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			{{For\|others with the same name\|Bhaskara (disambiguation)}}
			'''Bhāskara''' (c. 600 – c. 680) ({{lang-mr\|भास्कर}} commonly called '''Bhaskara I''' to avoid confusion with the 12th century mathematician [[Bhāskara II]]) was a 7th-century [[India]]n [[mathematician]], who was apparently the first to write numbers in the [[Hindu–Arabic numeral system\|Hindu]] [[decimal\|decimal system]] with a circle for the [[0 (number)\|zero]], and who gave a unique and remarkable rational [[approximation]] of the [[sine]] function in his commentary on [[Aryabhata]]'s work.<ref>[http://www.britannica.com/EBchecked/topic/853503/Bhaskara-I Bhaskara I], Britannica</ref> This commentary, ''Āryabhaṭīyabhāṣya'', written in 629 CE, is the oldest known prose work in Sanskrit on mathematics and astronomy. He also wrote two astronomical works in the line of Aryabhata's school, the Mahābhāskarīya and the Laghubhāskarīya.<ref>{{Harvtxt\|Keller\|2006\|p=xiii}}</ref>

			== Biography ==
			Little is known about Bhāskara's life. He was "probably a Marathi astronomer".<ref>{{Harvtxt\|Keller\|2006\|p=xiii}} cites [K S Shukla 1976; p. xxv-xxx], and [[David Pingree\|Pingree]], ''Census of the Exact Sciences in Sanskrit'', volume 4, p. 297.</ref>
			He was born at Bori, in Parbhani district of Maharashtra state in India in 7th century.

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			== Representation of numbers ==

			Chandua probably the most important mathematical contribution concerns the representation of numbers in a [[positional system]]. The first positional representations were known to Indian astronomers about 500 years ago. 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.

			His system is truly positional, since the same words representing, can also be used to represent the values 40 or 400.<ref>B. van der Waerden: ''Erwachende Wissenschaft. Ägyptische, babylonische und griechische Mathematik''. Birkäuser-Verlag Basel Stuttgart 1966 p. 90</ref> 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 numeral]]s, using a small circle for the [[0 (number)\|zero]] . 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 numeral]]s in a scientific contribution in [[Sanskrit]].

			== 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 formulae.

			His work ''Mahabhaskariya'' divides into eight chapters about mathematical astronomy. In chapter 7, he gives a remarkable approximation formula for sin x, that is
			:<math> \sin x \approx \frac{16x (\pi - x)}{5 \pi^2 - 4x (\pi - x)}, \qquad (0 \leq x \leq \frac{\pi}{2} )</math>
			which he assigns to [[Aryabhata]]. It reveals a relative error of less than 1.9% (the greatest deviation <math>\frac{16}{5\pi} - 1 \approx 1.859\%</math> at <math>x=0</math>). Moreover, relations between sine and cosine, as well as between the sine of an angle >90° >180° or >270° to the sine of an angle <90° are given.
			Parts of ''Mahabhaskariya'' were later translated into [[Arabic]].

			Bhaskara already dealt with the assertion that if ''p'' is a prime number, then 1 + (''p''–1)! is divisible by ''p''.{{Dubious\|date=May 2010}} It was proved later by [[Alhazen\|Al-Haitham]], also mentioned by [[Fibonacci]], and is now known as [[Wilson's theorem]].

			Moreover, Bhaskara stated theorems about the solutions of today so called [[Pell's equation\|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's equation\|Pell equation]] <math>8x^2 + 1 = y^2</math>. 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).

			==See also==

			[[Bhaskara I's sine approximation formula]]

			== References ==
			{{Reflist}}<!--added under references heading by script-assisted edit-->

			==Further reading==
			* 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]
			*{{Citation
			\| last=Keller
			\| first=Agathe
			\| year=2006
			\| title=Expounding the Mathematical Seed. Vol. 1: The Translation: A Translation of Bhaskara I on the Mathematical Chapter of the Aryabhatiya
			\| publisher=Basel, Boston, and Berlin: Birkhäuser Verlag, 172 pages
			\| isbn=3-7643-7291-5
			}}.
			*{{Citation
			\| last=Keller
			\| first=Agathe
			\| year=2006
			\| title=Expounding the Mathematical Seed. Vol. 2: The Supplements: A Translation of Bhaskara I on the Mathematical Chapter of the Aryabhatiya
			\| publisher=Basel, Boston, and Berlin: Birkhäuser Verlag, 206 pages
			\| isbn=3-7643-7292-3
			}}.
			* {{MacTutor Biography\|id=Bhaskara_I}}

			==Source texts==
			(From {{Harvtxt\|Keller\|2006}})
			* 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
			.

			{{Indian mathematics}}

			{{Persondata <!-- Metadata: see [[Wikipedia:Persondata]]. -->
			\| NAME = Bhaskara 1
			\| ALTERNATIVE NAMES =
			\| SHORT DESCRIPTION = Indian mathematician
			\| DATE OF BIRTH =08-04-1999
			\| PLACE OF BIRTH =lucknow
			\| DATE OF DEATH =
			\| PLACE OF DEATH =
			}}
			{{DEFAULTSORT:Bhaskara 1}}
			[[Category:Medieval Indian mathematicians]]
			[[Category:7th-century mathematicians]]
			[[Category:Medieval Indian astronomers]]
			[[Category:Year of birth uncertain]]
			[[Category:Year of death uncertain]]
			[[Category:7th-century deaths]]
			[[Category:7th-century Indian people]]

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2011-07-12T18:50:10Z

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