्वर:॥
</blockquote>

Transliteration:
<blockquote>
''nanyāvachaścha śūnyāni sankhyāḥ kaṭapayādayaḥ''<BR>
''miśre tūpāntyahal sankhyā na cha chintyo halasvaraḥ''
</blockquote>

Translation: ''na'' (न), ''ña'' (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 a vowel is to be ignored.

Explanation: The assignment of letters to the numerals are as per the following arrangement (In Devanagari, Kannada, Telugu & Malayalam scripts respectively)

{| class="wikitable" align="center" style="text-align:center" 
|-
! width="50pt"|1
! width="50pt"|2
! width="50pt"|3
! width="50pt"|4
! width="50pt"|5
! width="50pt"|6
! width="50pt"|7
! width="50pt"|8
! width="50pt"|9
! width="50pt"|0
|-
| ''ka''  क ಕ క ക
| ''kha''  ख ಖ ఖ ഖ
| ''ga''  ग ಗ గ ഗ
| ''gha''  घ ಘ ఘ ഘ
| ''nga''  ङ ಙ
ఙ  ങ
| ''ca''  च ಚ చ ച
| ''cha''  छ ಛ ఛ ഛ
| ''ja''  ज ಜ జ ജ
| ''jha''  झ ಝ ఝ ഝ
| ''nya''  ञ ಞ ఞ ഞ
|-
| ''ṭa''  ट ಟ ట ട    
| ''ṭha''  ठ ಠ ఠ ഠ
| ''ḍa''  ड ಡ డ ഡ
| ''ḍha''  ढ ಢ ఢ ഢ
| ''ṇa''  ण ಣ ణ ണ
| ''ta''  त ತ త ത
| ''tha''  थ ಥ థ ഥ
| ''da''  द ದ ద ദ
| ''dha''  ध ಧ ధ ധ
| ''na''  न ನ న ന
|-
| ''pa''  प ಪ ప പ    
| ''pha''  फ ಫ ఫ ഫ
| ''ba''  ब బ ബ
| ''bha''  भ ಭ భ ഭ
| ''ma''  म ಮ మ മ
| –
| –
| –
| –
| – 
|-
| ''ya''  य ಯ య യ       
| ''ra''  र ರ ర ര
| ''la''  ल ల ల ല
| ''va''  व ವ వ വ
| ''śa''  श ಶ శ ശ
| ''ṣa''  ष ಷ ష ഷ
| ''sa''  स ಸ స സ
| ''ha''  ह ಹ హ ഹ
| –
| –
|}

* Consonants have numerals assigned as per the above table. For example, ba (ब) is always 3 whereas 5 can be represented by either ''nga'' (ङ) or ''ṇa'' (ण) or ''ma'' (म) or ''śha'' (श).
* All stand-alone vowels like ''a'' (अ) and ''ṛ'' (ऋ) are assigned to zero.
* In case of a conjunct, consonants attached to a non-vowel will be valueless. For example, ''kya'' (क्य) is formed by, ''k'' (क्) + ''y'' (य्) + ''a'' (अ). The only consonant standing with a vowel is ''ya'' (य). So the corresponding numeral for ''kya'' (क्य) will be 1.
* There is no way of representing the [[decimal separator]] in the system.
* Indians used the [[Hindu–Arabic numeral system]] for numbering, traditionally written in increasing place values from left to right. This is as per the rule "अङ्कानां वामतो गतिः" which means numbers go from right to left.

=== Variations ===
* The [[consonant]], ḷ (Malayālam: ള, Devanāgarī: ळ, Kannada: ಳ) is employed in works using the Kaṭapayādi system, like [[Mādhava's sine table]].
* Late medieval practitioners do not map the stand-alone vowels to zero. But, it is sometimes considered valueless.

== Usage ==

=== Mathematics and astronomy ===
* [[Mādhava's sine table]] constructed by 14th century [[Kerala]] [[mathematician]]-[[astronomer]] [[Mādhava of Saṅgama·grāma]] employs the Kaṭapayādi system to list the trigonometric sines of angles.
* ''[[Karaṇa·paddhati]]'', written in the 15th century, has the following ''śloka'' for the value of [[pi]] (π)
<blockquote>
:അനൂനനൂന്നാനനനുന്നനിത്യൈ-
:സ്സമാഹതാശ്ചക്രകലാവിഭക്താഃ
:ചണ്ഡാംശുചന്ദ്രാധമകുംഭിപാലൈര്‍-
:വ്യാസസ്തദര്‍ദ്ധം ത്രിഭമൗര്‍വിക സ്യാത്‌
</blockquote>
:Transliteration
<blockquote>
:''anūnanūnnānananunnanityai''
:''ssmāhatāścakra kalāvibhaktoḥ''
:''caṇḍāṃśucandrādhamakuṃbhipālair''
''vyāsastadarddhaṃ tribhamaurvika syāt''
</blockquote>

:It gives the circumference of a circle of diameter, ''anūnanūnnānananunnanityai'' (10,000,000,000) as ''caṇḍāṃśucandrādhamakuṃbhipālair'' (31415926536).

* Śaṅkara·varman's ''[[Sad·ratna·mālā]]'' uses the Kaṭapayādi system. The first verse of Chapter 4 of the ''[[Sad·ratna·mālā]]'' ends with the line:<ref>Sarma (2001), [https://archive.org/details/sadratnamalaofsankaravarmaed.sarmak.v.ijhsvol3634_123_S/page/n31/mode/2up p. 26]</ref>
<blockquote>
:(स्याद्) भद्राम्बुधिसिद्धजन्मगणितश्रद्धा स्म यद् भूपगी:
</blockquote>
:Transliteration
<blockquote>
:''(syād) bhadrāmbudhisiddhajanmagaṇitaśraddhā sma yad bhūpagīḥ''
</blockquote>
:Splitting the consonants in the relevant phrase gives,
{| class="wikitable" align="center" style="text-align:center" 
|-
! width="20pt"|भ bha
! width="20pt"|द् d
! width="20pt"|रा rā
! width="20pt"|म् m
! width="20pt"|बु bu
! width="20pt"|द् d
! width="20pt"|धि dhi
! width="20pt"|सि si
! width="20pt"|द् d
! width="20pt"|ध dha
! width="20pt"|ज ja
! width="20pt"|न् n
! width="20pt"|म ma
! width="20pt"|ग ga
! width="20pt"|णि ṇi
! width="20pt"|त ta
! width="20pt"|श् ś
! width="20pt"|र ra
! width="20pt"|द् d
! width="20pt"|धा dhā
! width="20pt"|स् s
! width="20pt"|म ma
! width="20pt"|य ya
! width="20pt"|द् d
! width="20pt"|भू bhū
! width="20pt"|प pa
! width="20pt"|गी gī
|-
| 4
| –
| 2
| –
| 3
| –
| 9
| 7
| –
| 9
| 8
| –
| 5
| 3
| 5
| 6
| –
| 2
| –
| 9
| –
| 5
| 1
| –
| 4
| 1
| 3
|}

: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.

* This verse encrypts the value of [[pi]] (π) up to 31 decimal places.

  गोपीभाग्यमधुव्रात-शृङ्गिशोदधिसन्धिग॥
  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 ||
Traditionally, the order of digits are reversed to form the number, in katapayadi system. This rule is violated in this sloka.

=== Carnatic music ===
[[Image:Melakarta.katapayadi.sankhya.72.png|250px|thumb|[[Melakarta]] chart as per Kaṭapayādi system]]
* The [[melakarta]] [[ragas]] of the Carnatic music are named so that the first two syllables of the name will give its number. This system is sometimes called the [[Ka-ta-pa-ya-di sankhya]]. The [[Swara]]s 'Sa' and 'Pa' are fixed, and here is how to get the other swaras from the melakarta number.

# Melakartas 1 through 36 have Ma1 and those from 37 through 72 have Ma2.
# The other notes are derived by noting the (integral part of the) quotient and remainder when one less than the melakarta number is divided by 6. If the melakarta number is greater than 36, subtract 36 from the melakarta number before performing this step.
# 'Ri' and 'Ga' positions: the raga will have:
#* '''Ri1''' and '''Ga1''' if the quotient is 0
#* '''Ri1''' and '''Ga2''' if the quotient is 1
#* '''Ri1''' and '''Ga3''' if the quotient is 2
#* '''Ri2''' and '''Ga2''' if the quotient is 3
#* '''Ri2''' and '''Ga3''' if the quotient is 4
#* '''Ri3''' and '''Ga3''' if the quotient is 5
# 'Da' and 'Ni' positions: the raga will have:
#* '''Da1''' and '''Ni1''' if remainder is 0
#* '''Da1''' and '''Ni2''' if remainder is 1
#* '''Da1''' and '''Ni3''' if remainder is 2
#* '''Da2''' and '''Ni2''' if remainder is 3
#* '''Da2''' and '''Ni3''' if remainder is 4
#* '''Da3''' and '''Ni3''' if remainder is 5

*See [[Swara#Swaras in Carnatic music|swaras in Carnatic music]] for details on above notation.

====Raga [[Sankarabharanam (raga)|Dheerasankarabharanam]]====

The katapayadi scheme associates dha<math>\leftrightarrow</math>9 and ra<math>\leftrightarrow</math>2, hence the raga's melakarta number is 29 (92 reversed). 29 less than 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 [[Kalyani (raga)|MechaKalyani]]====

From the coding scheme Ma <math>\leftrightarrow</math> 5, Cha <math>\leftrightarrow</math> 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 [[Simhendramadhyamam]] ====
As per the above calculation, we should get Sa <math>\leftrightarrow</math> 7, Ha <math>\leftrightarrow</math> 8 giving the number 87 instead of 57 for Simhendramadhyamam. This should be ideally Sa <math>\leftrightarrow</math> 7, Ma <math>\leftrightarrow</math> 5 giving the number 57. So it is believed that the name should be written as ''Sihmendramadhyamam'' (as in the case of Bra'''hm'''ana in Sanskrit).

=== Representation of dates ===
Important 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''.

* The [[Malayalam calendar]] known as ''kollavarsham'' (Malayalam: കൊല്ലവര്‍ഷം) was adopted in Kerala beginning from 825 [[Common Era|CE]], revamping some calendars. This date is remembered as ''āchārya vāgbhadā'', converted using ''Kaṭapayādi'' into 1434160 days since the start of [[Kali Yuga]].<ref>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=Lilavati, gracious lady of arithmetic - India - A Mathematical Mystery Tour &#124; UNESCO Courier &#124; Find Articles at BNET |access-date=3 January 2010 |url-status=dead |archive-url=https://web.archive.org/web/20090906215627/http://findarticles.com/p/articles/mi_m1310/is_1989_Nov/ai_8171045/ |archive-date= 6 September 2009 }}</ref>
* [[Narayaniyam]], written by [[Melpathur Narayana Bhattathiri]], ends with the line, āyurārogyasaukhyam (ആയുരാരോഗ്യസൌഖ്യം) which means long-life, health and happiness.<ref>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|access-date=1 January 2010}}</ref>

{| class="wikitable" align="center" style="text-align:center" 
|-
! width="200pt" align="left"|In [[Malayalam]]
! width="200pt" align="left"|ആയുരാരോഗ്യസൌഖ്യം
|-
| |In [[Devanagari]]
| आयुरारोग्यसौख्यम्
|-
| |In [[IAST]]
| āyurārogyasaukhyam
|-
| |Value as per ''Kaṭapayādi''
| 1712210
|}

: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 ===
* Some people use the ''Kaṭapayādi'' system in naming newborns.<ref>[http://srigaruda.com/visti/attachments/061_name%5B1%5D.pdf Visti Larsen, Choosing the auspicious name]{{Dead link|date=October 2022 |bot=InternetArchiveBot |fix-attempted=yes }}</ref><ref>{{Cite web|url=http://varahamihira.blogspot.com/2004/06/principles-of-naming.html|title = The Principles of Naming}}</ref>
* The following verse compiled in Malayalam by Koduṅṅallur Kuññikkuṭṭan Taṃpurān using ''Kaṭapayādi'' is the number of days in the months of [[Gregorian Calendar]].
<blockquote>
:പലഹാരേ പാലു നല്ലൂ, പുലര്‍ന്നാലോ കലക്കിലാം
:ഇല്ലാ പാലെന്നു ഗോപാലന്‍ – ആംഗ്ലമാസദിനം ക്രമാല്‍
</blockquote>
:Transliteration
<blockquote>
:''palahāre pālu nallū, pularnnālo kalakkilāṃ''
:''illā pālennu gopālan – āṃgḷamāsadinaṃ kramāl''
</blockquote>
: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 ==
{{col div|colwidth=30em}}
* [[Abjad numerals]]
* [[Aksharapalli]]
* [[Aryabhata numeration]]
* [[Bhutasamkhya system]]
* [[Gematria]]
* [[Greek numerals]]
* [[Kerala school of astronomy and mathematics]]
* [[Madhava's sine table]]
* [[Mnemonic major system]]
* [[Notarikon]]
* [[Temurah (Kabbalah)]]
* [[Alphasyllabic numeral system]]
{{colend}}

== References ==
{{reflist}}

== External links ==

# [https://sanskrit.iitk.ac.in/jnanasangraha/sankhya/katapayaadi/ '''Kaṭapayādi Saṅkhyā,''' a Kaṭapayādi encoding-decoding system.] 

== Further reading ==
* A.A. Hattangadi, Explorations in Mathematics, Universities Press (India) Pvt. Ltd., Hyderabad (2001) {{ISBN|81-7371-387-1}} [https://books.google.com/books?id=4Tj3Sv1bKdIC&dq=katapayadi&pg=PA14]

{{Kerala School}}
{{Indian mathematics}}
{{Scientific Research in Kerala |state=collapsed}}

{{authority control}}

{{DEFAULTSORT:Katapayadi System}}
[[Category:Numeral systems]]
[[Category:Mnemonics]]
[[Category:Indian mathematics]]
[[Category:Kerala school of astronomy and mathematics]]