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cleaned up structure also for clarity

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{{For|others with the same name|Bhaskara (disambiguation)}}
{{Use dmy dates|date=June 2016}}
{{Use Indian English|date=June 2016}}
{{Infobox person
| name = Bhāskara I
| known_for = [[Bhaskara I's sine approximation formula]]
| birth_date = {{circa|600}} CE
| death_date = {{circa|680}} CE
| occupation = Mathematician; scientist
| nationality = [[Chalukya]]
}}
'''Bhāskara''' ({{circa|600|680}}) (commonly called '''Bhaskara I''' to avoid confusion with the 12th-century mathematician [[Bhāskara II]]) was a 7th-century mathematician and astronomer, who was the first to write [[number]]s 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.com</ref> This commentary, ''Āryabhaṭīyabhāṣya'', written in 629 CE, is among the oldest known prose works 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>

On 7 June 1979 the Indian Space Research Organisation launched [[Bhaskara (satellite)|Bhaskara I]] honouring the mathematician.<ref>[https://nssdc.gsfc.nasa.gov/nmc/spacecraft/display.action?id=1979-051A Bhaskara] NASA 16 September 2017</ref>

== Biography ==
Little is known about Bhāskara's life. He was probably an 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 in India in the 7th century.There are references to places in India in Bhaskara's writings. For example he mentions [[Vallabhi|Valabhi]] (today Vala), the capital of the Maitraka dynasty in the 7th century, and Sivarajapura, which were both in [[Saurashtra (region)|Saurastra]] which today is the Gujarat state of India on the west coast of the continent. Also mentioned are [[Bharuch]] (or Broach) in southern Gujarat and [[Thanesar]] in the eastern Punjab which was ruled by Harsa for 41 years from 606. Harsa was the pre-eminent ruler in north India through the first half of Bhaskara I's life. A reasonable guess would be that Bhaskara was born in [[Saurashtra (region)|Saurastra]] and later moved to [[Asmaka]].<ref>{{Cite web|title=Bhaskara I - Biography|url=https://mathshistory.st-andrews.ac.uk/Biographies/Bhaskara_I/|access-date=2021-05-05|website=Maths History|language=en}}</ref>

His astronomical education was given by his father. Bhaskara is considered the most important scholar of [[Aryabhata]]'s astronomical school. He and [[Brahmagupta]] are two of the most renowned Indian mathematicians who made considerable contributions to the study of fractions.
== Representation of numbers ==

Bhaskara's probably most important mathematical contribution concerns the representation of numbers in a [[positional system]]. The first positional representations had been known to Indian astronomers approximately 500 years prior to this work. However, these numbers, prior to Bhaskara, were written not 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 correspondings 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 system, however, the figures are written in descending values 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 ==

== Mathematics ==
'''Bhaskara''' wrote three astronomical contributions. In 629 he annotated 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 [[Bhaskara I's sine approximation formula|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 \pi )</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}}{{citation needed|date=March 2014}} 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).

== Astronomy ==
The Mahabhaskariya consists of eight chapters dealing with mathematical astronomy. The book deals with topics such as: the longitudes of the planets; association of the planets with each other and also with the bright stars; the lunar crescent; solar and lunar eclipses; and rising and setting of the planets.

==See also==
*[[Bhaskara I's sine approximation formula]]
*[[List of Indian mathematicians]]

== References ==
{{Reflist}}

==Sources==
(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.

==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}}

{{Indian mathematics}}

{{Authority control}}
{{DEFAULTSORT:Bhaskara 1}}
[[Category:7th-century Indian mathematicians]]
[[Category:7th-century Indian astronomers]]
[[Category:Year of birth uncertain]]
[[Category:Year of death uncertain]]
[[Category:7th-century deaths]]
[[Category:People from Parbhani district]]
[[Category:Scientists from Maharashtra]]
[[Category:Scholars from Maharashtra]]
[[Category:Acharyas]]

Bhāskara I - Revision history

MaximilianGreenb: cleaned up structure also for clarity

1.186.30.7 at 11:36, 2 July 2021