ka·ṭa·pa·yā·di (Devanagari: कटपयादि) system (also known as Paralppēru, Malayalam: പരല്പ്പേര്) of numerical notation is an ancient Indian alphasyllabic 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. It has been used in Laghu·bhāskarīya·vivaraṇa written by Śaṅkara·nārāyaṇa in 869 CE.
Some argue that the system originated from Vararuci. 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.
Aryabhata, in his treatise Ārya·bhaṭīya, is known to have used a similar, more complex system to represent astronomical numbers. There is no definitive evidence whether the Ka-ṭa-pa-yā-di system originated from Āryabhaṭa numeration.
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. 
नञावचश्च शून्यानि संख्या: कटपयादय:।
मिश्रे तूपान्त्यहल् संख्या न च चिन्त्यो हलस्वर:॥
nanyāvacaśca śūnyāni saṃkhyāḥ kaṭapayādayaḥ
miśre tūpāntyahal saṃkhyā na ca cintyo halasvaraḥ
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 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 respectively)
|ka क ಕ క ക||kha ख ಖ ఖ ഖ||ga ग ಗ గ ഗ||gha घ ಘ ఘ ഘ||nga ङ ಙ జ్ఞ 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||śha श ಶ శ ശ||sha ष ಷ ష ഷ||sa स ಸ స സ||ha ह ಹ హ ഹ||–||–|
- വ്യാസസ്തദര്ദ്ധം ത്രിഭമൗര്വിക സ്യാത്
- ssmāhatāścakra kalāvibhaktoḥ
vyāsastadarddhaṃ tribhamaurvika syāt
- (स्याद्) भद्राम्बुधिसिद्धजन्मगणितश्रद्धा स्म यद् भूपगी:
- (syād) bhadrāmbudhisiddhajanmagaṇitaśraddhā sma yad bhūpagīḥ
|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|
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.
The 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.
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.
|Value as per Kaṭapayādi||1712210|
- പലഹാരേ പാലു നല്ലൂ, പുലര്ന്നാലോ കലക്കിലാം
- ഇല്ലാ പാലെന്നു ഗോപാലന് – ആംഗ്ലമാസദിനം ക്രമാല്
- palahāre pālu nallū, pularnnālo kalakkilāṃ
- illā pālennu gopālan – āṃgḷamāsadinaṃ kramāl