Katapayadi system
Encyclopedia
Kaṭapayādi also called Nagari , is an abugida alphabet of India and Nepal...
: कटपयादि) system (also known as Paralppēru, Malayalam: പരല്പ്പേര്) of numerical notation is an ancient India
India
India , officially the Republic of India , is a country in South Asia. It is the seventh-largest country by geographical area, the second-most populous country with over 1.2 billion people, and the most populous democracy in the world...
n system to depict letters
Letter (alphabet)
A letter is a grapheme in an alphabetic system of writing, such as the Greek alphabet and its descendants. Letters compose phonemes and each phoneme represents a phone in the spoken form of the language....
to numerals
Number
A number is a mathematical object used to count and measure. In mathematics, the definition of number has been extended over the years to include such numbers as zero, negative numbers, rational numbers, irrational numbers, and complex numbers....
for easy remembrance of number
Number
A number is a mathematical object used to count and measure. In mathematics, the definition of number has been extended over the years to include such numbers as zero, negative numbers, rational numbers, irrational numbers, and complex numbers....
s as words or verses
Verse (poetry)
A verse is formally a single line in a metrical composition, e.g. poetry. However, the word has come to represent any division or grouping of words in such a composition, which traditionally had been referred to as a stanza....
. 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 System is from Grahacāraṇibandhana by HaridattaHaridatta
Haridatta was an astronomer-mathematician of Kerala, India, who is believed to be the promulgator of the Parahita system of astronomical computations. This system of computations is widely popular in Kerala and Tamil Nadu. According to legends, Haridatta promulgated the Parahita system on the...
in 683 CE
Common Era
Common Era ,abbreviated as CE, is an alternative designation for the calendar era originally introduced by Dionysius Exiguus in the 6th century, traditionally identified with Anno Domini .Dates before the year 1 CE are indicated by the usage of BCE, short for Before the Common Era Common Era...
. It has also been used in Laghubhāskariyavivarana written by Sankaranārāyana in 869 CE
Common Era
Common Era ,abbreviated as CE, is an alternative designation for the calendar era originally introduced by Dionysius Exiguus in the 6th century, traditionally identified with Anno Domini .Dates before the year 1 CE are indicated by the usage of BCE, short for Before the Common Era Common Era...
.
Some argue that the system originated from Vararuci
Vararuci
Vararuci is a name associated with several literary and scientific texts in Sanskrit and also with various legends in several parts of India. This Vararuci is often identified with Kātyāyana...
.
In some astronomical texts popular in Kerala planetary positions were encoded in Kaṭapayādi system. The first such work is considered to be the Chandra-vakyani of Vararuci, who is traditionally assigned to the fourth century CE
Common Era
Common Era ,abbreviated as CE, is an alternative designation for the calendar era originally introduced by Dionysius Exiguus in the 6th century, traditionally identified with Anno Domini .Dates before the year 1 CE are indicated by the usage of BCE, short for Before the Common Era Common Era...
.
Therefore, sometime in the early first millennium is a reasonable estimate for the origin of the Kaṭapayādi System.
Aryabhata
Aryabhata
Aryabhata was the first in the line of great mathematician-astronomers from the classical age of Indian mathematics and Indian astronomy...
, in his treatise Aryabhatiya
Aryabhatiya
Āryabhaṭīya or Āryabhaṭīyaṃ, a Sanskrit astronomical treatise, is the magnum opus and only extant work of the 5th century Indian mathematician, Āryabhaṭa.- Structure and style:...
, is known to have used a similar but more complex system to represent astronomical number
Number
A number is a mathematical object used to count and measure. In mathematics, the definition of number has been extended over the years to include such numbers as zero, negative numbers, rational numbers, irrational numbers, and complex numbers....
s. But there is not definitive evidence whether the Kaṭapayādi System originated from Āryabhaṭa numeration.
Geographical spread of the use
Almost all evidences of the use of Kaṭapayādi System is from south IndiaSouth India
South India is the area encompassing India's states of Andhra Pradesh, Karnataka, Kerala and Tamil Nadu as well as the union territories of Lakshadweep and Pondicherry, occupying 19.31% of India's area...
, especially Kerala
Kerala
or Keralam is an Indian state located on the Malabar coast of south-west India. It was created on 1 November 1956 by the States Reorganisation Act by combining various Malayalam speaking regions....
.
Not much is known about its use in North India. However, on Sanskrit
Sanskrit
Sanskrit , is a historical Indo-Aryan language and the primary liturgical language of Hinduism, Jainism and Buddhism.Buddhism: besides Pali, see Buddhist Hybrid Sanskrit Today, it is listed as one of the 22 scheduled languages of India and is an official language of the state of Uttarakhand...
astrolabe
Astrolabe
An astrolabe is an elaborate inclinometer, historically used by astronomers, navigators, and astrologers. Its many uses include locating and predicting the positions of the Sun, Moon, planets, and stars, determining local time given local latitude and longitude, surveying, triangulation, and to...
discovered in north India
North India
North India, known natively as Uttar Bhārat or Shumālī Hindustān , is a loosely defined region in the northern part of India. The exact meaning of the term varies by usage...
, the degrees of the altitude are marked in the Kaṭapayādi system. It is now preserved in the Sarasvathy Bhavan Library of Sampurnanand Sanskrit University
Sampurnanand Sanskrit University
Sampurnanand Sanskrit University is an institution of higher learning in Asian learning, Sanskrit allied areas, etc. that is located in Benares , Uttar Pradesh, India.- History :...
, Varanasi
Varanasi
-Etymology:The name Varanasi has its origin possibly from the names of the two rivers Varuna and Assi, for the old city lies in the north shores of the Ganga bounded by its two tributaries, the Varuna and the Asi, with the Ganges being to its south...
.
The use of the Kaṭapayādi system is not confined to India. Some Pali
Páli
- External links :* *...
chronogram
Chronogram
A chronogram is a sentence or inscription in which specific letters, interpreted as numerals, stand for a particular date when rearranged. The word, meaning "time writing", derives from the Greek words chronos and gramma . In the pure chronogram each word contains a numeral, the natural chronogram...
s based on the Kaṭapayādi system have been discovered from Burma.
Rules and practices
Following verse found in Śaṅkaravarman'sSankara Varman
Sankara Varman is an astronomer-mathematician belonging to the Kerala school of astronomy and mathematics. He is best known as the author of Sadratnamala, a treatise on astronomy and mathematics, composed in 1819...
Sadratnamāla
Sadratnamala
Sadratnamala is an astronomical-mathematical treatise in Sanskrit written by Sankara Varman, an astronomer-mathematician of the Kerala school of mathematics, in 1819. Even though the book has been written at a time when western mathematics and astronomy had been introduced in India, it is composed...
explains the mechanism of the system.
नज्ञावचश्च 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
Consonant
In articulatory phonetics, a consonant is a speech sound that is articulated with complete or partial closure of the vocal tract. Examples are , pronounced with the lips; , pronounced with the front of the tongue; , pronounced with the back of the tongue; , pronounced in the throat; and ,...
group beginning with ka, ṭa, pa, ya. In a conjunct
Conjunct
In linguistics, the term conjunct has three distinct uses:*A conjunct is an adjunct that adds information to the sentence that is not considered part of the propositional content but which connects the sentence with previous parts of the discourse...
consonant, the last of the consonants alone will count. A consonant without vowel
Vowel
In phonetics, a vowel is a sound in spoken language, such as English ah! or oh! , pronounced with an open vocal tract so that there is no build-up of air pressure at any point above the glottis. This contrasts with consonants, such as English sh! , where there is a constriction or closure at some...
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 ക | kha ख ഖ | ga ग ഗ | gha घ ഘ | nga ङ ങ | ca च ച | cha छ ഛ | ja ज ജ | jha झ ഝ | nya ञ ഞ |
| ṭa ट ട | ṭha ठ ഠ | ḍa ड ഡ | ḍha ढ ഢ | ṇa ण ണ | ta त ത | tha थ | da द ദ | dha ध ധ | na न ന |
| pa प പ | pha फ ഫ | ba ब ബ | bha भ ഭ | ma म മ | - | - | - | - | - |
| ya य യ | ra र ര | la ल ല | va व വ | śha श ശ | sha ष ഷ | sa स സ | ha ह ഹ | - | - |
- Consonants have numerals assigned as per the above table. For example, ba (ब) is always three 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 0.
- In case of a conjuct, 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 separatorDecimal separatorDifferent symbols have been and are used for the decimal mark. The choice of symbol for the decimal mark affects the choice of symbol for the thousands separator used in digit grouping. Consequently the latter is treated in this article as well....
in the system. - Indians used the Hindu-Arabic numeral systemHindu-Arabic numeral systemThe Hindu–Arabic numeral system or Hindu numeral system is a positional decimal numeral system developed between the 1st and 5th centuries by Indian mathematicians, adopted by Persian and Arab mathematicians , and spread to the western world...
for numbering, traditionally written in increasing place values from left to right. This is as per the rule aṅkānām vāmato gati (अङ्कानाम् वामतो गति) which means numbers go from left to right.
Variations
- The DravidianDravidian languagesThe Dravidian language family includes approximately 85 genetically related languages, spoken by about 217 million people. They are mainly spoken in southern India and parts of eastern and central India as well as in northeastern Sri Lanka, Pakistan, Nepal, Bangladesh, Afghanistan, Iran, and...
consonantConsonantIn articulatory phonetics, a consonant is a speech sound that is articulated with complete or partial closure of the vocal tract. Examples are , pronounced with the lips; , pronounced with the front of the tongue; , pronounced with the back of the tongue; , pronounced in the throat; and ,...
, ḷ (Malayalam: ള, DevanagariDevanagariDevanagari |deva]]" and "nāgarī" ), also called Nagari , is an abugida alphabet of India and Nepal...
: ळ), of the Dravidian languagesDravidian languagesThe Dravidian language family includes approximately 85 genetically related languages, spoken by about 217 million people. They are mainly spoken in southern India and parts of eastern and central India as well as in northeastern Sri Lanka, Pakistan, Nepal, Bangladesh, Afghanistan, Iran, and...
of south IndiaSouth IndiaSouth India is the area encompassing India's states of Andhra Pradesh, Karnataka, Kerala and Tamil Nadu as well as the union territories of Lakshadweep and Pondicherry, occupying 19.31% of India's area...
is employed in works using the Kaṭapayādi System, like the Madhava's sine tableMadhava's sine tableMadhava's sine table is the table of trigonometric sines of various angles constructed by the 14th century Kerala mathematician-astronomer Madhava of Sangamagrama. The table lists the trigonometric sines of the twenty-four angles 3.75°, 7.50°, 11.25°, ... , and 90.00°...
. - Some practicers doesn't map the stand-alone vowels to zero. But considers as valueless.
Mathematics and Astronomy
- Madhava's sine tableMadhava's sine tableMadhava's sine table is the table of trigonometric sines of various angles constructed by the 14th century Kerala mathematician-astronomer Madhava of Sangamagrama. The table lists the trigonometric sines of the twenty-four angles 3.75°, 7.50°, 11.25°, ... , and 90.00°...
constructed by 14th century KeralaKeralaor Keralam is an Indian state located on the Malabar coast of south-west India. It was created on 1 November 1956 by the States Reorganisation Act by combining various Malayalam speaking regions....
mathematicianMathematicianA mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
-astronomerAstronomerAn astronomer is a scientist who studies celestial bodies such as planets, stars and galaxies.Historically, astronomy was more concerned with the classification and description of phenomena in the sky, while astrophysics attempted to explain these phenomena and the differences between them using...
Madhava of SangamagramaMadhava of SangamagramaMādhava of Sañgamāgrama was a prominent Kerala mathematician-astronomer from the town of Irińńālakkuţa near Cochin, Kerala, India. He is considered the founder of the Kerala School of Astronomy and Mathematics...
, employs the Kaṭapayādi system to enlist the trigonometric sines of various angles.
- KaranapaddhatiKaranapaddhatiKaranapaddhati is an astronomical treatise in Sanskrit attributed to Puthumana Somayaji, an astronomer-mathematician of the Kerala school of astronomy and mathematics. The period of composition of the work is uncertain. C.M. Whish, a civil servant of the East India Company, brought this work to the...
, written in the 15th century, has the following śloka for the value of piPi' is a mathematical constant that is the ratio of any circle's circumference to its diameter. is approximately equal to 3.14. Many formulae in mathematics, science, and engineering involve , which makes it one of the most important mathematical constants...
(π)
- അനൂനനൂന്നാനനനുന്നനിത്യൈ-
- സ്സമാഹതാശ്ചക്രകലാവിഭക്താഃ
- ചണ്ഡാംശുചന്ദ്രാധമകുംഭിപാലൈര്-
- വ്യാസസ്തദര്ദ്ധം ത്രിഭമൗര്വിക സ്യാത്
- Transiliteration
- anūnanūnnānananunnanityai
- ssmāhatāścakra kalāvibhaktoḥ
- caṇḍāṃśucandrādhamakuṃbhipālair
vyāsastadarddhaṃ tribhamaurvika syāt
- It gives the circumference of a circle of diameter, anūnanūnnānananunnanityai (10,000,000,000) as caṇḍāṃśucandrādhamakuṃbhipālair (31415926536)
- Sankaravarman's SadratnamalaSadratnamalaSadratnamala is an astronomical-mathematical treatise in Sanskrit written by Sankara Varman, an astronomer-mathematician of the Kerala school of mathematics, in 1819. Even though the book has been written at a time when western mathematics and astronomy had been introduced in India, it is composed...
uses the Kaṭapayādi system. A famous verse found in SadratnamalaSadratnamalaSadratnamala is an astronomical-mathematical treatise in Sanskrit written by Sankara Varman, an astronomer-mathematician of the Kerala school of mathematics, in 1819. Even though the book has been written at a time when western mathematics and astronomy had been introduced in India, it is composed...
is
- भद्राम्बुद्धिसिद्धजन्मगणितश्रद्धा स्म यद् भूपगी:
- Transiliteration
- bhadrāṃbuddhisiddhajanmagaṇitaśraddhā sma yad bhūpagīḥ
- Splitting the consonants gives,
भ bha द् d रा rā म् ṃ बु ba द् d धि dh सि sa द् d ध dha ज ja न् n म ma ग ga णि ṇa त ta श् ṣ र ra द् d धा dha स् s म ma य ya द् d भू bha 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 piPi' is a mathematical constant that is the ratio of any circle's circumference to its diameter. is approximately equal to 3.14. Many formulae in mathematics, science, and engineering involve , which makes it one of the most important mathematical constants...
(π) to 17 decimal places, except the last digit might be rounded off to 4.
Carnatic music
- The melakartaMelakartaMelakarta is a collection of fundamental ragas in Carnatic music . Melakarta ragas are parent ragas from which other ragas may be generated. A melakarta raga is sometimes referred as mela, karta or sampurna as well.In Hindustani music the thaat is equivalent of 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 Katapayadi sankhya. The SwaraSwaraThe seven notes of the scale , in Indian music are named shadja, rishabh, gandhar, madhyam, pancham, dhaivat and nishad, and are shortened to Sa, Ri or Re , Ga, Ma, Pa, Dha, and Ni and written S, R, G, M, P, D, N. Collectively these notes are known as the sargam...
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.
- '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.
Examples
- Raga DheerasankarabharanamDheerasankarabharanamDheerashankarābharanam , commonly known as Shankarābharanam, is a rāga in Carnatic music . It is the 29th Melakarta rāga in the 72 melakarta rāga system of Carnatic music....
The katapayadi scheme associates dha
9 and ra
2, 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 quotientQuotient
In mathematics, a quotient is the result of division. For example, when dividing 6 by 3, the quotient is 2, while 6 is called the dividend, and 3 the divisor. The quotient further is expressed as the number of times the divisor divides into the dividend e.g. The quotient of 6 and 2 is also 3.A...
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 MechaKalyaniKalyani (raga)Kalyan or Kalyani, originally called Yaman, is a melakarta rāga in the Carnatic music of South India, and is also an important raga in Hindustani music.-Kalyani in Carnatic music:...
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 SimhendramadhyamamSimhendramadhyamamSimhendramadhyamam is a rāgam in Carnatic music . It is the 57th melakarta rāgam in the 72 melakarta rāgam system of Carnatic music. It is called Sumadhyuti in Muthuswami Dikshitar school of Carnatic music.Raganidhi by P. Subba Rao, Pub...
You can see that, 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 Sihmendramadhymam (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 YugaKali Yuga
Kali Yuga is the last of the four stages that the world goes through as part of the cycle of yugas described in the Indian scriptures. The other ages are Satya Yuga, Treta Yuga and Dvapara Yuga...
. It is sometimes called kalidina sankhya.
- The Malayalam calendarMalayalam calendarMalayalam calendar is a solar and sidereal Hindu calendar used in Kerala, India. The origin of the calendar has been dated as 825 CE....
known as kollavarsham (Malayalam: കൊല്ലവര്ഷം) was adopted in Kerala beginning from 825 CECommon EraCommon Era ,abbreviated as CE, is an alternative designation for the calendar era originally introduced by Dionysius Exiguus in the 6th century, traditionally identified with Anno Domini .Dates before the year 1 CE are indicated by the usage of BCE, short for Before the Common Era Common Era...
, revamping some existing calendars. This date is remembered as āchārya vāgbhadā, converted using Kaṭapayādi into 1434160 days since the start of Kali YugaKali YugaKali Yuga is the last of the four stages that the world goes through as part of the cycle of yugas described in the Indian scriptures. The other ages are Satya Yuga, Treta Yuga and Dvapara Yuga...
.
- NarayaniyamNarayaniyamNarayaniyam is a medieval Sanskrit text, comprising a summary study in poetic form of the Bhagavata Purana. It was composed by Melpathur Narayana Bhattathiri, a devotee-poet who lived in the Indian state of Kerala, in the sixteenth century AD...
, written by Melpathur Narayana BhattathiriMelpathur Narayana BhattathiriMelpathur Narayana Bhattathiri , third student of Achyuta Pisharati, was of Madhava of Sangamagrama's Kerala school of astronomy and mathematics. He was a mathematical linguist . His most important scholarly work, Prkriya-sarvawom, sets forth an axiomatic system elaborating on the classical system...
, ends with the line, āyurārogyasaukhyam (ആയുരാരോഗ്യസൌഖ്യം) which means long-life, health and happiness.
| In Malayalam | ആയുരാരോഗ്യസൌഖ്യം |
|---|---|
| In Devanagari Devanagari Devanagari |deva]]" and "nāgarī" ), also called Nagari , is an abugida alphabet of India and Nepal... |
आयुरारोग्यसौख्यम् |
| In IAST IAST The International Alphabet of Sanskrit Transliteration is a transliteration scheme that allows a lossless romanization of Indic scripts as employed by the Sanskrit language.-Popularity:... |
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 YugaKali YugaKali Yuga is the last of the four stages that the world goes through as part of the cycle of yugas described in the Indian scriptures. The other ages are Satya Yuga, Treta Yuga and Dvapara Yuga...
as per the Malayalam calendarMalayalam calendarMalayalam calendar is a solar and sidereal Hindu calendar used in Kerala, India. The origin of the calendar has been dated as 825 CE....
.
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 CalendarGregorian calendarThe Gregorian calendar, also known as the Western calendar, or Christian calendar, is the internationally accepted civil calendar. It was introduced by Pope Gregory XIII, after whom the calendar was named, by a decree signed on 24 February 1582, a papal bull known by its opening words Inter...
.
- പലഹാരേ പാലു നല്ലൂ, പുലര്ന്നാലോ കലക്കിലാം
- ഇല്ലാ പാലെന്നു ഗോപാലന് - ആംഗ്ലമാസദിനം ക്രമാല്
- Transiliteration
- 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
- Kerala school of astronomy and mathematics
- Madhava's sine tableMadhava's sine tableMadhava's sine table is the table of trigonometric sines of various angles constructed by the 14th century Kerala mathematician-astronomer Madhava of Sangamagrama. The table lists the trigonometric sines of the twenty-four angles 3.75°, 7.50°, 11.25°, ... , and 90.00°...
- Katapayadi sankhya
- Bhutasamkhya systemBhutasamkhya systemBhūtasaṃkhyā system is a method of recording numbers using ordinary words having connotations of numerical values. The method was popular among Indian astronomers and mathematicians since ancient times. Sanskrit was the language from which words were chosen to write numbers in the bhūtasaṃkhyā...
- Aryabhata numeration