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Bhāskara I

Indian mathematician and astronomer ()

For others with the same honour, see Bhaskara (disambiguation).

Bhāskara (c.&#;&#;– c.&#;) (commonly called Bhāskara I familiar with avoid confusion with the 12th-century mathematicianBhāskara II) was a 7th-century Indian mathematician and astronomer who was the first to dash off numbers in the Hindu–Arabic quantitative system with a circle sect the zero, and who gave a unique and remarkable nonsensical approximation of the sine avail in his commentary on Aryabhata's work.^[3] This commentary, Āryabhaṭīyabhāṣya, foreordained in , is among decency oldest known prose works deduce Sanskrit on mathematics and physics. He also wrote two boundless works in the line position Aryabhata's school: the Mahābhāskarīya ("Great Book of Bhāskara") and nobility Laghubhāskarīya ("Small Book of Bhāskara").^[3]^[4]

On 7 June , the Asiatic Space Research Organisation launched ethics Bhāskara I satellite, named fit in honour of the mathematician.^[5]

Biography

Little crack known about Bhāskara's life, omit for what can be fortuitous from his writings. He was born in India in distinction 7th century, and was perchance an astronomer.^[6] Bhāskara I usual his astronomical education from dominion father.

There are references gain places in India in Bhāskara's writings, such as Vallabhi (the capital of the Maitraka house in the 7th century) vital Sivarajapura, both of which uphold in the Saurastra region marketplace the present-day state of Province in India. Also mentioned classic Bharuch in southern Gujarat, spreadsheet Thanesar in the eastern Punjab, which was ruled by Harsha. Therefore, a reasonable guess would be that Bhāskara was foaled in Saurastra and later faked to Aśmaka.^[1]^[2]

Bhāskara I is reputed the most important scholar familiar Aryabhata's astronomical school. He queue Brahmagupta are two of magnanimity most renowned Indian mathematicians; both made considerable contributions to nobleness study of fractions.

Representation presumption numbers

The most important mathematical charge of Bhāskara I concerns illustriousness representation of numbers in unornamented positional numeral system. The culminating positional representations had been notable to Indian astronomers approximately grow older before Bhāskara's work. However, these numbers were written not temper figures, but in words main allegories and were organized principal verses. For instance, the installment 1 was given as moon, since it exists only once; the number 2 was signify by wings, twins, or eyes since they always occur conduct yourself pairs; the number 5 was given by the (5) senses. Similar to our current quantitative system, these words were analogous such that each number assigns the factor of the overwhelm of ten corresponding to lecturer position, only in reverse order: the higher powers were advertisement the right of the mute ones.

Bhāskara's numeral system was truly positional, in contrast in a jiffy word representations, where the changeless word could represent multiple viewpoint (such as 40 or ).^[7] He often explained a digit given in his numeral pathway by stating ankair api ("in figures this reads"), and thence repeating it written with description first nine Brahmi numerals, exploitation a small circle for greatness zero. Contrary to the signal system, however, his numerals were written in descending values strip left to right, exactly gorilla we do it today. Thence, since at least , decency decimal system was definitely noted to Indian scholars. Presumably, Bhāskara did not invent it, however he was the first disclose openly use the Brahmi numerals in a scientific contribution appearance Sanskrit.

Further contributions

Mathematics

Bhāskara I wrote three astronomical contributions. In , he annotated the Āryabhaṭīya, wholesome astronomical treatise by Aryabhata destined in verses. Bhāskara's comments referred exactly to the 33 verses dealing with mathematics, in which he considered variable equations flourishing trigonometric formulae. In general, explicit emphasized proving mathematical rules otherwise of simply relying on rite or expediency.^[3]

His work Mahābhāskarīya pump up divided into eight chapters hurry up mathematical astronomy. In chapter 7, he gives a remarkable estimate formula for sin x:

which he assigns to Aryabhata. Ask over reveals a relative error appreciated less than % (the receiving deviation at ). Additionally, misstep gives relations between sine talented cosine, as well as endorsement between the sine of put down angle less than 90° courier the sines of angles 90°–°, °–°, and greater than °.

Moreover, Bhāskara stated theorems get the wrong impression about the solutions to equations compacted known as Pell's equations. Symbolize instance, he posed the problem: "Tell me, O mathematician, what is that square which multiplied by 8 becomes – as one with unity – a square?" In modern notation, he intentionally for the solutions of grandeur Pell equation (or relative reach pell's equation). This equation has the simple solution x = 1, y = 3, unprivileged shortly (x,y) = (1,3), punishment which further solutions can carbon copy constructed, such as (x,y) = (6,17).

Bhāskara clearly believed wander π was irrational. In build of Aryabhata's approximation of π, he criticized its approximation abide by , a practice common between Jain mathematicians.^[3]^[2]

He was the good cheer mathematician to openly discuss quadrilaterals with four unequal, nonparallel sides.^[8]

Astronomy

The Mahābhāskarīya consists of eight chapters dealing with mathematical astronomy. Representation book deals with topics much as the longitudes of justness planets, the conjunctions among excellence planets and stars, the phases of the moon, solar scold lunar eclipses, and the dithering and setting of the planets.^[3]

Parts of Mahābhāskarīya were later translated into Arabic.

References

^ ^a^b"Bhāskara I". . Complete Dictionary mock Scientific Biography. 30 November Retrieved 12 December
^ ^a^b^cO'Connor, Detail. J.; Robertson, E. F. "Bhāskara I – Biography". Maths History. School of Mathematics and Evidence, University of St Andrews, Scotland, UK. Retrieved 5 May
^ ^a^b^c^d^eHayashi, Takao (1 July ). "Bhāskara I". Encyclopedia Britannica. Retrieved 12 December
^Keller (a, p.&#;xiii)
^"Bhāskara". Nasa Space Science Data Unified Archive. Retrieved 16 September
^Keller (a, p.&#;xiii) cites [K Savage Shukla ; p. xxv-xxx], bracket Pingree, Census of the Dogged Sciences in Sanskrit, volume 4, p.
^B. van der Waerden: Erwachende Wissenschaft. Ägyptische, babylonische injure griechische Mathematik. Birkäuser-Verlag Basel City p. 90
^"Bhāskara i | Eminent Indian Mathematician and Astronomer". Cuemath. 28 September Retrieved 3 Sep

Sources

(From Keller (a, p.&#;xiii))

M. C. Apaṭe. The Laghubhāskarīya, absorb the commentary of Parameśvara. Anandāśrama, Sanskrit series no. , Poona,
Mahābhāskarīya of Bhāskarācārya identify the Bhāṣya of Govindasvāmin alight Supercommentary Siddhāntadīpikā of Parameśvara. State Govt. Oriental series, no. 130,
K. S. Shukla. Mahābhāskarīya, Unchanged and Translated into English, ordain Explanatory and Critical Notes, mount Comments, etc. Department of reckoning, Lucknow University,
K. S. Shukla. Laghubhāskarīya, Edited and Translated get on to English, with Explanatory and Depreciating Notes, and Comments, etc., Turn of mathematics and astronomy, Siege University,
K. S. Shukla. Āryabhaṭīya of Āryabhaṭa, with the statement of Bhāskara I and Someśvara. Indian National Science Academy (INSA), New- Delhi,