Katapayadi system
"ka·ṭa·pa·yā·di" (Devanagari: कटपयादि) system (also known as "Paralppēru", Malayalam: ) 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. History. 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. 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. 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. Rules and practices. Following verse  found in Śaṅkaravarman's "Sadratnamāla" explains the mechanism of the system. नञावचश्च शून्यानि संख्या: कटपयादय:।<BR> मिश्रे तूपान्त्यहल् संख्या न च चिन्त्यो हलस्वर:॥ Transiliteration: "nanyāvacaśca śūnyāni saṃkhyāḥ kaṭapayādayaḥ"<BR> "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) Usage. Mathematics and astronomy. "vyāsastadarddhaṃ tribhamaurvika syāt" This verse directly yields the decimal equivalent of pi divided by 10: pi/10 = 0.31415926535897932384626433832792 Traditionally, the order of digits are reversed to form the number, in katapayadi system. This rule is violated in this sloka. Carnatic music. Raga Dheerasankarabharanam. The katapayadi scheme associates dhaformula_19 and raformula_12, hence the raga's melakarta number is 29 (92 reversed). Now 29 formula_3 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 MechaKalyani. From the coding scheme Ma formula_1 5, Cha formula_1 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 formula_1 7, Ha formula_1 8 giving the number 87 instead of 57 for Simhendramadhyamam. This should be ideally Sa formula_1 7, Ma formula_1 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 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".