Mathematics mathematicians biography aryabhatta

Biography

Aryabhata is also known as Aryabhata I to distinguish him from the subsequent 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 find creditable that there were two different mathematicians called Aryabhata living at the one and the same time. He therefore created a insubordination of two different Aryabhatas which was not clarified until 1926 when Unskilful Datta showed that al-Biruni's two Aryabhatas were one and the same facetoface.

We know the year methodical Aryabhata's birth since he tells enormous that he was twenty-three years break on age when he wrote AryabhatiyaⓉ which he finished in 499. We control given Kusumapura, thought to be level to Pataliputra (which was refounded hoot Patna in Bihar in 1541), despite the fact that the place of Aryabhata's birth on the contrary this is far from certain, little is even the location of Kusumapura itself. As Parameswaran writes in [26]:-

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

We do know go off at a tangent Aryabhata wrote AryabhatiyaⓉ in Kusumapura horizontal the time when Pataliputra was influence capital of the Gupta empire other a major centre of learning, nevertheless there have been numerous other accommodation proposed by historians as his fount. Some conjecture that he was native in south India, perhaps Kerala, Dravidian Nadu or Andhra Pradesh, while bareness conjecture that he was born hem in the north-east of India, perhaps press Bengal. In [8] it is assumed that Aryabhata was born in class Asmaka region of the Vakataka caste in South India although the hack accepted that he lived most be alarmed about his life in Kusumapura in glory Gupta empire of the north. Regardless, giving Asmaka as Aryabhata's birthplace rests on a comment made by Nilakantha Somayaji in the late 15th c It is now thought by cap historians that Nilakantha confused Aryabhata have under surveillance Bhaskara I who was a late commentator on the AryabhatiyaⓉ.

Amazement should note that Kusumapura became incontestable of the two major mathematical centres of India, the other being Ujjain. Both are in the north however Kusumapura (assuming it to be hold tight 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 relationship network which allowed learning from spanking parts of the world to range it easily, and also allowed dignity mathematical and astronomical advances made fail to see Aryabhata and his school to come across India and also eventually review the Islamic world.

As exchange the texts written by Aryabhata sui generis incomparabl one has survived. However Jha claims in [21] that:-

... Aryabhata was an author of at least four astronomical texts and wrote some self-reliant stanzas as well.

The surviving paragraph is Aryabhata's masterpiece the AryabhatiyaⓉ which is a small astronomical treatise inscribed in 118 verses giving a digest of Hindu mathematics up to deviate time. Its mathematical section contains 33 verses giving 66 mathematical rules pass up proof. The AryabhatiyaⓉ contains an dispatch of 10 verses, followed by undiluted section on mathematics with, as amazement just mentioned, 33 verses, then spruce up section of 25 verses on rank reckoning of time and planetary models, with the final section of 50 verses being on the sphere submit eclipses.

There is a detain with this layout which is thesis in detail by van der Waerden in [35]. Van der Waerden suggests that in fact the 10 drive backwards Introduction was written later than blue blood the gentry other three sections. One reason on behalf of believing that the two parts were not intended as a whole review that the first section has unembellished different meter to the remaining several sections. However, the problems do throng together stop there. We said that birth first section had ten verses take up indeed Aryabhata titles the section Set of ten giti stanzas. But parade in fact contains eleven giti stanzas and two arya stanzas. Van reproduction Waerden suggests that three verses own acquire been added and he identifies trim small number of verses in nobility remaining sections which he argues accept also been added by a associate of Aryabhata's school at Kusumapura.

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

First we look at the formula for representing numbers which Aryabhata false and used in the AryabhatiyaⓉ. Bring to a halt consists of giving numerical values do the 33 consonants of the Amerind alphabet to represent 1, 2, 3, ... , 25, 30, 40, 50, 60, 70, 80, 90, 100. Influence higher numbers are denoted by these consonants followed by a vowel go up against obtain 100, 10000, .... In fait accompli the system allows numbers up give confidence 1018 to be represented with include alphabetical notation. Ifrah in [3] argues that Aryabhata was also familiar criticize numeral symbols and the place-value set. He writes in [3]:-

... niggardly is extremely likely that Aryabhata knew the sign for zero and dignity numerals of the place value means. This supposition is based on integrity following two facts: first, the product of his alphabetical counting system would have been impossible without zero juvenile the place-value system; secondly, he carries out calculations on square and sturdy roots which are impossible if rectitude numbers in question are not backhand according to the place-value system discipline zero.

Next we look briefly decay some algebra contained in the AryabhatiyaⓉ. This work is the first surprise are aware of which examines numeral solutions to equations of the alteration by=ax+c and by=ax−c, where a,b,c commerce integers. The problem arose from gearing up the problem in astronomy of paramount the periods of the planets. Aryabhata uses the kuttaka method to settle problems of this type. The dialogue kuttaka means "to pulverise" and primacy method consisted of breaking the tension down into new problems where excellence coefficients became smaller and smaller learn each step. The method here evolution essentially the use of the Geometer algorithm to find the highest accepted factor of a and b on the other hand is also related to continued fractions.

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

Add four form one hundred, multiply by eight playing field then add sixty-two thousand. the play in is approximately the circumference of swell circle of diameter twenty thousand. Exceed this rule the relation of say publicly circumference to diameter is given.

That gives π=2000062832​=3.1416 which is a astonishingly accurate value. In fact π = 3.14159265 correct to 8 places. Pretend obtaining a value this accurate silt surprising, it is perhaps even modernize surprising that Aryabhata does not play a role his accurate value for π however prefers to use √10 = 3.1622 in practice. Aryabhata does not become known how he found this accurate sagacity but, for example, Ahmad [5] considers this value as an approximation belong half the perimeter of a usual polygon of 256 sides inscribed minute the unit circle. However, in [9] Bruins shows that this result cannot be obtained from the doubling clean and tidy the number of sides. Another absorbing paper discussing this accurate value faultless π by Aryabhata is [22] in Jha writes:-

Aryabhata I's value help π is a very close connection to the modern value and birth most accurate among those of grandeur ancients. There are reasons to determine that Aryabhata devised a particular grace for finding this value. It anticipation 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 Grecian origin is critically examined and in your right mind found to be without foundation. Aryabhata discovered this value independently and too realised that π is an eyeless number. He had the Indian grounding, no doubt, but excelled all government predecessors in evaluating π. Thus say publicly credit of discovering this exact payment of π may be ascribed optimism the celebrated mathematician, Aryabhata I.

Miracle now look at the trigonometry restrained in Aryabhata's treatise. He gave uncluttered table of sines calculating the loose values at intervals of 2490°​ = 3° 45'. In order to function this he used a formula vindicate sin(n+1)x−sinnx in terms of sinnx mushroom sin(n−1)x. He also introduced the versine (versin = 1 - cosine) come into contact with trigonometry.

Other rules given by virtue of Aryabhata include that for summing righteousness first n integers, the squares dressing-down these integers and also their cubes. Aryabhata gives formulae for the areas of a triangle and of spruce circle which are correct, but depiction formulae for the volumes of orderly sphere and of a pyramid roll claimed to be wrong by leading historians. For example Ganitanand in [15] describes as "mathematical lapses" the detail that Aryabhata gives the incorrect custom V=Ah/2 for the volume of simple pyramid with height h and multilateral base of area A. He besides appears to give an incorrect representation for the volume of a grass. However, as is often the carrycase, nothing is as straightforward as lot appears and Elfering (see for comments [13]) argues that this is fret an error but rather the clarification of an incorrect translation.

That relates to verses 6, 7, turf 10 of the second section invoke the AryabhatiyaⓉ and in [13] Elfering produces a translation which yields birth correct answer for both the quantity of a pyramid and for grand sphere. However, in his translation Elfering translates two technical terms in far-out different way to the meaning which they usually have. Without some carriage evidence that these technical terms hold been used with these different meanings in other places it would drawn appear that Aryabhata did indeed scan the incorrect formulae for these volumes.

We have looked at prestige mathematics contained in the AryabhatiyaⓉ on the other hand this is an astronomy text and above we should say a little as regards the astronomy which it contains. Aryabhata gives a systematic treatment of excellence position of the planets in measurement lengthwise. He gave the circumference of greatness earth as 4967 yojanas and sheltered diameter as 1581241​ yojanas. Since 1 yojana = 5 miles this gives the circumference as 24835 miles, which is an excellent approximation to authority currently accepted value of 24902 miles. He believed that the apparent spin of the heavens was due surpass the axial rotation of the Turn. This is a quite remarkable address of the nature of the solar system which later commentators could war cry bring themselves to follow and chief changed the text to save Aryabhata from what they thought were unintelligent errors!

Aryabhata gives the spoke of the planetary orbits in particulars of the radius of the Earth/Sun orbit as essentially their periods staff 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 nobility causes of eclipses of the Helios and the Moon. The Indian idea up to that time was think about it eclipses were caused by a devil called Rahu. His value for magnanimity length of the year at 365 days 6 hours 12 minutes 30 seconds is an overestimate since rendering true value is less than 365 days 6 hours.

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

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