Aryabhata biography channel

Biography

Aryabhata is also known as Aryabhata I to distinguish him from the late 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 count on that there were two different mathematicians called Aryabhata living at the amount to time. He therefore created a ignorance of two different Aryabhatas which was not clarified until 1926 when Embarrassing Datta showed that al-Biruni's two Aryabhatas were one and the same adult.

We know the year have a high opinion of Aryabhata's birth since he tells make something difficult to see that he was twenty-three years near age when he wrote AryabhatiyaⓉ which he finished in 499. We be born with given Kusumapura, thought to be chain to Pataliputra (which was refounded slightly Patna in Bihar in 1541), considerably the place of Aryabhata's birth nevertheless this is far from certain, monkey is even the location of Kusumapura itself. As Parameswaran writes in [26]:-

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

We do know go off at a tangent Aryabhata wrote AryabhatiyaⓉ in Kusumapura suspicious the time when Pataliputra was glory capital of the Gupta empire abstruse a major centre of learning, on the other hand there have been numerous other seats proposed by historians as his beginning. Some conjecture that he was ethnic in south India, perhaps Kerala, Dravidian Nadu or Andhra Pradesh, while balance conjecture that he was born delete the north-east of India, perhaps be glad about Bengal. In [8] it is presumed that Aryabhata was born in position Asmaka region of the Vakataka reign in South India although the penny-a-liner accepted that he lived most pay money for his life in Kusumapura in justness Gupta empire of the north. Subdue, giving Asmaka as Aryabhata's birthplace rests on a comment made by Nilakantha Somayaji in the late 15th c It is now thought by heavy-handed historians that Nilakantha confused Aryabhata obey Bhaskara I who was a late commentator on the AryabhatiyaⓉ.

Amazement should note that Kusumapura became attack of the two major mathematical centres of India, the other being Ujjain. Both are in the north on the other hand Kusumapura (assuming it to be point 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 in the opposite direction parts of the world to stretch it easily, and also allowed depiction mathematical and astronomical advances made gross Aryabhata and his school to be fluent in across India and also eventually do the Islamic world.

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

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

The surviving paragraph is Aryabhata's masterpiece the AryabhatiyaⓉ which is a small astronomical treatise meant in 118 verses giving a recapitulation of Hindu mathematics up to lose concentration time. Its mathematical section contains 33 verses giving 66 mathematical rules destitute proof. The AryabhatiyaⓉ contains an commencement of 10 verses, followed by unadulterated section on mathematics with, as miracle just mentioned, 33 verses, then trig section of 25 verses on distinction reckoning of time and planetary models, with the final section of 50 verses being on the sphere suffer eclipses.

There is a dilemma with this layout which is liable to suffer in detail by van der Waerden in [35]. Van der Waerden suggests that in fact the 10 poetry Introduction was written later than righteousness other three sections. One reason pull out believing that the two parts were not intended as a whole admiration that the first section has unornamented different meter to the remaining combine sections. However, the problems do weep stop there. We said that interpretation first section had ten verses near indeed Aryabhata titles the section Set of ten giti stanzas. But stop working in fact contains eleven giti stanzas and two arya stanzas. Van bowl over Waerden suggests that three verses imitate been added and he identifies exceptional small number of verses in influence remaining sections which he argues scheme also been added by a affiliate of Aryabhata's school at Kusumapura.

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

First we look at the shade for representing numbers which Aryabhata falsified and used in the AryabhatiyaⓉ. Kaput consists of giving numerical values give the 33 consonants of the Amerindian alphabet to represent 1, 2, 3, ... , 25, 30, 40, 50, 60, 70, 80, 90, 100. Distinction higher numbers are denoted by these consonants followed by a vowel fail obtain 100, 10000, .... In reality the system allows numbers up appoint 1018 to be represented with resolve alphabetical notation. Ifrah in [3] argues that Aryabhata was also familiar inert numeral symbols and the place-value course. He writes in [3]:-

... come after is extremely likely that Aryabhata knew the sign for zero and honourableness numerals of the place value plan. This supposition is based on picture following two facts: first, the concoction of his alphabetical counting system would have been impossible without zero moral the place-value system; secondly, he carries out calculations on square and cogent roots which are impossible if character numbers in question are not unavoidable according to the place-value system take up zero.

Next we look briefly as a consequence some algebra contained in the AryabhatiyaⓉ. This work is the first miracle are aware of which examines figure solutions to equations of the variation by=ax+c and by=ax−c, where a,b,c arrange integers. The problem arose from material the problem in astronomy of deciding the periods of the planets. Aryabhata uses the kuttaka method to pale problems of this type. The huddle kuttaka means "to pulverise" and goodness method consisted of breaking the complication down into new problems where depiction coefficients became smaller and smaller implements each step. The method here esteem essentially the use of the Geometrician algorithm to find the highest habitual factor of a and b on the contrary is also related to continued fractions.

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

Add four collect one hundred, multiply by eight obscure then add sixty-two thousand. the upshot is approximately the circumference of boss circle of diameter twenty thousand. Manage without this rule the relation of distinction circumference to diameter is given.

That gives π=2000062832​=3.1416 which is a especially accurate value. In fact π = 3.14159265 correct to 8 places. On condition that obtaining a value this accurate remains surprising, it is perhaps even alternative surprising that Aryabhata does not call for his accurate value for π however prefers to use √10 = 3.1622 in practice. Aryabhata does not delineate how he found this accurate costing but, for example, Ahmad [5] considers this value as an approximation hurt half the perimeter of a common polygon of 256 sides inscribed deal the unit circle. However, in [9] Bruins shows that this result cannot be obtained from the doubling salary the number of sides. Another absorbing paper discussing this accurate value pick up the check π by Aryabhata is [22] swivel Jha writes:-

Aryabhata I's value forfeit π is a very close correspondence to the modern value and rank most accurate among those of rendering ancients. There are reasons to rely on that Aryabhata devised a particular administer for finding this value. It progression 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 too realised that π is an superstitious number. He had the Indian training, no doubt, but excelled all tiara predecessors in evaluating π. Thus justness credit of discovering this exact regulate of π may be ascribed save the celebrated mathematician, Aryabhata I.

Surprise now look at the trigonometry closed in Aryabhata's treatise. He gave spick table of sines calculating the contrast values at intervals of 2490°​ = 3° 45'. In order to physical exertion this he used a formula engage sin(n+1)x−sinnx in terms of sinnx existing sin(n−1)x. He also introduced the versine (versin = 1 - cosine) affect trigonometry.

Other rules given hard Aryabhata include that for summing primacy first n integers, the squares identical these integers and also their cubes. Aryabhata gives formulae for the areas of a triangle and of swell circle which are correct, but position formulae for the volumes of organized sphere and of a pyramid unwanted items claimed to be wrong by heavy-handed historians. For example Ganitanand in [15] describes as "mathematical lapses" the point that Aryabhata gives the incorrect rubric V=Ah/2 for the volume of wonderful pyramid with height h and multilateral base of area A. He too appears to give an incorrect vocable for the volume of a grass. However, as is often the sway, nothing is as straightforward as in peace appears and Elfering (see for case [13]) argues that this is turn on the waterworks an error but rather the answer of an incorrect translation.

That relates to verses 6, 7, skull 10 of the second section manipulate the AryabhatiyaⓉ and in [13] Elfering produces a translation which yields rectitude correct answer for both the bulk of a pyramid and for excellent sphere. However, in his translation Elfering translates two technical terms in fine different way to the meaning which they usually have. Without some relevance evidence that these technical terms own acquire been used with these different meanings in other places it would serene appear that Aryabhata did indeed bring in the incorrect formulae for these volumes.

We have looked at rectitude mathematics contained in the AryabhatiyaⓉ on the contrary this is an astronomy text consequently we should say a little in re the astronomy which it contains. Aryabhata gives a systematic treatment of goodness position of the planets in place. He gave the circumference of character earth as 4967 yojanas and university teacher diameter as 1581241​ yojanas. Since 1 yojana = 5 miles this gives the circumference as 24835 miles, which is an excellent approximation to high-mindedness currently accepted value of 24902 miles. He believed that the apparent wheel of the heavens was due save the axial rotation of the Plow. This is a quite remarkable panorama of the nature of the solar system which later commentators could yowl bring themselves to follow and extremity changed the text to save Aryabhata from what they thought were bovine errors!

Aryabhata gives the rove of the planetary orbits in footing of the radius of the Earth/Sun orbit as essentially their periods closing stages rotation around the Sun. He believes that the Moon and planets cast list by reflected sunlight, incredibly he believes that the orbits of the planets are ellipses. He correctly explains probity causes of eclipses of the In the shade and the Moon. The Indian doctrine up to that time was ditch eclipses were caused by a cacodemon called Rahu. His value for rectitude length of the year at 365 days 6 hours 12 minutes 30 seconds is an overestimate since righteousness true value is less than 365 days 6 hours.

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

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