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ipedia jump to content main menu main menu move to sidebar hide navigation main page contents current events random article about wikipedia contact us contribute help learn to edit community portal recent changes upload file special pages search search appearance donate create account log in personal tools donate create account log in contents move to sidebar hide top 1 biography 2 representation of numbers 3 further contributions toggle further contributions subsection 3 1 mathematics 3 1 1 indeterminate equations kuṭṭaka and mathematical proof 3 2 astronomy 4 see also 5 notes 6 references 7 sources 8 further reading toggle the table of contents bhāskara i 26 languages العربية অসমীয়া català deutsch español euskara فارسی français galego हिन्दी հայերեն bahasa indonesia italiano ქართული मराठी नेपाली nederlands ଓଡ଼ିଆ português română русский தமிழ் ತುಳು తెలుగు oʻzbekcha ўзбекча 中文 edit links article talk english read edit view history tools tools move to sidebar hide actions read edit view history general what links here related changes upload file permanent link page information cite this page get shortened url switch to legacy parser print export download as pdf printable version in other projects wikiquote wikidata item appearance move to sidebar hide from wikipedia the free encyclopedia indian mathematician and astronomer 600 680 for others with the same name see bhaskara disambiguation not to be confused with bhāskara ii bhāskara i born c 600 ce possibly saurāṣṭra or asmaka 1 died c 680 ce possibly asmaka present day telangana and maharashtra 2 occupations mathematician scientist known for bhāskara i s sine approximation formula bhāskara i a c 600 c 680 was a 7th century indian mathematician and astronomer of the classical siddhānta school of jyotiṣa astronomy and applied mathematics one of the six vedāngas or auxillary disciplines of vedic scholarship he was the first to write numbers in the hindu arabic decimal system 3 with a circle for the zero and who gave a unique and remarkable rational approximation of the sine function jyā in his commentary on aryabhata s work 4 this commentary āryabhaṭīyabhāṣya written in 629 is among the oldest known prose works in sanskrit on mathematics and astronomy to be completely extant today he also wrote two astronomical works in the line of aryabhata s school the mahābhāskarīya great book of bhāskara and the laghubhāskarīya small book of bhāskara 4 5 on 7 june 1979 the indian space research organisation launched the bhāskara i satellite named in honour of the mathematician 6 biography edit little is known about bhāskara s life except for what can be deduced from his writings he was born in india in the 7th century and was probably an astronomer 7 bhāskara i received his astronomical education from his father there are references to places in india in bhāskara s writings such as vallabhi the capital of the maitraka dynasty in the 7th century and sivarajapura both of which are in the saurastra region of the present day state of gujarat in india also mentioned are bharuch in southern gujarat and thanesar in the eastern punjab which was ruled by harsha therefore a reasonable guess would be that bhāskara was born in saurastra and later moved to aśmaka 1 2 bhāskara i is considered the most important scholar of aryabhata s astronomical school he and brahmagupta are two of the most renowned indian mathematicians both made considerable contributions to the study of fractions representation of numbers edit the most important mathematical contribution of bhāskara i concerns the representation of numbers in a positional numeral system before bhāskara i astronomers and mathematicians of india used the bhūtasaṅkhyā numerical system of concrete word numerals for representing numbers in sanskrit literature 8 in this system specific objects or philosophical concepts were associated with specific numerals for example śaśin meaning moon represented the number 1 netra eyes or aśvin twins denoted the number 2 while bhūta as in pancha bhūta 5 elements or indirya 5 senses represented number 5 according to classical indian mathematical rule aṅkānāṁ vāmato gatiḥ numerals proceed from right to left digits are mentioned in the increasing order of tens evidencing the decimal place value system 9 in his āryabhaṭīyabhāṣya bhāskara i made combined use of poetic and technical arithmetics frequently in denoting numeral figures noting aṅkairapi in figure also it is written he recorded these numbers from left to right using nine brahmi numerals he used a small circle śūnya bindu to represent zero further contributions edit mathematics edit bhāskara i wrote three astronomical contributions in 629 he annotated the āryabhaṭīya an astronomical treatise by aryabhata written in verses bhāskara s comments referred exactly to the 33 verses dealing with mathematics in which he considered variable equations and trigonometric formulae in general he emphasized proving mathematical rules instead of simply relying on tradition or expediency 4 his work mahābhāskarīya is divided into eight chapters about mathematical astronomy in chapter 7 he gives a remarkable approximation formula for sin x sin x 16 x π x 5 π 2 4 x π x 0 x π displaystyle sin x approx frac 16x pi x 5 pi 2 4x pi x qquad 0 leq x leq pi which he assigns to aryabhata it reveals a relative error of less than 1 9 the greatest deviation 16 5 π 1 1 859 displaystyle frac 16 5 pi 1 approx 1 859 at x 0 displaystyle x 0 additionally he gives relations between sine and cosine as well as relations between the sine of an angle less than 90 and the sines of angles 90 180 180 270 and greater than 270 indeterminate equations kuṭṭaka and mathematical proof edit bhāskara s commentary on gaṇitapāda is the first known work of kuṭṭaka the pulverizer which was an algorithmic method proposed by āryabhata as a systemic approach for solving linear indeterminate equations in first degree like a x c b y displaystyle ax c by where a b displaystyle a b and c displaystyle c are integers 10 it can be effectively identified with the euclidean algorithm division and continued fractions this was utilized in indian astronomy to calculate planetary revolutions and conjunctions in cosmic cycles yugas 11 in his remarks bhāskara emphasizes the role of mathematical reasoning and demonstration upapatti affirming that rules must not be accepted solely based on traditions they must be accepted only if there is evidence or verification hetu to demonstrate the algorithm s practical use he provided several applied problems involving fractions of days adhimāsa and celestial revolutions 12 in indeterminate analysis bhāskara i focussed on linear indeterminate equations a x c b y displaystyle ax c by by providing commentary and solutions using the kuṭṭaka algorithm on āryabhata s verses quadratic indeterminate equations of the form n x 2 1 y 2 displaystyle nx 2 1 y 2 vargaprakṛti historically misattributed in western literature as pell s equations 13 were not handled by bhāskara i they were treated by his contemporary brahmagupta and later expanded by bhāskara ii 10 bhāskara clearly believed that π was irrational in support of aryabhata s approximation of π he criticized its approximation to 10 displaystyle sqrt 10 a practice common among jain mathematicians 4 2 he was the first mathematician to openly discuss quadrilaterals with four unequal nonparallel sides 14 astronomy edit the mahābhāskarīya consists of eight chapters dealing with mathematical astronomy the book deals with topics such as the longitudes of the planets the conjunctions among the planets and stars the phases of the moon solar and lunar eclipses and the rising and setting of the planets 4 parts of mahābhāskarīya were later translated into arabic see also edit bhāskara i s sine approximation formula list of astronomers list of indian mathematicians notes edit to avoid confusion with the 12th century mathematician bhāskara ii references edit 1 2 bhāskara i encyclopedia com complete dictionary of scientific biography 30 november 2022 retrieved 12 december 2022 1 2 3 o connor j j robertson e f bhāskara i biography maths history school of mathematics and statistics university of st andrews scotland uk retrieved 5 may 2021 keller agathe 2006 expounding the mathematical seed vol 1 the translation a translation of bhāskara i on the mathematical chapter of the aryabhatiya basel birkhäuser p xiii isbn 978 3 7643 7291 0 1 2 3 4 5 hayashi takao 1 july 2019 bhāskara i encyclopedia britannica retrieved 12 december 2022 keller 2006a p xiii bhāskara nasa space science data coordinated archive retrieved 16 september 2017 keller 2006a p xiii cites k s shukla 1976 p xxv xxx and pingree census of the exact sciences in sanskrit volume 4 p 297 datta bibhutibhusan singh avadhesh narayan 1935 history of hindu mathematics a source book part i numerical notation and arithmetic lahore motilal banarsidass pp 53 57 plofker kim 2009 mathematics in india princeton university press pp 73 75 isbn 978 0 691 12067 6 1 2 plofker kim 2009 mathematics in india princeton university press pp 133 140 150 153 isbn 978 0 691 12067 6 joseph george gheverghese 2011 the crest of the peacock non european roots of mathematics 3rd ed princeton university press pp 398 404 isbn 978 0 691 13526 7 keller agathe 2006 expounding the mathematical seed vol 2 the supplements basel birkhäuser pp 35 42 isbn 3 7643 7292 3 datta bibhutibhusan singh avadhesh narayan 1938 history of hindu mathematics a source book part ii algebra lahore motilal banarsidass pp 141 146 bhāskara i famous indian mathematician and astronomer cuemath 28 september 2020 retrieved 3 september 2022 sources edit from keller 2006a p xiii m c apaṭe the laghubhāskarīya with the commentary of parameśvara anandāśrama sanskrit series no 128 poona 1946 v harish mahābhāskarīya of bhāskarācārya with the bhāṣya of govindasvāmin and supercommentary siddhāntadīpikā of parameśvara madras govt oriental series no cxxx 1957 k s shukla mahābhāskarīya edited and translated into english with explanatory and critical notes and comments etc department of mathematics lucknow university 1960 k s shukla laghubhāskarīya edited and translated into english with explanatory and critical notes and comments etc department of mathematics and astronomy lucknow university 2012 k s shukla āryabhaṭīya of āryabhaṭa with the commentary of bhāskara i and someśvara indian national science academy insa new delhi 1999 further reading edit h w alten a djafari naini m folkerts h schlosser k h schlote h wußing 4000 jahre algebra springer verlag berlin heidelberg 2003 isbn 3 540 43554 9 3 2 1 s gottwald h j ilgauds k h schlote hrsg lexikon bedeutender mathematiker verlag harri thun frankfurt a m 1990 isbn 3 8171 1164 9 g ifrah the universal history of numbers john wiley sons new york 2000 isbn 0 471 39340 1 keller agathe 2006a expounding the mathematical seed vol 1 the translation a translation of bhāskara i on the mathematical chapter of the aryabhatiya basel boston and berlin birkhäuser verlag 172 pages isbn 3 7643 7291 5 keller agathe 2006b expounding the mathematical seed vol 2 the supplements a translation of bhāskara i on the mathematical chapter of the aryabhatiya basel boston and berlin birkhäuser verlag 206 pages isbn 3 7643 7292 3 o connor john j robertson edmund f bhāskara i mactutor history of mathematics archive university of st andrews v t e indian mathematics mathematicians ancient apastamba baudhayana katyayana manava pāṇini pingala yajnavalkya classical āryabhaṭa i āryabhaṭa ii bhāskara i bhāskara ii melpathur narayana bhattathiri brahmadeva brahmagupta govindasvāmi halayudha jyeṣṭhadeva kamalakara mādhava of saṅgamagrāma mahāvīra mahendra sūri munishvara narayana parameshvara achyuta pisharati jagannatha samrat nilakantha somayaji śrīpati sridhara gangesha upadhyaya varāhamihira sankara variar virasena modern srinivasa ramanujan satyendra nath bose p c mahalanobis subrahmanyan chandrasekhar c r rao veeravalli s varadarajan s r srinivasa varadhan k r parthasarathy probabilist m s narasimhan c s seshadri harish chandra subbayya sivasankaranarayana pillai tilak raj prabhakar manjul bhargava akshay venkatesh ravi vakil kannan soundararajan shanti swarup bhatnagar prize recipients in mathematical science treatises aryabhatiya bakhshali manuscript bijaganita brāhmasphuṭasiddhānta ganita kaumudi grahalaghava kanakkusaram karanapaddhati līlāvatī lokavibhaga pātīgaṇita paulisa siddhanta paitamaha siddhanta romaka siddhanta sadratnamala siddhānta shiromani śulba sūtras surya siddhanta tantrasamgraha vasishtha siddhanta veṇvāroha yuktibhāṣā yavanajataka pioneering innovations brahmi numerals hindu arabic numeral system symbol for zero 0 infinite series expansions for the trigonometric functions centres kerala school of astronomy and mathematics jantar mantar jaipur new delhi ujjain varanasi historians of mathematics bapudeva sastri 1821 1900 shankar balakrishna dikshit 1853 1898 sudhakara dvivedi 1855 1910 m rangacarya 1861 1916 p c sengupta 1876 1962 b b datta 1888 1958 t hayashi a a krishnaswamy ayyangar 1892 1953 a n singh 1901 1954 c t rajagopal 1903 1978 t a saraswati amma 1918 2000 s n sen 1918 1992 k s shukla 1918 2007 k v sarma 1919 2005 translators walter eugene clark henry thomas colebrooke david pingree other regions babylon china greece islamic mathematics europe modern institutions indian statistical institute bhaskaracharya pratishthana chennai mathematical institute institute of mathematical sciences indian institute of science harish chandra research institute homi bhabha centre for science education ramanujan institute for advanced study in mathematics tifr portals biography india mathematics astronomy stars spaceflight outer space solar system science authority control databases international isni viaf gnd fast worldcat national united states france bnf data netherlands israel academics zbmath people deutsche biographie other idref wikiquote has 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