In mathematics, the arctangent series , traditionally called Gregory's series , is the Taylor series expansion at the origin of the arctangent function :
81-572: Yuktibhāṣā ( Malayalam : യുക്തിഭാഷ , lit. 'Rationale'), also known as Gaṇita-yukti-bhāṣā and Gaṇitanyāyasaṅgraha ( English: Compendium of Astronomical Rationale ), is a major treatise on mathematics and astronomy , written by the Indian astronomer Jyesthadeva of the Kerala school of mathematics around 1530. The treatise, written in Malayalam, is a consolidation of
162-517: A Bhashya (language) where "Dravida and Sanskrit should combine together like ruby and coral, without the least trace of any discord". The scripts of Kolezhuthu and Malayanma were also used to write Middle Malayalam . In addition to Vatteluthu and Grantha script , those were used to write Old Malayalam . The literary works written in Middle Malayalam were heavily influenced by Sanskrit and Prakrit , while comparing them with
243-469: A canonical word order of SOV (subject–object–verb), as do other Dravidian languages . A rare OSV word order occurs in interrogative clauses when the interrogative word is the subject. Both adjectives and possessive adjectives precede the nouns they modify. Malayalam has 6 or 7 grammatical cases . Verbs are conjugated for tense, mood and aspect, but not for person, gender nor number except in archaic or poetic language. The modern Malayalam grammar
324-749: A distinct language due to geographical separation of Kerala from the Tamil country and the influence of Sanskrit and Prakrit from the Nambudiri Brahmins of the Malabar Coast . The Old Malayalam language was employed in several official records and transactions (at the level of the Chera Perumal kings, as well as the upper-caste ( Nambudiri ) village temples). Most of the inscriptions in Old Malayalam were found from
405-585: A literary language. The Malayalam script began to diverge from the Vatteluttu and the Western Grantha scripts in the 8th and 9th centuries of Common Era . By the end of the 13th century, a written form of the language emerged which was unique from the Vatteluttu script that was used to write Tamil on the eastern coast. Old Malayalam ( Paḻaya Malayāḷam ), an inscriptional language found in Kerala from circa 9th to circa 13th century CE,
486-592: A lot of its words from various foreign languages: mainly from the Semitic languages including Arabic , and the European languages including Dutch and Portuguese , due to the long heritage of Indian Ocean trade and the Portuguese-Dutch colonization of the Malabar Coast . Variations in intonation patterns, vocabulary, and distribution of grammatical and phonological elements are observable along
567-685: A spirit of brotherhood. മനുഷ്യരെല്ലാവരും തുല്യാവകാശങ്ങളോടും അന്തസ്സോടും സ്വാതന്ത്ര്യത്തോടുംകൂടി ജനിച്ചിട്ടുള്ളവരാണ്. അന്യോന്യം ഭ്രാതൃഭാവത്തോടെ പെരുമാറുവാനാണ് മനുഷ്യന് വിവേകബുദ്ധിയും മനസാക്ഷിയും സിദ്ധമായിരിക്കുന്നത്. manuṣyarellāvaruṁ tulyāvakāśaṅṅaḷōṭuṁ antassōṭuṁ svātantryattōṭuṅkūṭi janicciṭṭuḷḷavarāṇŭ. anyōnyaṁ bhrātr̥bhāvattōṭe perumāṟuvānāṇŭ manuṣyanŭ vivēkabuddhiyuṁ manasākṣiyuṁ siddhamāyirikkunnatŭ. /manuʂjaɾellaːʋaɾum t̪uljaːʋakaːʃaŋŋaɭoːʈum an̪t̪assoːʈum sʋaːt̪an̪tɾjat̪t̪oːʈuŋkuːʈi d͡ʒanit͡ʃt͡ʃiʈʈuɭɭaʋaɾaːɳɨ̆ ǁ anjoːnjam bʱraːt̪rɨ̆bʱaːʋat̪t̪oːʈe peɾumaːruʋaːnaːɳɨ̆ manuʂjanɨ̆ ʋiʋeːkabud̪d̪ʱijum manasaːkʂijum sid̪d̪ʱamaːjiɾikkun̪ːat̪ɨ̆ ǁ/ Malayalam has
648-1115: A subject of great mathematical interest. Isaac Newton accelerated the convergence of the arctangent series in 1684 (in an unpublished work; others independently discovered the result and it was later popularized by Leonhard Euler 's 1755 textbook; Euler wrote two proofs in 1779), yielding a series converging for | x | < ∞ , {\textstyle |x|<\infty ,} where | C ( x ) = 1 / 1 + x 2 = {\textstyle {\vphantom {\Big |}}C(x)=1{\big /}\!{\sqrt {1+x^{2}}}={}} cos ( arctan x ) {\displaystyle \cos(\arctan x)} and | S ( x ) = x / 1 + x 2 = {\textstyle {\vphantom {\Big |}}S(x)=x{\big /}\!{\sqrt {1+x^{2}}}={}} sin ( arctan x ) . {\textstyle \sin(\arctan x).} Each term of this modified series
729-500: A sum of arctangents of smaller values, eventually resulting in a variety of Machin-like formulas for π {\displaystyle \pi } . Isaac Newton (1684) and other mathematicians accelerated the convergence of the series via various transformations. If y = arctan x {\displaystyle y=\arctan x} then tan y = x . {\displaystyle \tan y=x.} The derivative
810-454: A western coastal dialect of early Middle Tamil and separated from it sometime between the 9th and 13th centuries. A second view argues for the development of the two languages out of "Proto-Dravidian" or "Proto-Tamil-Malayalam" either in the prehistoric period or in the middle of the first millennium A.D. , although this is generally rejected by historical linguists. The Quilon Syrian copper plates of 849/850 CE are considered by some to be
891-667: Is Taking the reciprocal, This sometimes is used as a definition of the arctangent: The Maclaurin series for x ↦ arctan ′ x = 1 / ( 1 + x 2 ) {\textstyle x\mapsto \arctan 'x=1{\big /}\left(1+x^{2}\right)} is a geometric series : One can find the Maclaurin series for arctan {\displaystyle \arctan } by naïvely integrating term-by-term: While this turns out correctly, integrals and infinite sums cannot always be exchanged in this manner. To prove that
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#1732793446154972-431: Is a pole : When x = ± 1 , {\displaystyle x=\pm 1,} the partial sums ∑ k = 0 n ( − x 2 ) k {\textstyle \sum _{k=0}^{n}(-x^{2})^{k}} alternate between the values 0 {\displaystyle 0} and 1 , {\displaystyle 1,} never converging to
1053-541: Is a rational function with its poles at x = ± i {\displaystyle x=\pm i} in the complex plane , the same place where the arctangent function has its poles. By contrast, a polynomial such as the Taylor series for arctangent forces all of its poles to infinity. The earliest person to whom the series can be attributed with confidence is Mādhava of Sangamagrāma (c. 1340 – c. 1425). The original reference (as with much of Mādhava's work)
1134-782: Is also a considerable Malayali population in the Persian Gulf regions, especially in Dubai , Kuwait and Doha . For the consonants and vowels, the International Phonetic Alphabet (IPA) symbol is given, followed by the Malayalam character and the ISO 15919 transliteration. The current Malayalam script bears high similarity with Tigalari script , which was used for writing the Tulu language , spoken in coastal Karnataka ( Dakshina Kannada and Udupi districts) and
1215-542: Is also credited with developing the Malayalam script into the current form through the intermixing and modification of the erstwhile scripts of Vatteluttu , Kolezhuthu , and Grantha script , which were used to write the inscriptions and literary works of Old and Middle Malayalam. He further eliminated excess and unnecessary letters from the modified script. Hence, Ezhuthachan is also known as The Father of modern Malayalam . The development of modern Malayalam script
1296-529: Is also spoken by the Malayali Diaspora worldwide, especially in the Persian Gulf countries , due to the large populations of Malayali expatriates there. They are a significant population in each city in India including Mumbai , Bengaluru , Chennai , Delhi , Hyderabad etc. The origin of Malayalam remains a matter of dispute among scholars. The mainstream view holds that Malayalam descends from
1377-412: Is based on the book Kerala Panineeyam written by A. R. Raja Raja Varma in late 19th century CE. The declensional paradigms for some common nouns and pronouns are given below. As Malayalam is an agglutinative language, it is difficult to delineate the cases strictly and determine how many there are, although seven or eight is the generally accepted number. Alveolar plosives and nasals (although
1458-445: Is divided into three parts: First part is an English translation of the relevant Malayalam part of Yuktibhasa , second part contains detailed explanatory notes on the translation, and in the third part the text in the Malayalam original is reproduced. The English translation is by K.V. Sarma and the explanatory notes are provided by K. Ramasubramanian, M. D. Srinivas, and M. S. Sriram. An open access edition of Yuktibhasa
1539-512: Is influenced by Tamil. Labels such as "Nampoothiri Dialect", "Mappila Dialect", and "Nasrani Dialect" refer to overall patterns constituted by the sub-dialects spoken by the subcastes or sub-groups of each such caste. The most outstanding features of the major communal dialects of Malayalam are summarized below: Malayalam has incorporated many elements from other languages over the years, the most notable of these being Sanskrit and later, English. According to Sooranad Kunjan Pillai who compiled
1620-471: Is lost, but he is credited with the discovery by several of his successors in the Kerala school of astronomy and mathematics founded by him. Specific citations to the series for arctan {\displaystyle \arctan } include Nīlakaṇṭha Somayāji 's Tantrasaṅgraha (c. 1500), Jyeṣṭhadeva 's Yuktibhāṣā (c. 1530), and the Yukti-dipika commentary by Sankara Variyar , where it
1701-660: Is phonemic and all of the vowels have minimal pairs for example kaṭṭi "thickness", kāṭṭi "showed", koṭṭi "tapped", kōṭṭi "twisted, stick, marble", er̠i "throw", ēr̠i "lots" Some speakers also have /æː/, /ɔː/, /ə/ from English loanwords e.g. /bæːŋgɨ̆/ "bank" but most speakers replace it with /aː/, /eː/ or /ja/; /oː/ or /aː/ and /e/ or /a/. The following text is Article 1 of the Universal Declaration of Human Rights . All human beings are born free and equal in dignity and rights. They are endowed with reason and conscience and should act towards one another in
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#17327934461541782-540: Is published by Sayahna Foundation in 2020. Malayalam language Malayalam ( / ˌ m æ l ə ˈ j ɑː l ə m / ; മലയാളം , Malayāḷam , IPA: [mɐlɐjaːɭɐm] ) is a Dravidian language spoken in the Indian state of Kerala and the union territories of Lakshadweep and Puducherry ( Mahé district ) by the Malayali people. It is one of 22 scheduled languages of India. Malayalam
1863-438: Is the earliest attested form of Malayalam. The beginning of the development of Old Malayalam from a western coastal dialect of Middle Tamil can be dated to circa 8th century CE. It remained a west coast dialect until circa 9th century CE or a little later. The origin of Malayalam calendar dates back to year 825 CE. It is generally agreed that the western coastal dialect of Tamil began to separate, diverge, and grow as
1944-727: The saṁvr̥tōkāram , which is not officially a vowel, and not as actual vocalic consonants) have been classified as vowels: vocalic r ( ഋ , /rɨ̆/ , r̥), long vocalic r ( ൠ , /rɨː/ , r̥̄), vocalic l ( ഌ , /lɨ̆/ , l̥) and long vocalic l ( ൡ , /lɨː/ , l̥̄). Except for the first, the other three have been omitted from the current script used in Kerala as there are no words in current Malayalam that use them. Some authors say that Malayalam has no diphthongs and /ai̯, au̯/ are clusters of V+glide j/ʋ while others consider all V+glide clusters to be diphthongs /ai̯, aːi̯, au̯, ei̯, oi̯, i̯a/ as in kai, vāypa, auṣadhaṁ, cey, koy and kāryaṁ Vowel length
2025-998: The Middle East , the United States, and Europe. There were 179,860 speakers of Malayalam in the United States, according to the 2000 census, with the highest concentrations in Bergen County, New Jersey , and Rockland County, New York . There are 144,000 of Malayalam speakers in Malaysia . There were 11,687 Malayalam speakers in Australia in 2016. The 2001 Canadian census reported 7,070 people who listed Malayalam as their mother tongue, mainly in Toronto . The 2006 New Zealand census reported 2,139 speakers. 134 Malayalam speaking households were reported in 1956 in Fiji . There
2106-684: The Namboothiri and Nair dialects have a common nature, the Arabi Malayalam is among the most divergent of dialects, differing considerably from literary Malayalam. Jeseri is a dialect of Malayalam spoken mainly in the Union territory of Lakshadweep and Beary is spoken in Tulu Nadu which are nearer to Kerala. Of the total 33,066,392 Malayalam speakers in India in 2001, 33,015,420 spoke
2187-536: The Pathinettara Kavikal (Eighteen and a half poets) in the court of the Zamorin of Calicut , also belong to Middle Malayalam. The literary works of this period were heavily influenced by Manipravalam , which was a combination of contemporary Tamil and Sanskrit . The word Mani-Pravalam literally means Diamond-Coral or Ruby-Coral . The 14th-century Lilatilakam text states Manipravalam to be
2268-583: The Voiced retroflex approximant (/ɻ/) which is seen in both Tamil and the standard form of Malayalam, are not seen in the northern dialects of Malayalam, as in Kannada . For example, the words Vazhi (Path), Vili (Call), Vere (Another), and Vaa (Come/Mouth), become Bayi , Bili , Bere , and Baa in the northern dialects of Malayalam. Similarly the Malayalam spoken in the southern districts of Kerala, i.e., Thiruvananthapuram - Kollam - Pathanamthitta area
2349-592: The Yerava dialect and 31,329 spoke non-standard regional variations like Eranadan . As per the 1991 census data, 28.85% of all Malayalam speakers in India spoke a second language and 19.64% of the total knew three or more languages. Just before independence, Malaya attracted many Malayalis. Large numbers of Malayalis have settled in Chennai , Bengaluru , Mangaluru , Hyderabad , Mumbai , Navi Mumbai , Pune , Mysuru and Delhi . Many Malayalis have also emigrated to
2430-521: The colonial period . Due to the geographical isolation of the Malabar Coast from the rest of the Indian peninsula due to the presence of the Western Ghats mountain ranges which lie parallel to the coast, the dialect of Old Tamil spoken in Kerala was different from that spoken in Tamil Nadu . The mainstream view holds that Malayalam began to grow as a distinct literary language from
2511-453: The complex disk | x | ≤ 1 , {\displaystyle |x|\leq 1,} except for x = ± i {\displaystyle x=\pm i} (where arctan ± i = ∞ {\displaystyle \arctan \pm i=\infty } ). It was first discovered in the 14th century by Indian mathematician Mādhava of Sangamagrāma ( c. 1340 – c. 1425),
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2592-517: The northern districts of Kerala , those lie adjacent to Tulu Nadu . Old Malayalam was mostly written in Vatteluttu script (with Pallava/Southern Grantha characters). Old Malayalam had several features distinct from the contemporary Tamil, which include the nasalisation of adjoining sounds, substitution of palatal sounds for dental sounds, contraction of vowels, and the rejection of gender verbs. Ramacharitam and Thirunizhalmala are
2673-423: The script and the region . According to Duarte Barbosa , a Portuguese visitor who visited Kerala in the early 16th century CE, the people in the southwestern Malabar coast of India from Kumbla in north to Kanyakumari in south had a unique language, which was called "Maliama" by them. Prior to this period , the people of Kerala usually referred to their language as "Tamil", and both terms overlapped into
2754-747: The 13th and 14th centuries of the Common Era . The Sandesha Kavya s of 14th century CE written in Manipravalam language include Unnuneeli Sandesam . Kannassa Ramayanam and Kannassa Bharatham by Rama Panikkar of the Niranam poets who lived between 1350 and 1450, are representative of this language. Ulloor has opined that Rama Panikkar holds the same position in Malayalam literature that Edmund Spenser does in English literature . The Champu Kavyas written by Punam Nambudiri, one among
2835-481: The 16th century CE, Thunchaththu Ezhuthachan from the Kingdom of Tanur and Poonthanam Nambudiri from the Kingdom of Valluvanad , followed the new trend initiated by Cherussery in their poems. The Adhyathmaramayanam Kilippattu and Mahabharatham Kilippattu , written by Ezhuthachan, and Jnanappana , written by Poonthanam, are also included in the earliest form of Modern Malayalam. Thunchaththu Ezhuthachan
2916-588: The Dravidian or South-Indian Family of Languages" , opined that literary Malayalam branched from Classical Tamil and over time gained a large amount of Sanskrit vocabulary and lost the personal terminations of verbs. As the language of scholarship and administration, Old-Tamil, which was written in Tamil-Brahmi and the Vatteluttu alphabet later, greatly influenced the early development of Malayalam as
2997-483: The approximation of π to 13 decimal places of accuracy when n =76. Apart from these, the Yuktibhāṣā contains many elementary and complex mathematical topics, including, Chapters eight to seventeen deal with subjects of astronomy: planetary orbits , celestial spheres , ascension , declination , directions and shadows, spherical triangles , ellipses , and parallax correction. The planetary theory described in
3078-721: The arctangent of 1 {\displaystyle 1} is traditionally called the Leibniz formula for π , or recently sometimes the Mādhava–Leibniz formula : The extremely slow convergence of the arctangent series for | x | ≈ 1 {\displaystyle |x|\approx 1} makes this formula impractical per se. Kerala-school mathematicians used additional correction terms to speed convergence. John Machin (1706) expressed 1 4 π {\displaystyle {\tfrac {1}{4}}\pi } as
3159-515: The authoritative Malayalam lexicon, the other principal languages whose vocabulary was incorporated over the ages were Arabic , Dutch , Hindustani , Pali , Persian , Portuguese , Prakrit , and Syriac . Malayalam is a language spoken by the native people of southwestern India and the islands of Lakshadweep in the Arabian Sea . According to the Indian census of 2011, there were 32,413,213 speakers of Malayalam in Kerala, making up 93.2% of
3240-533: The authorship of the medieval work Keralolpathi , which describes the Parashurama legend and the departure of the final Cheraman Perumal king to Mecca , to Thunchaththu Ramanujan Ezhuthachan. Kunchan Nambiar introduced a new literary form called Thullal , and Unnayi Variyar introduced reforms in Attakkatha literature . The printing, prose literature, and Malayalam journalism , developed after
3321-496: The book Kerala Panineeyam written by A. R. Raja Raja Varma in late 19th century CE. The first travelogue in any Indian language is the Malayalam Varthamanappusthakam , written by Paremmakkal Thoma Kathanar in 1785. Robert Caldwell describes the extent of Malayalam in the 19th century as extending from the vicinity of Kumbla in the north where it supersedes with Tulu to Kanyakumari in
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3402-467: The book is similar to that later adopted by Danish astronomer Tycho Brahe . The topics covered in the eight chapters are computation of mean and true longitudes of planets, Earth and celestial spheres, fifteen problems relating to ascension, declination, longitude, etc., determination of time, place, direction, etc., from gnomonic shadow, eclipses, Vyatipata (when the sun and moon have the same declination), visibility correction for planets and phases of
3483-429: The discoveries by Madhava of Sangamagrama , Nilakantha Somayaji , Parameshvara , Jyeshtadeva , Achyuta Pisharati , and other astronomer-mathematicians of the Kerala school. It also exists in a Sanskrit version, with unclear author and date, composed as a rough translation of the Malayalam original. The work contains proofs and derivations of the theorems that it presents. Modern historians used to assert, based on
3564-507: The early Middle Tamil period, thus making independent descent impossible. For example, Old Tamil lacks the first and second person plural pronouns with the ending kaḷ . It is in the Early Middle Tamil stage that kaḷ first appears: Indeed, most features of Malayalam morphology are derivable from a form of speech corresponding to early Middle Tamil. Robert Caldwell , in his 1856 book " A Comparative Grammar of
3645-586: The entire Malayalam text, alongside an English translation and detailed explanatory notes, was published in two volumes by Springer in 2008. A third volume, containing a critical edition of the Sanskrit Ganitayuktibhasa, was published by the Indian Institute of Advanced Study , Shimla in 2009. This edition of Yuktibhasa has been divided into two volumes: Volume I deals with mathematics and Volume II treats astronomy. Each volume
3726-481: The first 21 terms of the infinite series was used to compute the approximation to 11 decimal places. The other method was to add a remainder term to the original series of π. The remainder term n 2 + 1 4 n 3 + 5 n {\textstyle {\frac {n^{2}+1}{4n^{3}+5n}}} was used in the infinite series expansion of π 4 {\displaystyle {\frac {\pi }{4}}} to improve
3807-417: The first term is repeatedly multiplied by the square of the sine and divided by the square of the cosine. All the terms are then divided by the odd numbers 1, 3, 5, .... The arc is obtained by adding and subtracting respectively the terms of odd rank and those of even rank. It is laid down that the sine of the arc or that of its complement whichever is the smaller should be taken here as the given sine. Otherwise
3888-584: The founder of the Kerala school , and is described in extant works by Nīlakaṇṭha Somayāji (c. 1500) and Jyeṣṭhadeva (c. 1530). Mādhava's work was unknown in Europe, and the arctangent series was independently rediscovered by James Gregory in 1671 and by Gottfried Leibniz in 1673. In recent literature the arctangent series is sometimes called the Mādhava–Gregory series to recognize Mādhava's priority (see also Mādhava series ). The special case of
3969-415: The integral on the left converges to the sum on the right for real | x | ≤ 1 , {\displaystyle |x|\leq 1,} arctan ′ {\displaystyle \arctan '} can instead be written as the finite sum, Again integrating both sides, In the limit as N → ∞ , {\displaystyle N\to \infty ,}
4050-672: The integral on the right above tends to zero when | x | ≤ 1 , {\displaystyle |x|\leq 1,} because Therefore, The series for arctan ′ {\textstyle \arctan '} and arctan {\displaystyle \arctan } converge within the complex disk | x | < 1 {\displaystyle |x|<1} , where both functions are holomorphic . They diverge for | x | > 1 {\displaystyle |x|>1} because when x = ± i {\displaystyle x=\pm i} , there
4131-534: The latter with astronomy. Beyond this, the continuous text does not have any further division into subjects or topics, so published editions divide the work into chapters based on editorial judgment. This subjects treated in the mathematics part of the Yuktibhāṣā can be divided into seven chapters: The first four chapters of the contain elementary mathematics, such as division, the Pythagorean theorem , square roots , etc. Novel ideas are not discussed until
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#17327934461544212-528: The latter-half of the 18th century CE. Modern literary movements in Malayalam literature began in the late 19th century with the rise of the famous Modern Triumvirate consisting of Kumaran Asan , Ulloor S. Parameswara Iyer and Vallathol Narayana Menon . In the second half of the 20th century, Jnanpith winning poets and writers like G. Sankara Kurup , S. K. Pottekkatt , Thakazhi Sivasankara Pillai , M. T. Vasudevan Nair , O. N. V. Kurup , and Akkitham Achuthan Namboothiri , had made valuable contributions to
4293-468: The modern Malayalam literature . The Middle Malayalam was succeeded by Modern Malayalam ( Aadhunika Malayalam ) by 15th century CE. The poem Krishnagatha written by Cherusseri Namboothiri , who was the court poet of the king Udaya Varman Kolathiri (1446–1475) of Kolathunadu , is written in modern Malayalam. The language used in Krishnagatha is the modern spoken form of Malayalam. During
4374-573: The modern Malayalam script does not distinguish the latter from the dental nasal ) are underlined for clarity, following the convention of the National Library at Kolkata romanization . Vocative forms are given in parentheses after the nominative , as the only pronominal vocatives that are used are the third person ones, which only occur in compounds. വിഭക്തി സംബോധന പ്രതിഗ്രാഹിക സംബന്ധിക ഉദ്ദേശിക പ്രായോജിക ആധാരിക സംയോജിക Gregory%27s series This series converges in
4455-399: The modern Malayalam literature. The life and works of Edasseri Govindan Nair have assumed greater socio-literary significance after his death and Edasseri is now recognised as an important poet of Malayalam. Later, writers like O. V. Vijayan , Kamaladas , M. Mukundan , Arundhati Roy , and Vaikom Muhammed Basheer , have gained international recognition. Malayalam has also borrowed
4536-550: The moon. Specifically, The importance of Yuktibhāṣā was brought to the attention of modern scholarship by C. M. Whish in 1832 through a paper published in the Transactions of the Royal Asiatic Society of Great Britain and Ireland . The mathematics part of the text, along with notes in Malayalam, was first published in 1948 by Rama Varma Thampuran and Akhileswara Aiyar. The first critical edition of
4617-408: The name of its language. The language Malayalam was alternatively called Alealum , Malayalani , Malayali , Malabari , Malean , Maliyad , Mallealle , and Kerala Bhasha until the early 19th century CE. The earliest extant literary works in the regional language of present-day Kerala probably date back to as early as the 12th century . At that time, the language
4698-673: The northernmost Kasargod district of Kerala. Tigalari script was also used for writing Sanskrit in Malabar region . Malayalam has also borrowed the Sanskrit diphthongs of /ai̯/ (represented in Malayalam as ഐ , ai) and /au̯/ (represented in Malayalam as ഔ , au) although these mostly occur only in Sanskrit loanwords. Traditionally (as in Sanskrit), four vocalic consonants (usually pronounced in Malayalam as consonants followed by
4779-432: The oldest available inscription written in Old Malayalam . However, the existence of Old Malayalam is sometimes disputed by scholars. They regard the Chera Perumal inscriptional language as a diverging dialect or variety of contemporary Tamil . The oldest extant literary work in Malayalam distinct from the Tamil tradition is Ramacharitam (late 12th or early 13th century). The earliest script used to write Malayalam
4860-643: The parameters of region, religion, community, occupation, social stratum, style and register. According to the Dravidian Encyclopedia, the regional dialects of Malayalam can be divided into fifteen dialect areas. They are as follows: According to Ethnologue, the dialects are: Malabar, Nagari-Malayalam, North Kerala, Central Kerala, South Kerala, Kayavar, Namboodiri , Nair , Mappila , Beary , Jeseri , Yerava , Pulaya, Nasrani , and Kasargod . The community dialects are: Namboodiri , Nair , Arabi Malayalam , Pulaya, and Nasrani . Whereas both
4941-573: The point ± 1 , {\displaystyle \pm 1,} so the difference in integrals can be made arbitrarily small by taking sufficiently many terms: Because of its exceedingly slow convergence (it takes five billion terms to obtain 10 correct decimal digits), the Leibniz formula is not a very effective practical method for computing 1 4 π . {\textstyle {\tfrac {1}{4}}\pi .} Finding ways to get around this slow convergence has been
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#17327934461545022-410: The possible literary works of Old Malayalam found so far. Old Malayalam gradually developed into Middle Malayalam ( Madhyakaala Malayalam ) by the 13th century CE. Malayalam literature also completely diverged from Tamil literature during this period. Works including Unniyachi Charitham , Unnichiruthevi Charitham , and Unniyadi Charitham , are written in Middle Malayalam , and date back to
5103-500: The prehistoric period from a common ancestor, "Proto-Tamil-Malayalam", and that the notion of Malayalam being a "daughter" of Tamil is misplaced. This is based on the fact that Malayalam and several Dravidian languages on the Western Coast have common archaic features which are not found even in the oldest historical forms of literary Tamil. Despite this, Malayalam shares many common innovations with Tamil that emerged during
5184-464: The results in Nilakantha's Tantra Samgraha . It is considered an early text to give some ideas of calculus like Taylor and infinity series, predating Newton and Leibniz by two centuries. The treatise was largely unnoticed outside India, as it was written in the local language of Malayalam. In modern times, due to wider international cooperation in mathematics, the wider world has taken notice of
5265-597: The single largest linguistic group accounting for 35.5% in the Virajpet Taluk. Around one-third of the Malayalis in Kodagu district speak the Yerava dialect according to the 2011 census, which is native to Kodagu and Wayanad . In all, Malayalis made up 3.22% of the total Indian population in 2011. Of the total 34,713,130 Malayalam speakers in India in 2011, 33,015,420 spoke the standard dialects, 19,643 spoke
5346-423: The sixth chapter on circumference of a circle . Yuktibhāṣā contains a derivation and proof for the power series of inverse tangent , discovered by Madhava. In the text, Jyesthadeva describes Madhava's series in the following manner: The first term is the product of the given sine and radius of the desired arc divided by the cosine of the arc. The succeeding terms are obtained by a process of iteration when
5427-509: The south, where it begins to be superseded by Tamil , beside the inhabited islands of Lakshadweep in the Arabian Sea . In a 7th century poem written by the Tamil poet Sambandar the people of Kerala are referred to as malaiyāḷar (mountain people). The word Malayalam is also said to originate from the words mala , meaning ' mountain ', and alam , meaning ' region ' or '-ship' (as in "township"); Malayalam thus translates directly as 'the mountain region'. The term Malabar
5508-490: The standard dialects, 19,643 spoke the Yerava dialect and 31,329 spoke non-standard regional variations like Eranadan . The dialects of Malayalam spoken in the districts like Kasaragod , Kannur , Wayanad , Kozhikode , and Malappuram in the former Malabar District have few influences from Kannada . For example, the words those start with the sound "V" in Malayalam become "B" in these districts as in Kannada . Also
5589-427: The terms obtained by this above iteration will not tend to the vanishing magnitude. In modern mathematical notation, or, expressed in terms of tangents, which in Europe was conventionally called Gregory's series after James Gregory , who rediscovered it in 1671. The text also contains Madhava's infinite series expansion of π which he obtained from the expansion of the arc-tangent function. which in Europe
5670-590: The total number of Malayalam speakers in India, and 97.03% of the total population of the state. There were a further 701,673 (1.14% of the total number) in Karnataka , 957,705 (2.70%) in Tamil Nadu , and 406,358 (1.2%) in Maharashtra . The number of Malayalam speakers in Lakshadweep is 51,100, which is only 0.15% of the total number, but is as much as about 84% of the population of Lakshadweep. Malayalam
5751-514: The value arctan ′ ( ± 1 ) = 1 2 . {\textstyle \arctan '(\pm 1)={\tfrac {1}{2}}.} However, its term-by-term integral, the series for arctan , {\textstyle \arctan ,} (barely) converges when x = ± 1 , {\displaystyle x=\pm 1,} because arctan ′ {\displaystyle \arctan '} disagrees with its series only at
5832-467: The western coastal dialect of Middle Tamil and the linguistic separation completed sometime between the 9th and 13th centuries. The renowned poets of Classical Tamil such as Paranar (1st century CE), Ilango Adigal (2nd–3rd century CE), and Kulasekhara Alvar (9th century CE) were Keralites . The Sangam works can be considered as the ancient predecessor of Malayalam. Some scholars however believe that both Tamil and Malayalam developed during
5913-474: The work. For example, both Oxford University and the Royal Society of Great Britain have given attribution to pioneering mathematical theorems of Indian origin that predate their Western counterparts. Yuktibhāṣā contains most of the developments of the earlier Kerala school, particularly Madhava and Nilakantha . The text is divided into two parts – the former deals with mathematical analysis and
5994-539: The works of Indian mathematics that first became available, that early Indian scholars in astronomy and computation lacked in proofs, but Yuktibhāṣā demonstrates otherwise. Some of its important topics include the infinite series expansions of functions; power series , including of π and π/4; trigonometric series of sine , cosine , and arctangent ; Taylor series , including second and third order approximations of sine and cosine ; radii, diameters and circumferences. Yuktibhāṣā mainly gives rationale for
6075-494: Was also heavily influenced by the Tigalari script , which was used to write Sanskrit , due to the influence of Tuluva Brahmins in Kerala. The language used in the Arabi Malayalam works of the 16th–17th century CE is a mixture of Modern Malayalam and Arabic . They follow the syntax of modern Malayalam, though written in a modified form of Arabic script , which is known as Arabi Malayalam script . P. Shangunny Menon ascribes
6156-443: Was conventionally called Leibniz's series , after Gottfried Leibniz who rediscovered it in 1673. Using a rational approximation of this series, he gave values of the number π as 3.14159265359, correct to 11 decimals, and as 3.1415926535898, correct to 13 decimals. The text describes two methods for computing the value of π. First, obtain a rapidly converging series by transforming the original infinite series of π. By doing so,
6237-514: Was designated a " Classical Language of India " in 2013. Malayalam has official language status in Kerala, Lakshadweep and Puducherry ( Mahé ), and is also the primary spoken language of Lakshadweep. Malayalam is spoken by 35 million people in India. Malayalam is also spoken by linguistic minorities in the neighbouring states; with a significant number of speakers in the Kodagu and Dakshina Kannada districts of Karnataka , and Kanyakumari , Coimbatore and Nilgiris district of Tamil Nadu. It
6318-418: Was differentiated by the name Kerala Bhasha . The earliest mention of Malayalam as a language is found outside of Kerala in the 15th century Telugu work Śrībhīmēśvarapurāṇamu by Śrīnātha. The distinctive "Malayalam" named identity of this language appears to have come into existence in Kerala only around the 16th century , when it was known as "Malayayma" or "Malayanma"; the words were also used to refer to
6399-465: Was the Vatteluttu script . The current Malayalam script is based on the Vatteluttu script, which was extended with Grantha script letters to adopt Indo-Aryan loanwords. It bears high similarity with the Tigalari script , a historical script that was used to write the Tulu language in South Canara , and Sanskrit in the adjacent Malabar region . The modern Malayalam grammar is based on
6480-554: Was the most spoken language in erstwhile Gudalur taluk (now Gudalur and Panthalur taluks) of Nilgiris district in Tamil Nadu which accounts for 48.8% population and it was the second most spoken language in Mangalore and Puttur taluks of South Canara accounting for 21.2% and 15.4% respectively according to 1951 census report. 25.57% of the total population in the Kodagu district of Karnataka are Malayalis , and they form
6561-457: Was used as an alternative term for Malayalam in foreign trade circles to denote the southwestern coast of the Indian peninsula, which also means The land of hills . The term originally referred to the western hilly land of the Chera dynasty (later Zamorins and the Kingdom of Cochin ), Kingdom of Ezhimala (later Kolathunadu ), and Ay kingdom (later Travancore ), and only later became
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