** 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.

## History[edit]

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]}

Aryabhata, in his treatise *Ārya·bhaṭīya*, is known to have used a similar, more complex system to represent [[astronomical number]]s. There is no definitive evidence whether the *Ka-ṭa-pa-yā-di* system originated from Āryabhaṭa numeration.^{[5]}

## Geographical spread of the use[edit]

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 Sarasvathy Bhavan Library of Sampurnanand Sanskrit University, Varanasi.^{[1]}

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.^{[6]}

## Rules and practices[edit]

Following verse found in Śaṅkaravarman's *Sadratnamāla* explains the mechanism of the system.^{[7]}^{[8]}

नञावचश्च शून्यानि संख्या: कटपयादय:।

मिश्रे तूपान्त्यहल् संख्या न च चिन्त्यो हलस्वर:॥

Transiliteration:

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 vowel without consonant is to be ignored.

Explanation: The assignment of letters to the numerals are as per the following arrangement.

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 0 |
---|---|---|---|---|---|---|---|---|---|

ka क క ക | kha ख ఖ ഖ | ga ग గ ഗ | gha घ ఘ ഘ | nga ङ జ్ఞ ങ | ca च చ ച | cha छ ఛ ഛ | ja ज జ ജ | jha झ ఝ ഝ | nya ञ ఞ ഞ |

ṭa ट ట ട | ṭha ठ ఠ ഠ | ḍa ड డ ഡ | ḍha ढ ఢ ഢ | ṇa ण ణ ണ | ta त త ത | tha थ థ ഥ | da द ద ദ | dha ध ధ ധ | na न న ന |

pa प ప പ | pha फ ఫ ഫ | ba ब బ ബ | bha भ భ ഭ | ma म మ മ | – | – | – | – | – |

ya य య യ | ra र ర ര | la ल ల ല | va व వ വ | śha श శ ശ | sha ष ష ഷ | sa स స സ | ha ह 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 not be valueless. For example,
*kya*(क्या) is formed by*k*(क्) +*ya*(य) +*a*(अ). The only consonant standing with a vowel is*ya*(य). So the corresponding numeral for*kya*(क्या) will be 1. - There is no way of representing 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
*gyan-ki-baatein*(ज्ञान की बातें) which means numbers go from left to right.^{[9]}

### Variations[edit]

- The consonant, ḷ (Malayālam: ള, Devanāgarī: ळ) is employed in works using the Kaṭapayādi system, like Mādhava's sine table.
- Some
^{[who?]}practitioners do not map the stand-alone vowels to zero. But, it is sometimes considered valueless.

## Usage[edit]

### Mathematics and astronomy[edit]

- Mādhava's sine table constructed by 14th century Kerala mathematician-astronomer Mādhava of Saṅgama·grāma employs the Kaṭapayādi system to enlist the trigonometric sines of angles.
*Karaṇa·paddhati*, written in the 15th century, has the following*śloka*for the value of pi (π)

- അനൂനനൂന്നാനനനുന്നനിത്യൈ-
- സ്സമാഹതാശ്ചക്രകലാവിഭക്താഃ
- ചണ്ഡാംശുചന്ദ്രാധമകുംഭിപാലൈര്-
- വ്യാസസ്തദര്ദ്ധം ത്രിഭമൗര്വിക സ്യാത്

- Transliteration

anūnanūnnānananunnanityaissmāhatāścakra kalāvibhaktoḥcaṇḍāṃśucandrādhamakuṃbhipālair

vyāsastadarddhaṃ tribhamaurvika syāt

- 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. A famous verse found in*Sad·ratna·mālā*is

- भद्राम्बुद्धिसिद्धजन्मगणितश्रद्धा स्म यद् भूपगी:

- Transliteration

bhadrāṃbuddhisiddhajanmagaṇitaśraddhā sma yad bhūpagīḥ

- Splitting the consonants gives,

भ bha | द् d | रा rā | म् ṃ | 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 ṣ | र ra | द् d | धा dha | स् s | म ma | य ya | द् d | भू bhu | प pa | गि gi |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

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-श्रुग्ङिशोदधिसन्धिग॥ 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 ||

### Carnatic music[edit]

- The melakarta ragas of the Carnatic music is 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.
- '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 swaras in Carnatic music for details on above notation.

#### Raga Dheerasankarabharanam[edit]

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*.

#### Raga MechaKalyani[edit]

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*.

#### Exception for Simhendramadhyamam[edit]

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 Bra**hm**ana in Sanskrit).

### Representation of dates[edit]

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.^{[10]} - Narayaniyam, written by Melpathur Narayana Bhattathiri, ends with the line, āyurārogyasaukhyam (ആയുരാരോഗ്യസൌഖ്യം) which means long-life, health and happiness.
^{[11]}

In Malayalam | 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[edit]

- Some people use the
*Kaṭapayādi*system in naming newborns.^{[12]}^{[13]} - 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

- Transiliteration

palahāre pālu nallū, pularnnālo kalakkilāṃillā 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.

## See also[edit]

- 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

## References[edit]

- ^
^{a}^{b}Sreeramamula Rajeswara Sarma (1999), Kaṭapayādi Notation on a Sanskrit Astrolabe [1]^{[permanent dead link]} **^**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.**^**Usenet Discussion. "Aryabhatta's numerical encoding". Archived from the original on 17 July 2011. Retrieved 1 January 2010.**^**Plofker, Kim (2008).*Mathematics in India*. Princeton University Press. p. 384. ISBN 978-0-691-12067-6.**^**J. F. Fleet (April 1912). "The Ka-ta-pa-ya-di Notation of the Second Arya-Siddhanta".*The Journal of the Royal Asiatic Society of Great Britain and Ireland*. Royal Asiatic Society of Great Britain and Ireland: 459–462. JSTOR 25190035.**^**J.F. Fleet (July 1911). "The Katapayadi System of Expressing Numbers".*The Journal of the Royal Asiatic Society of Great Britain and Ireland*. Royal Asiatic Society of Great Britain and Ireland: 788–794. JSTOR 25189917.**^**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.CS1 maint: archived copy as title (link)**^**Anand Raman. "The Ancient Katapayadi Formula and the Modern Hashing Method" (PDF). Archived from the original (PDF) on 16 June 2011. Cite journal requires`|journal=`

(help)**^**gyan-ki-baatein^{[dead link]}**^**Francis Zimmerman, 1989, Lilavati, gracious lady of arithmetic – India – A Mathematical Mystery Tour "Archived copy". Archived from the original on 6 September 2009. Retrieved 3 January 2010.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.**^**Visti Larsen, Choosing the auspicious name**^**[2]

## Further reading[edit]

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