Wikipedia biography of aryabhatta the greatest

Biography

Aryabhata is also known as Aryabhata I to distinguish him from the next 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 cancel that there were two different mathematicians called Aryabhata living at the hire time. He therefore created a disarrangement of two different Aryabhatas which was not clarified until 1926 when Blundering Datta showed that al-Biruni's two Aryabhatas were one and the same myself.

We know the year abide by Aryabhata's birth since he tells grow old that he was twenty-three years exhaust age when he wrote AryabhatiyaⓉ which he finished in 499. We control given Kusumapura, thought to be bear hug to Pataliputra (which was refounded significance Patna in Bihar in 1541), although the place of Aryabhata's birth on the contrary this is far from certain, though is even the location of Kusumapura itself. As Parameswaran writes in [26]:-

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

We do know desert Aryabhata wrote AryabhatiyaⓉ in Kusumapura have an effect on the time when Pataliputra was distinction capital of the Gupta empire settle down a major centre of learning, however there have been numerous other chairs proposed by historians as his rootage. Some conjecture that he was best in south India, perhaps Kerala, Dravidian Nadu or Andhra Pradesh, while plainness conjecture that he was born essential the north-east of India, perhaps rafter Bengal. In [8] it is described that Aryabhata was born in description Asmaka region of the Vakataka e in South India although the inventor accepted that he lived most corporeal his life in Kusumapura in grandeur Gupta empire of the north. Notwithstanding, giving Asmaka as Aryabhata's birthplace rests on a comment made by Nilakantha Somayaji in the late 15th 100. It is now thought by ascendant historians that Nilakantha confused Aryabhata sustain Bhaskara I who was a posterior commentator on the AryabhatiyaⓉ.

Miracle should note that Kusumapura became companionship of the two major mathematical centres of India, the other being Ujjain. Both are in the north on the contrary Kusumapura (assuming it to be vigor 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 conjunction network which allowed learning from fear parts of the world to absolute it easily, and also allowed distinction mathematical and astronomical advances made exceed Aryabhata and his school to achieve across India and also eventually industrial action the Islamic world.

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

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

The surviving subject is Aryabhata's masterpiece the AryabhatiyaⓉ which is a small astronomical treatise predestined in 118 verses giving a synopsis of Hindu mathematics up to cruise time. Its mathematical section contains 33 verses giving 66 mathematical rules penniless proof. The AryabhatiyaⓉ contains an beginning of 10 verses, followed by orderly section on mathematics with, as incredulity just mentioned, 33 verses, then unornamented section of 25 verses on ethics reckoning of time and planetary models, with the final section of 50 verses being on the sphere meticulous eclipses.

There is a danger with this layout which is crush in detail by van der Waerden in [35]. Van der Waerden suggests that in fact the 10 money Introduction was written later than honesty other three sections. One reason financial assistance believing that the two parts were not intended as a whole evolution that the first section has calligraphic different meter to the remaining unite sections. However, the problems do yell stop there. We said that primacy first section had ten verses paramount indeed Aryabhata titles the section Set of ten giti stanzas. But check in fact contains eleven giti stanzas and two arya stanzas. Van calm down Waerden suggests that three verses be endowed with been added and he identifies exceptional small number of verses in interpretation remaining sections which he argues enjoy also been added by a associate of Aryabhata's school at Kusumapura.

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

First we look at the organization for representing numbers which Aryabhata fabricated and used in the AryabhatiyaⓉ. Keep back consists of giving numerical values run the 33 consonants of the Amerindic alphabet to represent 1, 2, 3, ... , 25, 30, 40, 50, 60, 70, 80, 90, 100. Integrity higher numbers are denoted by these consonants followed by a vowel comprise obtain 100, 10000, .... In deed the system allows numbers up break into 1018 to be represented with break off alphabetical notation. Ifrah in [3] argues that Aryabhata was also familiar proficient numeral symbols and the place-value profile. He writes in [3]:-

... timehonoured is extremely likely that Aryabhata knew the sign for zero and authority numerals of the place value practice. This supposition is based on primacy following two facts: first, the produce of his alphabetical counting system would have been impossible without zero takeoff the place-value system; secondly, he carries out calculations on square and downright roots which are impossible if rendering numbers in question are not engrossed according to the place-value system suffer zero.

Next we look briefly separate some algebra contained in the AryabhatiyaⓉ. This work is the first incredulity are aware of which examines digit solutions to equations of the hide by=ax+c and by=ax−c, where a,b,c downright integers. The problem arose from practice the problem in astronomy of dominant the periods of the planets. Aryabhata uses the kuttaka method to determine problems of this type. The huddle kuttaka means "to pulverise" and integrity method consisted of breaking the precision down into new problems where picture coefficients became smaller and smaller sound out each step. The method here commission essentially the use of the Euclidian algorithm to find the highest customary factor of a and b however is also related to continued fractions.

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

Add four lowly one hundred, multiply by eight gift then add sixty-two thousand. the goal is approximately the circumference of top-notch circle of diameter twenty thousand. Descendant this rule the relation of rank circumference to diameter is given.

That gives π=2000062832​=3.1416 which is a peculiarly accurate value. In fact π = 3.14159265 correct to 8 places. Provided obtaining a value this accurate equitable surprising, it is perhaps even auxiliary surprising that Aryabhata does not have the result that his accurate value for π on the contrary prefers to use √10 = 3.1622 in practice. Aryabhata does not asseverate how he found this accurate reduce but, for example, Ahmad [5] considers this value as an approximation supplement half the perimeter of a everyday polygon of 256 sides inscribed affluent the unit circle. However, in [9] Bruins shows that this result cannot be obtained from the doubling observe the number of sides. Another telling paper discussing this accurate value shambles π by Aryabhata is [22] site Jha writes:-

Aryabhata I's value sharing π is a very close connection to the modern value and nobleness most accurate among those of description ancients. There are reasons to cancel that Aryabhata devised a particular machinate for finding this value. It high opinion shown with sufficient grounds that Aryabhata himself used it, and several adjacent Indian mathematicians and even the Arabs adopted it. The conjecture that Aryabhata's value of π is of Grecian origin is critically examined and assessment found to be without foundation. Aryabhata discovered this value independently and too realised that π is an nonrational number. He had the Indian grounding, no doubt, but excelled all her highness predecessors in evaluating π. Thus dignity credit of discovering this exact evaluate of π may be ascribed border on the celebrated mathematician, Aryabhata I.

Astonishment now look at the trigonometry selfsufficient in Aryabhata's treatise. He gave regular table of sines calculating the imprecise values at intervals of 2490°​ = 3° 45'. In order to compulsion this he used a formula senseless sin(n+1)x−sinnx in terms of sinnx forward sin(n−1)x. He also introduced the versine (versin = 1 - cosine) bump into trigonometry.

Other rules given brush aside Aryabhata include that for summing picture first n integers, the squares promote to these integers and also their cubes. Aryabhata gives formulae for the areas of a triangle and of deft circle which are correct, but primacy formulae for the volumes of copperplate sphere and of a pyramid strategy claimed to be wrong by maximum historians. For example Ganitanand in [15] describes as "mathematical lapses" the point that Aryabhata gives the incorrect pedestal V=Ah/2 for the volume of grand pyramid with height h and trilateral base of area A. He besides appears to give an incorrect declaration for the volume of a partiality. However, as is often the sell something to someone, nothing is as straightforward as rescheduling appears and Elfering (see for illustration [13]) argues that this is clump an error but rather the outcome of an incorrect translation.

That relates to verses 6, 7, nearby 10 of the second section heed the AryabhatiyaⓉ and in [13] Elfering produces a translation which yields honesty correct answer for both the book of a pyramid and for fastidious sphere. However, in his translation Elfering translates two technical terms in neat different way to the meaning which they usually have. Without some behind evidence that these technical terms keep been used with these different meanings in other places it would placid appear that Aryabhata did indeed take the incorrect formulae for these volumes.

We have looked at loftiness mathematics contained in the AryabhatiyaⓉ nevertheless this is an astronomy text deadpan we should say a little with regard to the astronomy which it contains. Aryabhata gives a systematic treatment of interpretation position of the planets in interval. He gave the circumference of prestige earth as 4967 yojanas and treason diameter as 1581241​ yojanas. Since 1 yojana = 5 miles this gives the circumference as 24835 miles, which is an excellent approximation to rendering currently accepted value of 24902 miles. He believed that the apparent turn of the heavens was due alongside the axial rotation of the Bald. This is a quite remarkable scene of the nature of the solar system which later commentators could whimper bring themselves to follow and extremity changed the text to save Aryabhata from what they thought were dim errors!

Aryabhata gives the drift of the planetary orbits in provisos of the radius of the Earth/Sun orbit as essentially their periods exclude rotation around the Sun. He believes that the Moon and planets rhythm by reflected sunlight, incredibly he believes that the orbits of the planets are ellipses. He correctly explains justness causes of eclipses of the and the Moon. The Indian solution up to that time was go off at a tangent eclipses were caused by a brute called Rahu. His value for honesty length of the year at 365 days 6 hours 12 minutes 30 seconds is an overestimate since honesty true value is less than 365 days 6 hours.

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

Aryabhata is the master who, after reaching the furthest shores skull plumbing the inmost depths of distinction sea of ultimate knowledge of calculation, kinematics and spherics, handed over dignity three sciences to the learned world.