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Encyclopedia > Bhaskara I

Bhāskara, or Bhāskara I, (c. 600 - c. 680) was a 7th century Indian mathematician, who was apparently the first to write numbers in the decimal system with a circle for the zero, and who gave a unique and remarkable rational approximation of the sine function in his commentary on Aryabhata's work. For other uses, see number 600. ... Events October 10 - Battle of Kerbela November 12 - The Sixth Ecumenical Council opens in Constantinople The Bulgars subjugate the country of current-day Bulgaria Pippin of Herstal becomes Mayor of the Palace Umayyad caliph Muawiyah I succeeded by Yazid I ibn Muawiyah Erwig deposes Wamba to become king of the... // Overview Events The Roman-Persian Wars end. ... This article is in need of attention from an expert on the subject. ... The decimal (base ten or occasionally denary) numeral system has ten as its base. ... 0 (zero), alternatively called naught, nil, nada, ought, zilch, zip, nothing or nought, is both a number and a numeral. ... An approximation is an inexact representation of something that is still close enough to be useful. ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... Aryabhata (आर्यभट) Ä€ryabhaá¹­a) (476 - 550) is the first of the great astronomers of the classical age of India. ...

Contents


Biography

We know little about Bhāskaras life. Presumably he was born near Saurashtra in Gujarat and died in Ashmaka. His astronomical education was given by his father. Bhaskara is considered as the most important scholar of the Aryabhata's astronomical school. Saurashtra in between Gulf of Kutch and Gulf of Khambat. ... Gujarat (Gu: , Hi: ; , IPA ; also spelled Gujrat and sometimes (incorrectly) Gujarath) contained many of the former Princely states of India, and is the second-most industrialized state in the Republic of India after Maharashtra. ... Ashmaka is also known as Ashmakadesa, though not completely identified but is supposed to be in Kerala, South India. ... Aryabhata (आर्यभट) Ä€ryabhaá¹­a) (476 - 550) is the first of the great astronomers of the classical age of India. ...


Representation of numbers

Bhaskara's probably most important mathematical contribution concerns the representation of numbers in a positional system. The first positional representations were known to Indian astronomers about 500. However, the numbers were not written in figures, but in words or allegories, and were organized in verses. For instance, the number 1 was given as moon, since it exists only once; the number 2 was represented by wings, twins, or eyes, since they always occur in pairs; the number 5 was given by the (5) senses. Similar to our current decimal system, these words were aligned such that each number assigns the factor of the power of ten corresponding to its position, only in reverse order: the higher powers were right from the lower ones. For example, Positional notation is a system in which each position has a value represented by a unique symbol or character. ... Events Possible date for the Battle of Mons Badonicus: Romano-British and Celts defeat an Anglo-Saxon army that may have been led by the bretwalda Aelle of Sussex (approximate date; suggested dates range from 490 to 510) Note: This battle may have influenced the legend of King Arthur. ... The decimal (base ten or occasionally denary) numeral system has ten as its base. ...

1052 = wings senses void moon.

Why did the Indian scientists used words instead of the already known Brahmi numerals? The texts were written in Sanskrit, the "language of the gods", which played a similar role as Latin in Europe, the spoken languages were quite different dialects. Presumably, the Brahmi numerals which were used in every-day life were regarded as too vulgar for the gods (Ifrah 2000, p. 431). The Brahmi numerals are an indigenous Indian numeral system attested from the 3rd century BCE (somewhat later in the case of most of the tens). ... Sanskrit ( संस्कृतम्) is an Indo-European classical language of India and a liturgical language of Hinduism, Buddhism, and Jainism. ... Latin is an ancient Indo-European language originally spoken in the region around Rome called Latium. ...


About 510, Aryabhata used a different method ("Aryabhata cipher") assigning syllables to the numbers. His number system has the basis 100, and not 10 (Ifrah 2000, p. 449). In his commentary to Aryabhata's Aryabhatiya in 629, Bhaskara modified this system to a true positional system with the base 10, containing a zero. He used properly defined words for the numbers, began with the ones, then writes the tens, etc. For instance, he wrote the number 4,320,000 as Events Anicius Manlius Severinus Boëthius is appointed a consul by Theoderic Births Gildas, Celtic monk Deaths Hashim, great-grandfather of Muhammad and ancestor of the Hashemites Categories: 510 ... Aryabhata (आर्यभट) Ä€ryabhaá¹­a) (476 - 550) is the first of the great astronomers of the classical age of India. ... Aryabhata cipher is a code to represent numbers by words. ... Aryabhata (आर्यभट) Ä€ryabhaá¹­a) (476 - 550) is the first of the great astronomers of the classical age of India. ... Events Jerusalem reconquered by Byzantine Empire from the Persian Empire (September). ... Positional notation is a system in which each position has a value represented by a unique symbol or character. ... 0 (zero), alternatively called naught, nil, nada, ought, zilch, zip, nothing or nought, is both a number and a numeral. ...

viyat ambara akasha sunya yama rama veda
sky atmosphere ether void primordial couple (Yama & Yami) Rama Veda
0 0 0 0 2 3 4

His system is truely positional, since the same words representing, e.g., the number 4 (like veda), can also be used to represent the values 40 or 400 (van der Waerden 1966, p. 90). Quite remarkably, he often explains a number given in this system, using the formula ankair api ("in figures this reads"), by repeating it written with the first nine Brahmi numerals, using a small circle for the zero (Ifrah 2000, p. 415). Contrary to his word number system, however, the figures are written in descending valuedness from left to right, exactly as we do it today. Therefore, at least since 629 the decimal system is definitely known to the Indian scientists. Presumably, Bhaskara did not invent it, but he was the first having no compunctions to use the Brahmi numerals in a scientific contribution in Sanskrit. Tibetan Dharmapala at the Field Museum in Chicago, Illinois Yama (Sanskrit: यम) is the lord of death, whose first recorded appearance is in the Vedas. ... In Vedic beliefs, Yami is the first woman, along with her twin brother, Yama. ... Lord Rama (center) with wife Sita, brother Lakshmana and devotee Hanuman. ... The Vedas are part of the Hindu Shruti; these religious scriptures form part of the core of the Brahminical and Vedic traditions within Hinduism and are the inspirational, metaphysical and mythological foundation for later Vedanta, Yoga, Tantra and even Bhakti forms of Hinduism. ... The Brahmi numerals are an indigenous Indian numeral system attested from the 3rd century BCE (somewhat later in the case of most of the tens). ... 0 (zero), alternatively called naught, nil, nada, ought, zilch, zip, nothing or nought, is both a number and a numeral. ... Events Jerusalem reconquered by Byzantine Empire from the Persian Empire (September). ... The decimal (base ten or occasionally denary) numeral system has ten as its base. ... The Brahmi numerals are an indigenous Indian numeral system attested from the 3rd century BCE (somewhat later in the case of most of the tens). ... Sanskrit ( संस्कृतम्) is an Indo-European classical language of India and a liturgical language of Hinduism, Buddhism, and Jainism. ...


The first, however, to compute with the zero as a number and to know negative numbers, was Bhaskara's contemporary Brahmagupta. 0 (zero), alternatively called naught, nil, nada, ought, zilch, zip, nothing or nought, is both a number and a numeral. ... Brahmagupta (ब्रह्मगुप्त) (598_668) was an Indian mathematician and astronomer. ...


Further contributions

Bhaskara wrote three astronomical contributions. In 629 he commented the Aryabhatiya, written in verses, about mathematical astronomy. The comments referred exactly to the 33 verses dealing with mathematics. There he considered variable equations and trigonometric formulas. Events Jerusalem reconquered by Byzantine Empire from the Persian Empire (September). ...


His work Mahabhaskariya divides into eight chapters about mathematical astronomy. In chapter 7, he gives a remarkable approximation formula for sinx, that is

sin x approx frac{16x (pi - x)}{5 pi^2 - 4x (pi - x)}, qquad (0 leq x leq frac{pi}{2} )

which he assigns to Aryabhata. It reveals a relative error of less than 1.9% (the greatest deviation frac{16}{5pi} - 1 approx 1,859% at x = 0). Moreover, relations between sine and cosine, as well as between the sine of an angle >90^circ, >180^circ or >270^circ to the sine of an angle <90^circ are given. Parts of Mahabhaskariya were later translated into Arabic. Aryabhata (आर्यभट) Āryabhaṭa) (476 - 550) is the first of the great astronomers of the classical age of India. ... Arabic can mean: From or related to Arabia From or related to the Arabs The Arabic language; see also Arabic grammar The Arabic alphabet, used for expressing the languages of Arabic, Persian, Malay ( Jawi), Kurdish, Panjabi, Pashto, Sindhi and Urdu, among others. ...


Bhaskara already dealt with the assertion: If p is a prime number, then 1 + (p − 1)! is divisible by p. It was proved later by Al-Haitham, also mentioned by Fibonacci, and is now known as Wilson's theorem. Alhazen Abu Ali al-Hasan Ibn Al-Haitham (also: Ibn al Haythen), (965-1040), was an Persian mathematician; he is sometimes called al-Basri, after his birthplace. ... Portrait of Fibonacci, probably not authentic Leonardo of Pisa or Leonardo Pisano (Pisa, c. ... In mathematics, Wilsons Theorem states that for a prime number p, (see factorial and modular arithmetic for the notation). ...


Moreover, Bhaskara stated theorems about the solutions of today so called Pell equations. For instance, he posed the problem: "Tell me, O mathematician, what is that square which multiplied by 8 becomes - together with unity - a square?" In modern notation, he asked for the solutions of the Pell equation 8x2 + 1 = y2. It has the simple solution x = 1, y = 3, or shortly (x,y) = (1,3), from which further solutions can be constructed, e.g., (x,y) = (6,17). Pells equation is any Diophantine equation of the form where n is a nonsquare integer. ... Pells equation is any Diophantine equation of the form where n is a nonsquare integer. ...


See also

Bhāskara (1114-1185), also called Bhāskara II and Bhāskarācārya (Bhaskara the teacher) was an Indian mathematician. ...

Links

BITCH!111 ...

References

  • 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]
  • B. van der Waerden: Erwachende Wissenschaft. Ägyptische, babylonische und griechische Mathematik. Birkäuser-Verlag Basel Stuttgart 1966

  Results from FactBites:
 
Bhaskaracharya (229 words)
Bhaskara was born in 1114 A.D. according to a statement he recorded in one of his own works.
Bhaskara wrote his famous Siddhanta Siroman in the year 1150 A.D. It is divided into four parts; Lilavati (arithmetic), Bijaganita (algebra), Goladhyaya (celestial globe), and Grahaganita (mathematics of the planets).
Much of Bhaskara's work in the Lilavati and Bijaganita was derived from earlier mathematicians; hence it is not surprising that Bhaskara is best in dealing with indeterminate analysis.
  More results at FactBites »

 
 

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