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{{Short description|Indian mathematician and astronomer (600–680)}}
{{For|others with the same name|Bhaskara (disambiguation)}}
{{Distinguish|Bhāskara II}}
{{Use dmy dates|date=June 2016}}
{{Use Indian English |date=June 2016}}
{{Infobox person
| name = Bhāskara I
| known for = [[Bhāskara I's sine approximation formula]]
| birth_date = {{circa|600}} CE
| birth_place = possibly [[Saurashtra (region)|Saurāṣṭra]] or [[Asmaka]]<ref name=":1" />
| death_date = {{circa|680}} CE
| death_place = possibly [[Asmaka]] (present-day [[Telangana]] and [[Maharashtra]])<ref name=":2" />
| occupation = Mathematician, scientist
}}
'''Bhāskara I{{Efn|to avoid confusion with the 12th century mathematician [[Bhāskara II]]}}''' ({{circa|600|680}}) was a 7th-century Indian mathematician and [[astronomer]] who was the first to write [[number]]s in the [[Hindu–Arabic numeral system|Hindu–Arabic 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 name=":0">{{Cite web |last=Hayashi |first=Takao |date=1 July 2019 |title=Bhāskara I |url=https://www.britannica.com/biography/Bhaskara-I |access-date=2022-12-12 |website=Encyclopedia Britannica |language=en}}</ref> 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").<ref name=":0" /><ref>{{Harvtxt|Keller|2006a|p=xiii}}</ref>

On 7 June 1979, the [[ISRO|Indian Space Research Organisation]] launched the [[Bhaskara (satellite)|Bhāskara I satellite]], named in honour of the mathematician.<ref>{{Cite web |title=Bhāskara |url=https://nssdc.gsfc.nasa.gov/nmc/spacecraft/display.action?id=1979-051A |access-date=16 September 2017 |website=Nasa Space Science Data Coordinated Archive}}</ref>

== Biography ==
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]].<ref>{{Harvtxt|Keller|2006a|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> Bhāskara I received his [[Astronomy|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 [[Saurashtra (region)|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 [[Saurashtra (region)|Saurastra]] and later moved to [[Aśmaka]].<ref name=":1">{{Cite web |date=30 November 2022 |title=Bhāskara I |url=https://www.encyclopedia.com/science/dictionaries-thesauruses-pictures-and-press-releases/bhaskara-i |access-date=2022-12-12 |website=Encyclopedia.com |series=Complete Dictionary of Scientific Biography}}</ref><ref name=":2">{{Cite web |last=O'Connor |first=J. J. |last2=Robertson |first2=E. F. |orig-date=November 2000 |title=Bhāskara I – Biography |url=https://mathshistory.st-andrews.ac.uk/Biographies/Bhaskara_I/ |access-date=2021-05-05 |website=Maths History |publisher=School of Mathematics and Statistics, University of St Andrews, Scotland, UK |language=en}}</ref>

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 ==
The most important mathematical contribution of Bhāskara I concerns the representation of numbers in a [[positional numeral system]]. The first positional representations had been known to Indian astronomers approximately 500 years before Bhāskara's work. However, these numbers 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 corresponding to its position, only in reverse order: the higher powers were to the right of the lower ones.

Bhāskara's numeral system was truly positional, in contrast to word representations, where the same word could represent multiple values (such as 40 or 400).<ref>B. van der Waerden: ''Erwachende Wissenschaft. Ägyptische, babylonische und griechische Mathematik''. Birkäuser-Verlag Basel Stuttgart 1966 p. 90</ref> He often explained a number given in his numeral system by stating ''ankair api'' ("in figures this reads"), and then repeating it written with the first nine [[Brahmi numeral]]s, using a small circle for the [[0 (number)|zero]]. Contrary to the word system, however, his numerals were written in descending values from left to right, exactly as we do it today. Therefore, since at least 629, the [[decimal]] system was definitely known to Indian scholars. Presumably, Bhāskara did not invent it, but he was the first to openly use the [[Brahmi numeral]]s in a scientific contribution in [[Sanskrit]].

== Further contributions ==

=== Mathematics ===
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.<ref name=":0" />

His work ''Mahābhāskarīya'' is divided into eight chapters about mathematical astronomy. In chapter 7, he gives [[Bhāskara I's sine approximation formula|a remarkable approximation formula for sin x]]:
: <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>). 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°.

Moreover, Bhāskara stated theorems about the solutions to equations now known as [[Pell's equation]]s. 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 <math> 8x^2 + 1 = y^2 </math> (or <math>y^2 - 8x^2 = 1 </math> relative to pell's equation). This equation has the simple solution x = 1, y = 3, or shortly (x,y) = (1,3), from which further solutions can be constructed, such as (x,y) = (6,17).

Bhāskara clearly believed that [[Pi|''{{pi}}'']] was irrational. In support of [[Aryabhata]]'s approximation of {{Pi}}, he criticized its approximation to <math>\sqrt{10}</math>, a practice common among [[Jainism|Jain]] mathematicians.<ref name=":0" /><ref name=":2" />

He was the first mathematician to openly discuss [[quadrilateral]]s with four unequal, nonparallel sides.<ref>{{Cite web |date=28 September 2020 |title=Bhāskara i {{!}} Famous Indian Mathematician and Astronomer |url=https://www.cuemath.com/learn/bhaskara-i/ |access-date=2022-09-03 |website=Cuemath |language=en}}</ref>

=== Astronomy ===
The ''Mahābhāskarīya'' consists of eight chapters dealing with mathematical astronomy. The book deals with topics such as the longitudes of the planets, the [[Conjunction (astronomy)|conjunctions]] among the planets and stars, the phases of the moon, solar and lunar [[eclipse]]s, and the rising and setting of the planets.<ref name=":0" />

Parts of ''Mahābhāskarīya'' were later translated into [[Arabic]].

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

== Notes ==
<references group="lower-alpha" />

== References ==
{{Reflist}}

== Sources ==
(From {{Harvtxt|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 ==
* 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=2006a
| title=Expounding the Mathematical Seed. Vol. 1: The Translation: A Translation of Bhāskara 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=2006b
| title=Expounding the Mathematical Seed. Vol. 2: The Supplements: A Translation of Bhāskara 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}}

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{{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:Scientists from Gujarat]]
[[Category:Scholars from Gujarat]]
[[Category:Scientists from Maharashtra]]
[[Category:Scholars from Maharashtra]]
[[Category:Acharyas]]

Bhāskara I - Revision history

MaximilianGreenb at 06:04, 6 February 2026