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Kaṭapayādi system (Devanagari: कटपयादि, also known asParalppēru, Malayalam:പരല്‍പ്പേര്) of numerical notation is anancientIndianalphasyllabic numeral system to depictletters tonumerals for easy remembrance ofnumbers aswords orverses. 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 ofKaṭapayādi (Sanskrit: कटपयादि) system is fromGrahacāraṇibandhana byHaridatta in 683CE.^[1] It has been used inLaghu·bhāskarīya·vivaraṇa written byŚaṅkara·nārāyaṇa in 869CE.^[2]

In some astronomical texts popular in Kerala planetary positions were encoded in the Kaṭapayādi system. The first such work is considered to be theChandra-vakyani ofVararuci, who is traditionally assigned to the fourth centuryCE. Therefore, sometime in the early first millennium is a reasonable estimate for the origin of theKaṭapayādi system.^[3]

Aryabhata, in his treatiseĀrya·bhaṭīya, is known to have used a similar, more complex system to representastronomical numbers. There is no definitive evidence whether theKa-ṭa-pa-yā-di system originated fromĀryabhaṭa numeration.^[4]

Almost all evidences of the use ofKa-ṭa-pa-yā-di system is fromSouth India, especiallyKerala. Not much is known about its use in North India. However, on aSanskritastrolabe discovered inNorth India, the degrees of the altitude are marked in theKaṭapayādi system. It is preserved in the Sarasvati Bhavan Library ofSampurnanand Sanskrit University,Varanasi.^[5]

TheKa-ṭa-pa-yā-di system is not confined to India. SomePalichronograms based on theKa-ṭa-pa-yā-di system have been discovered inBurma.^[6]

Following verse found inŚaṅkaravarman'sSadratnamāla explains the mechanism of the system.^[7][8]

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) anda (अ)-s, i.e.,vowels representzero. The nineintegers are represented byconsonant group beginning withka,ṭa,pa,ya. In aconjunct 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)

• Consonants have numerals assigned as per the above table. For example, ba (ब) is always 3 whereas 5 can be represented by eithernga (ङ) orṇa (ण) orma (म) orśha (श).
• All stand-alone vowels likea (अ) andṛ (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 isya (य). So the corresponding numeral forkya (क्य) will be 1.
• There is no way of representing thedecimal separator in the system.
• Indians used theHindu–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.

• Theconsonant, ḷ (Malayālam: ള, Devanāgarī: ळ, Kannada: ಳ) is employed in works using the Kaṭapayādi system, likeMādhava's sine table.
• Late medieval practitioners do not map the stand-alone vowels to zero. But, it is sometimes considered valueless.

• Mādhava's sine table constructed by 14th centuryKeralamathematician-astronomerMādhava of Saṅgama·grāma employs the Kaṭapayādi system to list the trigonometric sines of angles.
• Karaṇa·paddhati, written in the 15th century, has the followingśloka for the value ofpi (π)

അനൂനനൂന്നാനനനുന്നനിത്യൈ-

സ്സമാഹതാശ്ചക്രകലാവിഭക്താഃ

ചണ്ഡാംശുചന്ദ്രാധമകുംഭിപാലൈര്‍-

വ്യാസസ്തദര്‍ദ്ധം ത്രിഭമൗര്‍വിക സ്യാത്‌

Transliteration

anūnanūnnānananunnanityai

ssmāhatāścakra kalāvibhaktoḥ

caṇḍāṃśucandrādhamakuṃbhipālair

vyāsastadarddhaṃ tribhamaurvika syāt

It gives the circumference of a circle of diameter,anūnanūnnānananunnanityai (10,000,000,000) ascaṇḍāṃśucandrādhamakuṃbhipālair (31415926536).

• Śaṅkara·varman'sSad·ratna·mālā uses the Kaṭapayādi system. The first verse of Chapter 4 of theSad·ratna·mālā ends with the line:^[9]

(स्याद्) भद्राम्बुधिसिद्धजन्मगणितश्रद्धा स्म यद् भूपगी:

Transliteration

(syād) bhadrāmbudhisiddhajanmagaṇitaśraddhā sma yad bhūpagīḥ

Splitting the consonants in the relevant phrase gives,

Reversing the digits to modern-day usage of descending order of decimal places, we get314159265358979324 which is the value ofpi (π) to 17 decimal places, except the last digit might be rounded off to 4.

• This verse encrypts the value ofpi (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.

• Themelakartaragas of the Carnatic music are named so that the first two syllables of the name will give its number. This system is sometimes called theKa-ta-pa-ya-di sankhya. TheSwaras 'Sa' and 'Pa' are fixed, and here is how to get the other swaras from the melakarta number.

1. Melakartas 1 through 36 have Ma1 and those from 37 through 72 have Ma2.
2. 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.
3. 'Ri' and 'Ga' positions: the raga will have:
   • Ri1 andGa1 if the quotient is 0
   • Ri1 andGa2 if the quotient is 1
   • Ri1 andGa3 if the quotient is 2
   • Ri2 andGa2 if the quotient is 3
   • Ri2 andGa3 if the quotient is 4
   • Ri3 andGa3 if the quotient is 5
4. 'Da' and 'Ni' positions: the raga will have:
   • Da1 andNi1 if remainder is 0
   • Da1 andNi2 if remainder is 1
   • Da1 andNi3 if remainder is 2
   • Da2 andNi2 if remainder is 3
   • Da2 andNi3 if remainder is 4
   • Da3 andNi3 if remainder is 5

• Seeswaras in Carnatic music for details on above notation.

The katapayadi scheme associates dha${\displaystyle \leftrightarrow }$9 and ra${\displaystyle \leftrightarrow }$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, thequotient 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 isSa Ri2 Ga3 Ma1 Pa Da2 Ni3 SA.

From the coding scheme Ma${\displaystyle \leftrightarrow }$ 5, Cha${\displaystyle \leftrightarrow }$ 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 notesSa Ri2 Ga3 Ma2 Pa Da2 Ni3 SA.

As per the above calculation, we should get Sa${\displaystyle \leftrightarrow }$ 7, Ha${\displaystyle \leftrightarrow }$ 8 giving the number 87 instead of 57 for Simhendramadhyamam. This should be ideally Sa${\displaystyle \leftrightarrow }$ 7, Ma${\displaystyle \leftrightarrow }$ 5 giving the number 57. So it is believed that the name should be written asSihmendramadhyamam (as in the case of Brahmana in Sanskrit).

Important dates were remembered by converting them usingKaṭapayādi system. These dates are generally represented as number of days since the start ofKali Yuga. It is sometimes calledkalidina sankhya.

• TheMalayalam calendar known askollavarsham (Malayalam: കൊല്ലവര്‍ഷം) was adopted in Kerala beginning from 825CE, revamping some calendars. This date is remembered asāchārya vāgbhadā, converted usingKaṭapayādi into 1434160 days since the start ofKali Yuga.^[10]
• Narayaniyam, written byMelpathur Narayana Bhattathiri, ends with the line, āyurārogyasaukhyam (ആയുരാരോഗ്യസൌഖ്യം) which means long-life, health and happiness.^[11]

This number is the time at which the work was completed represented as number of days since the start ofKali Yuga as per theMalayalam calendar.

• Some people use theKaṭapayādi system in naming newborns.^[12][13]
• The following verse compiled in Malayalam by Koduṅṅallur Kuññikkuṭṭan Taṃpurān usingKaṭapayādi is the number of days in the months ofGregorian 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. ButGopālan says there is no milk – the number of days of English months in order.

Converting pairs of letters usingKaṭapayādi 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.

• Kaṭapayādi Saṅkhyā, a Kaṭapayādi encoding-decoding system.

• Saṅkhyāpaddhatiḥ (सङ्ख्यापद्धतिः), a collection of ancient Indian alpha-syllabic numeral systems

• A.A. Hattangadi, Explorations in Mathematics, Universities Press (India) Pvt. Ltd., Hyderabad (2001)ISBN 81-7371-387-1[3]

• Numeral systems
• Mnemonics
• Indian mathematics
• Kerala school of astronomy and mathematics

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