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Bhaskara Ii: more detail

Bhaskara II
People From Bijapur: Bhaskara Ii, Basava, Aluru Venkata Rao, Basappa Danappa Jatti, Venkanna H. Naik, Ramesh Jigajinagi, Sunil Kumar Desai
Expounding the Mathematical Seed. Vol. 2: The Supplements: A Translation of Bhaskara I on the Mathematical Chapter of the Aryabhatiya (Science Networks. Historical Studies) by Agathe Keller, 2006-04-28
1114 Births: Bhaskara Ii, Fujiwara No Shunzei, Henry, Earl of Northumbria, Mor O'toole, Queen of Leinster, Al-Suhayli
Indian Astronomers: Subrahmanyan Chandrasekhar, Aryabhata, Bhaskara Ii, Nilakantha Somayaji, Venkatraman Radhakrishnan, Jyesthadeva, Vasistha
1185 Deaths: Andronikos I Komnenos, Afonso I of Portugal, Pope Lucius Iii, Baldwin Iv of Jerusalem, Bhaskara Ii, Ibn Tufail, Emperor Antoku
12th-Century Mathematicians: Omar Khayyám, Adelard of Bath, Abraham Bar Hiyya, Al-Khazini, Bhaskara Ii, Sharaf Al-Din Al-usi
Indian Mathematics: Swami Bharati Krishna Tirtha's Vedic Mathematics, Bhaskara Ii, Tantrasamgraha, Shulba Sutras, Aryabhata's Sine Table
12th-Century Scientists: 12th-Century Mathematicians, Omar Khayyám, Adelard of Bath, Abraham Bar Hiyya, Al-Khazini, Bhaskara Ii
Bhaskara I and His Works: Part II: Maha-Bhaskariya by Kripa Shankar Shukla (Ed. & Trs. ), 1960-01-01

lists with details

Bhaskara II -- Encyclopædia Britannica
bhaskara ii was the lineal successor of the noted Indian mathematician Brahmagupta MLA style " bhaskara ii." Encyclopædia Britannica. 2004. Encyclopædia Britannica Premium Service
http://www.britannica.com/eb/article?eu=81187
História Da Matemática Na Índia - Bhaskara II
bhaskara ii. Bhaskara nasceu em 1114, na Índia, numa terra chamada Vijalavida (da qual se desconhece a astrónomo e o seu professor. Bhaskara escreveu o. Siddhanta Siromani, aos 36
http://www.malhatlantica.pt/mathis/India/BhaskaraII.htm

Extractions: O seu livro foi usado em toda a Índia, tendo substituído maior parte dos textos que eram utilizados até então, como o do astrónomo indiano Lalla (720 - 790), mas só saiu as fronteiras da Índia no século XVI. Nessa altura foi traduzido para persa por Faizi (1587). Foi este tradutor que introduziu a história de que Lilavati era o nome da filha de Bhaskara. De acordo com essa história, a partir do seu horóscopo, Bhaskara tinha previsto o dia e a hora propícia para o casamento da sua filha. Para saber a hora exacta tinha construído um relógio, colocando um copo com um pequeno orifício, por onde entrava água, numa vasilha cheia de água. De tal forma que ao início da hora exacta do casamento o copo afundar-se-ia. Quando tudo estava pronto, Lilavati, cheia de curiosidade, inclinou-se sobre a vasilha e uma perola do seu vestido caiu no copo e bloqueou o orifício. A hora do casamento passou sem que o copo se afundasse. Lilavati nunca se casou. Para consolar a sua filha Bhaskara prometeu escrever-lhe um livro de matemática!
Bhaskara - Wikipedija
bhaskara ii., imenovan Aarja ( sanskrtsko uitelj, ueni), indijski matematik in astronom, * 1114 Bhaskara je deloval v Udainu in bil med drugim vodja tamkajnjega znamenitega
http://sl.wikipedia.org/wiki/Bhaskara

Extractions: Bhaskara II. , imenovan AÄarja sanskrtsko uÄitelj, uÄeni), indijski matematik in astronom Biddur Indija , verjetno UdÅ¾ain Bhaskara je deloval v UdÅ¾ainu in bil med drugim vodja tamkajÅ¡njega znamenitega observatorija , kjer sta delovala pred njim Varahamihira in Brahmagupta . Med Å¡tevilnimi izjemnimi indijskimi matematiki je bil najbrÅ¾ najpomembnejÅ¡i in najvplivnejÅ¡i. Pri njem lahko najdemo prvo sploÅ¡no reÅ¡itev nedoloÄenih enaÄb prve stopnje: Pri tem delamo napako, da imenujemo linearne nedoloÄene enaÄbe diofantske enaÄbe . Medtem ko je Diofant Å¡e dopuÅ¡Äal reÅ¡itev z ulomki , so bili indijski matematiki zadovoljni samo s celoÅ¡tevilskimi reÅ¡itvami. Dalj kot Diofant so Å¡li tudi v tem, ker so priznavali negativna Å¡tevila za korene enaÄb, Äeprav je bilo to brÅ¾kone v navadi Å¾e prej v babilonski astronomiji . Bhaskara je na primer dobil za reÅ¡itev enaÄbe: korena x = 50 in tudi x = 5, ter za reÅ¡itev enaÄbe: korena x = 50 in x = - 5, je pa dvomil o negativnem korenu x . IzboljÅ¡al je metodo iskanja celoÅ¡tevilÄnih reÅ¡itev enaÄbe: Vedel je, da ima druga grÅ¡ka kanonska oblika
Bhaskara_II
Bhaskara is also known as bhaskara ii or as Bhaskaracharya, this lattername meaning Bhaskara the Teacher . Since he is known in
http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Bhaskara_II.html

Extractions: Bhaskara is also known as Bhaskara II or as Bhaskaracharya, this latter name meaning "Bhaskara the Teacher". Since he is known in India as Bhaskaracharya we will refer to him throughout this article by that name. Bhaskaracharya's father was a Brahman named Mahesvara. Mahesvara himself was famed as an astrologer. This happened frequently in Indian society with generations of a family being excellent mathematicians and often acting as teachers to other family members. Bhaskaracharya became head of the astronomical observatory at Ujjain, the leading mathematical centre in India at that time. Outstanding mathematicians such as Varahamihira and Brahmagupta had worked there and built up a strong school of mathematical astronomy. In many ways Bhaskaracharya represents the peak of mathematical knowledge in the 12th century. He reached an understanding of the number systems and solving equations which was not to be achieved in Europe for several centuries. Six works by Bhaskaracharya are known but a seventh work, which is claimed to be by him, is thought by many historians to be a late forgery. The six works are:
Pell's Equation
Next bhaskara ii knows that there are infinitely many m such that am k = b2. bhaskara ii knows (almost certainly by experience rather than by having a proof) that the
http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Pell.html

Extractions: We will discuss below whether Pell 's equation is properly named. By this we mean simply: did Pell contribute at all to the study of Pell 's equation? There is no doubt that the equation had been studied in depth for hundreds of years before Pell was born. In fact the first contribution by Brahmagupta was made around 1000 years before Pell 's time and it is with Brahmagupta 's contribution that we begin our historical study. First let us say what Pell 's equation is. We are talking about the indeterminate quadratic equation nx y which we can also write as y nx where n is a given integer and we are looking for integer solutions ( x y Now, although it is fair to say that Brahmagupta was the first to study this equation, it is equally possible to see that earlier authors had studied problems related to Pell 's equation. To mention some briefly: Diophantus examines problems related to Pell 's equation and we can reduce Archimedes ' "cattle problem" to solving Pell 's equation although there is no evidence that Archimedes made this connection.
Pell's Equation
The next step forward was taken by bhaskara ii in 1150. bhaskara ii now appliesthe method of composition to the pairs (a, b) and (1, m) to get
http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Pell.html

Extractions: We will discuss below whether Pell 's equation is properly named. By this we mean simply: did Pell contribute at all to the study of Pell 's equation? There is no doubt that the equation had been studied in depth for hundreds of years before Pell was born. In fact the first contribution by Brahmagupta was made around 1000 years before Pell 's time and it is with Brahmagupta 's contribution that we begin our historical study. First let us say what Pell 's equation is. We are talking about the indeterminate quadratic equation nx y which we can also write as y nx where n is a given integer and we are looking for integer solutions ( x y Now, although it is fair to say that Brahmagupta was the first to study this equation, it is equally possible to see that earlier authors had studied problems related to Pell 's equation. To mention some briefly: Diophantus examines problems related to Pell 's equation and we can reduce Archimedes ' "cattle problem" to solving Pell 's equation although there is no evidence that Archimedes made this connection.
Bhaskara II -- Britannica Student Encyclopedia
bhaskara ii Britannica Student Encyclopedia. (111485?), Indianmathematician. bhaskara ii was born in 1114 in Biddur, India. He
http://www.britannica.com/ebi/article?eu=342278&query=decimal number system&ct=e
Bhaskara
Biography of Bhaskara (11141185) Bhaskara is also known as bhaskara ii or as Bhaskaracharya, this latter name meaning "Bhaskara the Teacher". Since www-history.mcs.st-andrews.ac.uk/ Mathematicians/Bhaskara_II.html
http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Bhaskara.html
Swaveda - Background - Mathematics In Ancient India - Introduction
1150 AD Bijaganita by bhaskara ii. 1150 AD - Karana-Kutuhala by bhaskara ii. 1150 AD - Lilavati by bhaskara ii.
http://www.theinquirer.net/clickthru.aspx?id=566
Winners
of equations was first solved by Brahmaguptaand bhaskara ii. The principle is basically the same as his "Chakravala" method. Later, bhaskara ii ( AD 12th century) simplified this
http://www.angelfire.com/ak/ashoksandhya/winners2.html

Extractions: the remaining people ate for $41 each. Basically, you want me to solve the equation in integers . Look at that equation. 12n is divisible by 3, 9 is divisible 3, so 41m also must be divisible by 3, which means m is divisible by 3. Putting m = 3k, we get a simpler equation Such equations are called indeterminate or Diophantine equations and there are many methods to solve them. I give two of them. Correct Answer was sent by : Balaji R, CA, USA I am using the symbol == to denote "is congruent to" as there is no symbol for congruence in standard ASCII set. So, 41k == 3 (mod 4) 4n == 3 (mod 41) Since 41 and 4 are mutually prime, this is soluble by considering the complete residue system. Thus, by putting k = 0, 1, 2, 3, we get 41k is congruent to 0, 1, 2, 3 respectively (mod 4).
História Da Matemática Na Índia - Bhaskara II - Teorema De Pitágoras
Translate this page História da Matemática na Índia. Lilavati de bhaskara ii. Medições.Medidas de lados (Teorema de Pitágoras). Verso 144 Descobre
http://www.malhatlantica.pt/mathis/India/BhaskaraII-7.htm

Extractions: Se um bambu medindo 32 cúbitos e estando em pé, se partisse, num local, por acção do vento, e a sua extremidade encontrasse o chão a 16 cúbitos da base do bambu. Diz, matemático, a quantos cúbitos da raiz é que ele se partiu? (citado por Shen Kangshen et al. Nota: Para conhecer a história deste problema consulte a página Problemas Pitagóricos Verso 158 O buraco de uma cobra está na base de um pilar que tem 9 cúbitos de altura. Um pavão está empoleirado no seu cume. Vendo uma cobra, a uma distância igual ao triplo da altura do pilar, a deslizar para o seu buraco, precipita-se obliquamente sobre a cobra. Diz depressa a quantos cúbitos do buraco da cobra é que eles se encontram, ambos
9 II. Mathematicians Of Kerala
He was strongly influenced by the work of bhaskara ii, which proves work from theclassic period was known to Keralese mathematicians and was thus influential
http://www-history.mcs.st-andrews.ac.uk/history/Projects/Pearce/Chapters/Ch9_2.h

Extractions: (9 III. Madhava of Sangamagramma) Narayana Pandit (c. 1340-1400), the earliest of the notable Keralese mathematicians, is known to have definitely written two works, an arithmetical treatise called Ganita Kaumudi and an algebraic treatise called Bijganita Vatamsa . He was strongly influenced by the work of Bhaskara II, which proves work from the classic period was known to Keralese mathematicians and was thus influential in the continued progress of the subject. Due to this influence Narayana is also thought to be the author of an elaborate commentary of Bhaskara II 's Lilavati , titled Karmapradipika (or Karma-Paddhati ). It has been suggested that this work was written in conjunction with another scholar, Sankara Variyar , while others attribute the work to Madhava (see later). Although the Karmapradipika contains very little original work, seven different methods for squaring numbers are found within it, a contribution that is wholly original to the author. Narayana 's other major works contain a variety of mathematical developments, including a rule to calculate approximate values of square roots, using the second order indeterminate equation
Military Implications Of Indias Space Program
In November 1981, bhaskara ii, Indias second earth observation satellite, was launched from a Unlike its predecessor, the bhaskara ii experienced no problems with its cameras
http://www.airpower.maxwell.af.mil/airchronicles/aureview/1983/May-Jun/frederick

Extractions: Development Programs Dr. Abdul Kalam, head of launch vehicle development in the ISRO, has declared that by 1990 India will be able to position a 2500 kg communication satellite into geosynchronous orbit at 36,000 km. He has further asserted that the ISRO can produce a cryogenic rocket engine (using liquid oxygen and liquid hydrogen) during the l980s. The Indians consider cryogenic engines more cost effective than rockets employing solid or storable liquid fuel because of greater thrust generation and the possibility of reduced vehicle size. Military Implications of Space Research The Indian leadership has emphasized that advances in rocket and satellite technology will not be translated into an enhanced military capability. Thus, for example, the Minister of State in the Ministry of Defense apprised Parliament after the launching of Rohini I that no plans existed for the manufacture of IRBMs. Despite such pronouncements potential military applications of Indian space technology are manifold, to include development of reconnaissance satellites, improvements in command and control, greater precision in operational planning based on satellite-derived meteorological data, and IRBM production.
8 V. Bhaskaracharya II
Bhaskaracharya, or bhaskara ii, is regarded almost without question as the greatestHindu mathematician of all time and his contribution to not just Indian
http://www-history.mcs.st-andrews.ac.uk/history/Projects/Pearce/Chapters/Ch8_5.h

Extractions: (8 VI. Pell's equation) Bhaskaracharya , or Bhaskara II, is regarded almost without question as the greatest Hindu mathematician of all time and his contribution to not just Indian, but world mathematics is undeniable. As L Gurjar states: ...Because of his work India gave a definite 'quota' to the forward world march of the science. [LG, P 104] Born in 1114 AD (in Vijayapura, he belonged to Bijjada Bida) he became head of the Ujjain school of mathematical astronomy ( Varahamihira and Brahmagupta had helped to found this school or at least 'build it up'). There is some confusion amongst the texts I have referred to as to the works that he wrote. C Srinivasiengar claims he wrote Siddhanta Siromani in 1150 AD, which contained four sections: Lilavati (arithmetic)
The Date Of Mahabharata Based On The Indian Astronomical Works
of (Maha)bharat (battle)". Therefore, Bhaskara and Someswara had known the Siddhanta of Aryabhata II (i.5), Siddhantasekhara (i.10), Siddhanta-siromani of bhaskara ii (I.i.15
http://www.hindunet.org/saraswati/colloquium/astronomy01.htm

Extractions: Mahabharata as the sheet-anchor of bharatiya itihasa International Colloquium The Date of Mahabharata Based on the Indian Astronomical Works K.V. Ramakrishna Rao, B.Sc., M.A., A.M.I.E., C.Eng.(I)., B.L., Introduction The date of Mahabharat is analyzed for determination only based on the Indian astronomical works. The following facts are taken into consideration for such critical study: The Indian astronomers of Siddhantic works and followers have recorded the date of Bharata implying Mahabharat war in particular and starting of Kaliyuga or Era, that is used to reckon the dates of themselves at many places and in conjunction with Saka era in some places later. Aryabhata makes a specific mention about Bharata in his Aryabhatiyam. Most of the scholars including westerners have taken the connotation of it as referring to Mahabharat and in particular Mahabharat war, because, that is considered as the staring point of Kaliyuga / era in Indian astronomy and history too. Therefore, taking the astronomical works - Siddhantas, Tantras and Karanas like - Aryabhatiyam, Mahabhaskariyam, Vatesvara - Siddhanta
Swaveda - Background - Mathematics In Ancient India - Introduction
1150 AD Bijaganita by bhaskara ii. 1150 AD - Karana-Kutuhala by bhaskara ii. 1150AD - Lilavati by bhaskara ii. 1150 AD - Siddhanta-Siromani by bhaskara ii.
http://www.swaveda.com/background.php?category=science&title=Mathematics in Anci
Kuttaka
23 + 137m, for m an integer. bhaskara ii later made a modificationto Aryabhata s Kuttaka. Go back to the Indian History timeline.
http://www.math.sfu.ca/histmath/India/5thCenturyAD/Kuttaka.html

Extractions: Aryabhata noted that any factor common to a and b should be a factor of c, otherwise the equation has no solution. Dividing a, b and c by the greatest common factor of (a,b) we can reduce the equation to the form where a and b are relatively prime. In the discussion that follows we can assume that (a,b)=1 and without loss of generality c>0.
Bhaskara
Translate this page inspiré. On parle parfois de Bhaskara I pour évoquer ce mathématicienhomonyme, bhaskara ii désignant celui qui nous intéresse ici.
http://www.sciences-en-ligne.com/momo/chronomath/chrono1/Bhaskara.html
Gacetilla Matematica
Translate this page Este problema muy posiblemente haya pasado de La China a la India y en efecto esasí pues bhaskara ii (llamado también Bhaskaracharya ) en su Lilavati expone
http://www.arrakis.es/~mcj/
Bhaskara - Encyclopedia Article About Bhaskara. Free Access, No Registration Nee
See also. Bhaskara I. External link. bhaskara ii MacTutor History of Mathematicsarchive. preview not available. Click the link for more information.
http://encyclopedia.thefreedictionary.com/Bhaskara

Extractions: Dictionaries: General Computing Medical Legal Encyclopedia Word: Word Starts with Ends with Definition (1114-1185), also called and ("Bhaskara the teacher") was an Indian The Republic of India , located in South Asia and comprising most of the Indian subcontinent is the second most populous country in the world and is the world's largest democracy, with over one billion people speaking more than one hundred distinct languages. The Indian economy is the fourth-largest in the world, in terms of purchasing power parity. India borders Bangladesh, Myanmar, China, Bhutan, Nepal and Pakistan, with Sri Lanka and the Maldives just across the Indian mainland in the Indian Ocean. Click the link for more information. mathematician A mathematician is a person whose area of study and research is mathematics. Mathematicians not only study, but also research, and this must be given prominent mention here, because a misconception that everything in mathematics is already known is widespread among persons not learned in that field. In fact, the publication of new discoveries in mathematics continues at an immense rate in hundreds of scientific journals, many of them devoted to mathematics and many devoted to subjects to which mathematics is applied (such as theoretical computer science, physics or quantum mechanics).

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