Katapayadi Sankhya
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''Kaṭapayādi'' system (
Devanagari Devanagari ( ; in script: , , ) is an Indic script used in the Indian subcontinent. It is a left-to-right abugida (a type of segmental Writing systems#Segmental systems: alphabets, writing system), based on the ancient ''Brāhmī script, Brā ...
: कटपयादि, also known as ''Paralppēru'', Malayalam: പരല്‍പ്പേര്) of numerical notation is an
ancient Ancient history is a time period from the beginning of writing and recorded human history through late antiquity. The span of recorded history is roughly 5,000 years, beginning with the development of Sumerian cuneiform script. Ancient h ...
India India, officially the Republic of India, is a country in South Asia. It is the List of countries and dependencies by area, seventh-largest country by area; the List of countries by population (United Nations), most populous country since ...
n
alphasyllabic numeral system Alphasyllabic numeral systems are a type of numeral systems, developed mostly in India starting around 500 AD. Based on various alphasyllabary, alphasyllabic scripts, in this type of numeral systems glyphs of the numerals are not abstract signs, bu ...
to depict letters to
numerals A numeral is a figure (symbol), word, or group of figures (symbols) or words denoting a number. It may refer to: * Numeral system used in mathematics * Numeral (linguistics), a part of speech denoting numbers (e.g. ''one'' and ''first'' in English ...
for easy remembrance of
number A number is a mathematical object used to count, measure, and label. The most basic examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers can ...
s as
words A word is a basic element of language that carries meaning, can be used on its own, and is uninterruptible. Despite the fact that language speakers often have an intuitive grasp of what a word is, there is no consensus among linguists on its ...
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.Sreeramamula Rajeswara Sarma, THE ''KATAPAYADI'' SYSTEM OF NUMERICAL NOTATION AND ITS SPREAD OUTSIDE KERALA, ''Rev. d'Histoire de Mathmatique'' 18 (2012

/ref> It has been used in ''Laghu·bhāskarīya·vivaraṇa'' written by '' Sankara Narayana, Śaṅkara·nārāyaṇa'' in 869 CE. 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 Aryabhata ( ISO: ) or Aryabhata I (476–550 CE) was the first of the major mathematician-astronomers from the classical age of Indian mathematics and Indian astronomy. His works include the '' Āryabhaṭīya'' (which mentions that in 3600 ' ...
, 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 Āryabhaṭa numeration is an alphasyllabic numeral system based on Sanskrit phonemes. It was introduced in the early 6th century in India by Āryabhaṭa, in the first chapter titled ''Gītika Padam'' of his '' Aryabhatiya''. It attribute ...
.


Geographical spread of the use

Almost all evidences of the use of ''Ka-ṭa-pa-yā-di'' system is from
South India South India, also known as Southern India or Peninsular India, is the southern part of the Deccan Peninsula in India encompassing the states of Andhra Pradesh, Karnataka, Kerala, Tamil Nadu and Telangana as well as the union territories of ...
, especially
Kerala Kerala ( , ) is a States and union territories of India, state on the Malabar Coast of India. It was formed on 1 November 1956, following the passage of the States Reorganisation Act, by combining Malayalam-speaking regions of the erstwhile ...
. Not much is known about its use in North India. However, on a
Sanskrit Sanskrit (; stem form ; nominal singular , ,) is a classical language belonging to the Indo-Aryan languages, Indo-Aryan branch of the Indo-European languages. It arose in northwest South Asia after its predecessor languages had Trans-cultural ...
astrolabe An astrolabe (; ; ) is an astronomy, astronomical list of astronomical instruments, instrument dating to ancient times. It serves as a star chart and Model#Physical model, physical model of the visible celestial sphere, half-dome of the sky. It ...
discovered in
North India North India is a geographical region, loosely defined as a cultural region comprising the northern part of India (or historically, the Indian subcontinent) wherein Indo-Aryans (speaking Indo-Aryan languages) form the prominent majority populati ...
, the degrees of the altitude are marked in the ''Kaṭapayādi'' system. It is preserved in the Sarasvati Bhavan Library of
Sampurnanand Sanskrit University Sampurnanand Sanskrit Vishwavidyalaya (IAST: ; formerly Varanaseya Sanskrit Vishwavidyalaya and Government Sanskrit College, Varanasi) is an Indian university and institution of higher learning located in Varanasi, Uttar Pradesh. It is one of t ...
,
Varanasi Varanasi (, also Benares, Banaras ) or Kashi, is a city on the Ganges river in northern India that has a central place in the traditions of pilgrimage, death, and mourning in the Hindu world.* * * * The city has a syncretic tradition of I ...
. The ''Ka-ṭa-pa-yā-di'' system is not confined to India. Some
Pali Pāli (, IAST: pāl̤i) is a Classical languages of India, classical Middle Indo-Aryan languages, Middle Indo-Aryan language of the Indian subcontinent. It is widely studied because it is the language of the Buddhist ''Pali Canon, Pāli Can ...
chronograms based on the ''Ka-ṭa-pa-yā-di'' system have been discovered in
Burma Myanmar, officially the Republic of the Union of Myanmar; and also referred to as Burma (the official English name until 1989), is a country in northwest Southeast Asia. It is the largest country by area in Mainland Southeast Asia and ha ...
.


Rules and practices

Following verse found in Śaṅkaravarman's '' Sadratnamāla'' explains the mechanism of the system.
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 A vowel is a speech sound pronounced without any stricture in the vocal tract, forming the nucleus of a syllable. Vowels are one of the two principal classes of speech sounds, the other being the consonant. Vowels vary in quality, in loudness a ...
represent
zero 0 (zero) is a number representing an empty quantity. Adding (or subtracting) 0 to any number leaves that number unchanged; in mathematical terminology, 0 is the additive identity of the integers, rational numbers, real numbers, and compl ...
. The nine
integers An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
are represented by
consonant In articulatory phonetics, a consonant is a speech sound that is articulated with complete or partial closure of the vocal tract, except for the h sound, which is pronounced without any stricture in the vocal tract. Examples are and pronou ...
group beginning with ''ka'', ''ṭa'', ''pa'', ''ya''. In a
conjunct {{unreferenced, date=August 2024 In linguistics, the term conjunct has three distinct uses: *A conjunct is an adverbial that adds information to the sentence that is not considered part of the propositional content (or at least not essential) b ...
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 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 FIle:Decimal separators.svg, alt=Four types of separating decimals: a) 1,234.56. b) 1.234,56. c) 1'234,56. d) ١٬٢٣٤٫٥٦., Both a comma and a full stop (or period) are generally accepted decimal separators for international use. The apost ...
in the system. * Indians used the
Hindu–Arabic numeral system The Hindu–Arabic numeral system (also known as the Indo-Arabic numeral system, Hindu numeral system, and Arabic numeral system) is a positional notation, positional Decimal, base-ten numeral system for representing integers; its extension t ...
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.


Variations

* The
consonant In articulatory phonetics, a consonant is a speech sound that is articulated with complete or partial closure of the vocal tract, except for the h sound, which is pronounced without any stricture in the vocal tract. Examples are and pronou ...
, ḷ (Malayālam: ള, Devanāgarī: ळ, Kannada: ಳ) is employed in works using the Kaṭapayādi system, like Mādhava's sine table. * Late medieval practitioners do not map the stand-alone vowels to zero. But, it is sometimes considered valueless.


Usage


Mathematics and astronomy

* Mādhava's sine table constructed by 14th century
Kerala Kerala ( , ) is a States and union territories of India, state on the Malabar Coast of India. It was formed on 1 November 1956, following the passage of the States Reorganisation Act, by combining Malayalam-speaking regions of the erstwhile ...
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...
-
astronomer An astronomer is a scientist in the field of astronomy who focuses on a specific question or field outside the scope of Earth. Astronomers observe astronomical objects, such as stars, planets, natural satellite, moons, comets and galaxy, galax ...
Mādhava of Saṅgama·grāma employs the Kaṭapayādi system to list 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
:''anūnanūnnānananunnanityai'' :''ssmāhatāścakra kalāvibhaktoḥ'' :''caṇḍāṃśucandrādhamakuṃbhipālair'' ''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. The first verse of Chapter 4 of the '' Sad·ratna·mālā'' ends with the line:
:(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 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-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.


Carnatic music

* The
melakarta Mēḷakartā is a collection of fundamental musical scales ( ragas) in Carnatic music (South Indian classical music). ''Mēḷakartā'' ragas are parent ragas (hence known as ''janaka'' ragas) from which other ragas may be derived. A ''melaka ...
ragas A raga ( ; , ; ) is a melodic framework for improvisation in Indian classical music akin to a melodic mode. It is central to classical Indian music. Each raga consists of an array of melodic structures with musical motifs; and, from the perspec ...
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
Swara Swara () or svara is an Indian classical music term that connotes simultaneously a breath, a vowel, a note, the sound of a musical note corresponding to its name, and the successive steps of the octave, or ''saptanka''. More comprehensively ...
s '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 Dhīraśankarābharaṇaṃ, commonly known as ''Śankarābharaṇaṃ'', is a rāga in Carnatic music. It is the 29th '' Melakarta'' rāga in the 72 ''Melakarta'' rāga system of Carnatic music. Since this raga has many Gamakās (ornamentations) ...

The katapayadi scheme associates dha\leftrightarrow9 and ra\leftrightarrow2, 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 In arithmetic, a quotient (from 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics. It has two definitions: either the integer part of a division (in th ...
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 Kalyani is a melakarta raga (parent musical scale) in the Carnatic music. It is the prati madhyama equivalent of the raga Sankarabharanam (raga), Sankarabharanam. It was called Kalyan but is now more popularly called Yaman (raga), Yaman in Hindu ...

From the coding scheme Ma \leftrightarrow 5, Cha \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 notes ''Sa Ri2 Ga3 Ma2 Pa Da2 Ni3 SA''.


Exception for Simhendramadhyamam

As per the above calculation, we should get Sa \leftrightarrow 7, Ha \leftrightarrow 8 giving the number 87 instead of 57 for Simhendramadhyamam. This should be ideally Sa \leftrightarrow 7, Ma \leftrightarrow 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 ''Kali Yuga'' (Devanagari: कलियुग), in Hinduism, is the fourth, shortest, and worst of the four '' yugas'' (world ages) in a '' Yuga cycle'', preceded by '' Dvapara Yuga'' and followed by the next cycle's '' Krita (Satya) Yuga''. I ...
. It is sometimes called ''kalidina sankhya''. * The
Malayalam calendar The Malayalam Calendar, or the Kollam Era (), is a sidereal solar calendar used in Kerala. The origin of the calendar has been dated to 825 CE, commemorating the establishment of Kollam. There are many theories regarding the origin of t ...
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 ''Kali Yuga'' (Devanagari: कलियुग), in Hinduism, is the fourth, shortest, and worst of the four '' yugas'' (world ages) in a '' Yuga cycle'', preceded by '' Dvapara Yuga'' and followed by the next cycle's '' Krita (Satya) Yuga''. I ...
. * Narayaniyam, written by
Melpathur Narayana Bhattathiri Melpathur Narayana Bhattathiri (Mēlpattūr Nārāyaṇa Bhaṭṭatiri; 1560–1646/1666), third student of Achyuta Pisharati, was a member of Madhava of Sangamagrama's Kerala school of astronomy and mathematics. He was a mathematical linguist ...
, ends with the line, āyurārogyasaukhyam (ആയുരാരോഗ്യസൌഖ്യം) which means long-life, health and happiness. :This number is the time at which the work was completed represented as number of days since the start of
Kali Yuga ''Kali Yuga'' (Devanagari: कलियुग), in Hinduism, is the fourth, shortest, and worst of the four '' yugas'' (world ages) in a '' Yuga cycle'', preceded by '' Dvapara Yuga'' and followed by the next cycle's '' Krita (Satya) Yuga''. I ...
as per the
Malayalam calendar The Malayalam Calendar, or the Kollam Era (), is a sidereal solar calendar used in Kerala. The origin of the calendar has been dated to 825 CE, commemorating the establishment of Kollam. There are many theories regarding the origin of t ...
.


Others

* Some people use the ''Kaṭapayādi'' system in naming newborns. * The following verse compiled in Malayalam by Koduṅṅallur Kuññikkuṭṭan Taṃpurān using ''Kaṭapayādi'' is the number of days in the months of
Gregorian Calendar The Gregorian calendar is the calendar used in most parts of the world. It went into effect in October 1582 following the papal bull issued by Pope Gregory XIII, which introduced it as a modification of, and replacement for, the Julian cale ...
.
: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.


See also

*
Abjad numerals The Abjad numerals, also called Hisab al-Jummal (, ), are a decimal alphabetic numeral system/alphanumeric code, in which the 28 letters of the Arabic alphabet are assigned numerical values. They have been used in the Arab world, Arabic-speaking ...
*
Aksharapalli Aksharapalli () is a certain type of alphasyllabic numeration scheme extensively used in the pagination of manuscripts produced in India in pre-modern times. The name ''Aksharapalli'' can be translated as the ''letter system''. In this system ...
*
Aryabhata numeration Aryabhata ( ISO 15919, ISO: ) or Aryabhata I (476–550 Common Era, CE) was the first of the major mathematician-astronomers from the classical age of Indian mathematics and Indian astronomy. His works include the ''Āryabhaṭīya'' (which me ...
*
Bhutasamkhya system The Bhūtasaṃkhyā system is a method of recording numbers in Sanskrit using common nouns having connotations of numerical values. The method was introduced already in astronomical texts in antiquity, but it was expanded and developed during th ...
*
Gematria In numerology, gematria (; or , plural or ) is the practice of assigning a numerical value to a name, word, or phrase by reading it as a number, or sometimes by using an alphanumeric cipher. The letters of the alphabets involved have standar ...
*
Greek numerals Greek numerals, also known as Ionic, Ionian, Milesian, or Alexandrian numerals, is a numeral system, system of writing numbers using the letters of the Greek alphabet. In modern Greece, they are still used for ordinal number (linguistics), ordi ...
*
Kerala school of astronomy and mathematics The Kerala school of astronomy and mathematics or the Kerala school was a school of Indian mathematics, mathematics and Indian astronomy, astronomy founded by Madhava of Sangamagrama in Kingdom of Tanur, Tirur, Malappuram district, Malappuram, K ...
* Madhava's sine table *
Mnemonic major system The mnemonic major system (also called the phonetic number system, phonetic mnemonic system, or Hérigone's mnemonic system) is a mnemonic technique used to help in memorizing numbers. The system works by converting numbers into consonants, then i ...
* Notarikon * Temurah (Kabbalah) *
Alphasyllabic numeral system Alphasyllabic numeral systems are a type of numeral systems, developed mostly in India starting around 500 AD. Based on various alphasyllabary, alphasyllabic scripts, in this type of numeral systems glyphs of the numerals are not abstract signs, bu ...


References


External links


Kaṭapayādi Saṅkhyā, a Kaṭapayādi encoding-decoding system.


Further reading

* A.A. Hattangadi, Explorations in Mathematics, Universities Press (India) Pvt. Ltd., Hyderabad (2001)

{{DEFAULTSORT:Katapayadi System Numeral systems Mnemonics Indian mathematics Kerala school of astronomy and mathematics