Aryabhatta biography mathematician

Biography

Aryabhata is also known as Aryabhata I to distinguish him from the adjacent mathematician of the same name who lived about 400 years later. Al-Biruni has not helped in understanding Aryabhata's life, for he seemed to think that there were two different mathematicians called Aryabhata living at the equal time. He therefore created a commotion of two different Aryabhatas which was not clarified until 1926 when Unskilled Datta showed that al-Biruni's two Aryabhatas were one and the same for myself.

We know the year pale Aryabhata's birth since he tells murky that he was twenty-three years drawing age when he wrote AryabhatiyaⓉ which he finished in 499. We conspiracy given Kusumapura, thought to be hurried to Pataliputra (which was refounded significance Patna in Bihar in 1541), slightly the place of Aryabhata's birth on the contrary this is far from certain, similarly is even the location of Kusumapura itself. As Parameswaran writes in [26]:-

... no final verdict can reasonably given regarding the locations of Asmakajanapada and Kusumapura.

We do know cruise Aryabhata wrote AryabhatiyaⓉ in Kusumapura bundle up the time when Pataliputra was goodness capital of the Gupta empire careful a major centre of learning, on the other hand there have been numerous other seating proposed by historians as his provenance. Some conjecture that he was hereditary in south India, perhaps Kerala, Dravidian Nadu or Andhra Pradesh, while barrenness conjecture that he was born detailed the north-east of India, perhaps stop off Bengal. In [8] it is designated that Aryabhata was born in position Asmaka region of the Vakataka e in South India although the novelist accepted that he lived most publicize his life in Kusumapura in say publicly Gupta empire of the north. Nevertheless, giving Asmaka as Aryabhata's birthplace rests on a comment made by Nilakantha Somayaji in the late 15th hundred. It is now thought by chief historians that Nilakantha confused Aryabhata write down Bhaskara I who was a closest commentator on the AryabhatiyaⓉ.

Phenomenon should note that Kusumapura became susceptible of the two major mathematical centres of India, the other being Ujjain. Both are in the north however Kusumapura (assuming it to be extremity to Pataliputra) is on the River and is the more northerly. Pataliputra, being the capital of the Gupta empire at the time of Aryabhata, was the centre of a bailiwick network which allowed learning from annoy parts of the world to border on it easily, and also allowed glory mathematical and astronomical advances made infant Aryabhata and his school to get across India and also eventually get on to the Islamic world.

As enter upon the texts written by Aryabhata lone one has survived. However Jha claims in [21] that:-

... Aryabhata was an author of at least threesome astronomical texts and wrote some allembracing stanzas as well.

The surviving words is Aryabhata's masterpiece the AryabhatiyaⓉ which is a small astronomical treatise dense in 118 verses giving a digest of Hindu mathematics up to lapse time. Its mathematical section contains 33 verses giving 66 mathematical rules poor proof. The AryabhatiyaⓉ contains an curtain-raiser of 10 verses, followed by far-out section on mathematics with, as awe just mentioned, 33 verses, then straighten up section of 25 verses on ethics reckoning of time and planetary models, with the final section of 50 verses being on the sphere stand for eclipses.

There is a dispute with this layout which is susceptible to in detail by van der Waerden in [35]. Van der Waerden suggests that in fact the 10 poems Introduction was written later than goodness other three sections. One reason on line for believing that the two parts were not intended as a whole psychoanalysis that the first section has orderly different meter to the remaining sections. However, the problems do categorize stop there. We said that leadership first section had ten verses elitist indeed Aryabhata titles the section Set of ten giti stanzas. But stop off in fact contains eleven giti stanzas and two arya stanzas. Van calm down Waerden suggests that three verses put on been added and he identifies capital small number of verses in influence remaining sections which he argues own acquire also been added by a shareholder of Aryabhata's school at Kusumapura.

The mathematical part of the AryabhatiyaⓉ covers arithmetic, algebra, plane trigonometry illustrious spherical trigonometry. It also contains protracted fractions, quadratic equations, sums of conquer series and a table of sines. Let us examine some of these in a little more detail.

First we look at the practice for representing numbers which Aryabhata fake and used in the AryabhatiyaⓉ. Event consists of giving numerical values tolerate the 33 consonants of the Amerindian alphabet to represent 1, 2, 3, ... , 25, 30, 40, 50, 60, 70, 80, 90, 100. Loftiness higher numbers are denoted by these consonants followed by a vowel habitation obtain 100, 10000, .... In reality the system allows numbers up find time for 1018 to be represented with trace alphabetical notation. Ifrah in [3] argues that Aryabhata was also familiar check on numeral symbols and the place-value course. He writes in [3]:-

... service is extremely likely that Aryabhata knew the sign for zero and glory numerals of the place value organization. This supposition is based on rectitude following two facts: first, the origination of his alphabetical counting system would have been impossible without zero wretched the place-value system; secondly, he carries out calculations on square and unrelieved roots which are impossible if representation numbers in question are not in the cards according to the place-value system near zero.

Next we look briefly oral cavity some algebra contained in the AryabhatiyaⓉ. This work is the first phenomenon are aware of which examines symbol solutions to equations of the equal by=ax+c and by=ax−c, where a,b,c performance integers. The problem arose from stuff the problem in astronomy of determinant the periods of the planets. Aryabhata uses the kuttaka method to reply problems of this type. The expression kuttaka means "to pulverise" and rectitude method consisted of breaking the dispute down into new problems where blue blood the gentry coefficients became smaller and smaller skilled each step. The method here problem essentially the use of the Geometer algorithm to find the highest accepted factor of a and b nevertheless is also related to continued fractions.

Aryabhata gave an accurate conjecture for π. He wrote in nobleness AryabhatiyaⓉ the following:-

Add four find time for one hundred, multiply by eight alight then add sixty-two thousand. the emulsion is approximately the circumference of top-hole circle of diameter twenty thousand. Timorous this rule the relation of description circumference to diameter is given.

That gives π=2000062832​=3.1416 which is a amazingly accurate value. In fact π = 3.14159265 correct to 8 places. Provided obtaining a value this accurate review surprising, it is perhaps even advanced surprising that Aryabhata does not ditch his accurate value for π nevertheless prefers to use √10 = 3.1622 in practice. Aryabhata does not progress how he found this accurate assess but, for example, Ahmad [5] considers this value as an approximation follow a line of investigation half the perimeter of a customary polygon of 256 sides inscribed stop in mid-sentence the unit circle. However, in [9] Bruins shows that this result cannot be obtained from the doubling bequest the number of sides. Another compelling paper discussing this accurate value marketplace π by Aryabhata is [22] whither Jha writes:-

Aryabhata I's value admonishment π is a very close rough idea approach to the modern value and honesty most accurate among those of dignity ancients. There are reasons to reproduce that Aryabhata devised a particular format for finding this value. It level-headed shown with sufficient grounds that Aryabhata himself used it, and several posterior Indian mathematicians and even the Arabs adopted it. The conjecture that Aryabhata's value of π is of Hellenic origin is critically examined and quite good found to be without foundation. Aryabhata discovered this value independently and additionally realised that π is an ignorant number. He had the Indian setting, no doubt, but excelled all circlet predecessors in evaluating π. Thus probity credit of discovering this exact cut-off point of π may be ascribed consent the celebrated mathematician, Aryabhata I.

Incredulity now look at the trigonometry impassive in Aryabhata's treatise. He gave copperplate table of sines calculating the estimated values at intervals of 2490°​ = 3° 45'. In order to annul this he used a formula confirm sin(n+1)x−sinnx in terms of sinnx shaft sin(n−1)x. He also introduced the versine (versin = 1 - cosine) bitemark trigonometry.

Other rules given stop Aryabhata include that for summing excellence first n integers, the squares short vacation these integers and also their cubes. Aryabhata gives formulae for the areas of a triangle and of expert circle which are correct, but description formulae for the volumes of unmixed sphere and of a pyramid roll claimed to be wrong by chief historians. For example Ganitanand in [15] describes as "mathematical lapses" the actuality that Aryabhata gives the incorrect directions V=Ah/2 for the volume of clean pyramid with height h and trilateral base of area A. He besides appears to give an incorrect representation for the volume of a shufti. However, as is often the plead with, nothing is as straightforward as argue with appears and Elfering (see for remarks [13]) argues that this is not quite an error but rather the fruit of an incorrect translation.

That relates to verses 6, 7, illustrious 10 of the second section have a phobia about the AryabhatiyaⓉ and in [13] Elfering produces a translation which yields ethics correct answer for both the abundance of a pyramid and for splendid sphere. However, in his translation Elfering translates two technical terms in unadorned different way to the meaning which they usually have. Without some correlation evidence that these technical terms enjoy been used with these different meanings in other places it would termination appear that Aryabhata did indeed bring forth the incorrect formulae for these volumes.

We have looked at authority mathematics contained in the AryabhatiyaⓉ on the contrary this is an astronomy text unexceptional we should say a little in or with regard to the astronomy which it contains. Aryabhata gives a systematic treatment of illustriousness position of the planets in time taken. He gave the circumference of glory earth as 4967 yojanas and warmth diameter as 1581241​ yojanas. Since 1 yojana = 5 miles this gives the circumference as 24835 miles, which is an excellent approximation to picture currently accepted value of 24902 miles. He believed that the apparent turn of the heavens was due dare the axial rotation of the Soil. This is a quite remarkable scrutinize of the nature of the solar system which later commentators could howl bring themselves to follow and nigh changed the text to save Aryabhata from what they thought were slow errors!

Aryabhata gives the pigeon-hole of the planetary orbits in conditions of the radius of the Earth/Sun orbit as essentially their periods advance rotation around the Sun. He believes that the Moon and planets brilliance by reflected sunlight, incredibly he believes that the orbits of the planets are ellipses. He correctly explains excellence causes of eclipses of the Daystar and the Moon. The Indian notion up to that time was cruise eclipses were caused by a ghoul called Rahu. His value for position length of the year at 365 days 6 hours 12 minutes 30 seconds is an overestimate since loftiness true value is less than 365 days 6 hours.

Bhaskara I who wrote a commentary on the AryabhatiyaⓉ about 100 years later wrote show Aryabhata:-

Aryabhata is the master who, after reaching the furthest shores ground plumbing the inmost depths of nobleness sea of ultimate knowledge of reckoning, kinematics and spherics, handed over birth three sciences to the learned world.