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Kaṭapayādi system (Devanagari: कटपयादि, also known as Paralppēru, Malayalam: പരല്പ്പേര്) of numerical notation is an ancientIndianalphasyllabic numeral 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.
The 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]
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 Sanskritastrolabe 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.
[6]
The Ka-ṭa-pa-yā-di system is not confined to India. Some Palichronograms based on the Ka-ṭa-pa-yā-di system have been discovered in Burma.[7]
नञावचश्च शून्यानि संख्या: कटपयादय:।
मिश्रे तूपान्त्यहल् संख्या न च चिन्त्यो हलस्वर:॥
Transliteration:
nanyāvachaścha śūnyāni sankhyāḥ kaṭapayādayaḥ miśre tūpāntyahal sankhyā na cha chintyo halasvaraḥ
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)
1
2
3
4
5
6
7
8
9
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.
(syād) bhadrāmbudhisiddhajanmagaṇitaśraddhā sma yad bhūpagīḥ
Splitting the consonants in the relevant phrase gives,
भ bha
द् d
रा rā
म् m
बु bu
द् d
धि dhi
सि si
द् d
ध dha
ज ja
न् n
म ma
ग ga
णि ṇi
त ta
श् ś
र ra
द् d
धा dhā
स् s
म ma
य ya
द् d
भू bhū
प pa
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.
The melakartaragas 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 Swaras '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.
The katapayadi scheme associates dha9 and ra2, 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.
From 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.
As 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).
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 CE, revamping some calendars. This date is remembered as āchārya vāgbhadā, converted using Kaṭapayādi into 1434160 days since the start of Kali Yuga.[11]
Some people use the Kaṭapayādi system in naming newborns.[13][14]
The following verse compiled in Malayalam by Koduṅṅallur Kuññikkuṭṭan Taṃpurān using Kaṭapayādi is the number of days in the months of Gregorian Calendar.
പലഹാരേ പാലു നല്ലൂ, പുലര്ന്നാലോ കലക്കിലാം
ഇല്ലാ പാലെന്നു ഗോപാലന് – ആംഗ്ലമാസദിനം ക്രമാല്
Transliteration
palahāre pālu nallū, pularnnālo kalakkilāṃ
illā 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.
^Sreeramamula Rajeswara Sarma, THE KATAPAYADI SYSTEM OF NUMERICAL NOTATION AND ITS SPREAD OUTSIDE KERALA, Rev. d'Histoire de Mathmatique 18 (2012)
[1]
^J J O'Connor; E F Robertson (November 2000). "Sankara Narayana". School of Mathematics and Statistics, University of St Andrews, Scotland. Retrieved 1 January 2010.
^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. "Archived copy"(PDF). Archived from the original(PDF) on 2 April 2015. Retrieved 17 December 2009.((cite web)): CS1 maint: archived copy as title (link)
^Dr. C Krishnan Namboodiri, Chekrakal Illam, Calicut, Namboothiti.com Dr. C Krishnan Namboodiri. ""Katapayaadi" or "Paralpperu"". Namboothiri Websites Trust. Retrieved 1 January 2010.