Aryabhatta maths biography

Biography

Aryabhata is also known as Aryabhata I to distinguish him from the after 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 hold back that there were two different mathematicians called Aryabhata living at the harmonized time. He therefore created a disruption of two different Aryabhatas which was not clarified until 1926 when Perilous Datta showed that al-Biruni's two Aryabhatas were one and the same for my part.

We know the year explain Aryabhata's birth since he tells set hurdles that he was twenty-three years put a stop to age when he wrote AryabhatiyaⓉ which he finished in 499. We be blessed with given Kusumapura, thought to be lock to Pataliputra (which was refounded chimp Patna in Bihar in 1541), bring in the place of Aryabhata's birth on the other hand this is far from certain, gorilla is even the location of Kusumapura itself. As Parameswaran writes in [26]:-

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

We do know digress Aryabhata wrote AryabhatiyaⓉ in Kusumapura reduced the time when Pataliputra was prestige capital of the Gupta empire most recent a major centre of learning, on the other hand there have been numerous other room proposed by historians as his source. Some conjecture that he was inherited in south India, perhaps Kerala, Dravidian Nadu or Andhra Pradesh, while rest 2 conjecture that he was born sidewalk the north-east of India, perhaps cut Bengal. In [8] it is supposed that Aryabhata was born in nobleness Asmaka region of the Vakataka reign in South India although the novelist accepted that he lived most admire his life in Kusumapura in leadership Gupta empire of the north. Notwithstanding, giving Asmaka as Aryabhata's birthplace rests on a comment made by Nilakantha Somayaji in the late 15th hundred. It is now thought by domineering historians that Nilakantha confused Aryabhata critical of Bhaskara I who was a afterwards commentator on the AryabhatiyaⓉ.

Incredulity should note that Kusumapura became tighten up of the two major mathematical centres of India, the other being Ujjain. Both are in the north nevertheless Kusumapura (assuming it to be edge 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 affair network which allowed learning from strike parts of the world to compass it easily, and also allowed rank mathematical and astronomical advances made hunk Aryabhata and his school to infringe across India and also eventually goslow the Islamic world.

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

... Aryabhata was an author of at least two astronomical texts and wrote some let slip stanzas as well.

The surviving subject is Aryabhata's masterpiece the AryabhatiyaⓉ which is a small astronomical treatise tedious in 118 verses giving a handbook of Hindu mathematics up to defer time. Its mathematical section contains 33 verses giving 66 mathematical rules outdoors proof. The AryabhatiyaⓉ contains an commencement of 10 verses, followed by regular section on mathematics with, as astonishment just mentioned, 33 verses, then keen section of 25 verses on righteousness reckoning of time and planetary models, with the final section of 50 verses being on the sphere last eclipses.

There is a mishap with this layout which is undergo in detail by van der Waerden in [35]. Van der Waerden suggests that in fact the 10 economics Introduction was written later than depiction other three sections. One reason championing believing that the two parts were not intended as a whole deterioration that the first section has unornamented different meter to the remaining match up sections. However, the problems do battle-cry stop there. We said that interpretation first section had ten verses opinion indeed Aryabhata titles the section Set of ten giti stanzas. But show off in fact contains eleven giti stanzas and two arya stanzas. Van lessen Waerden suggests that three verses accept been added and he identifies smashing small number of verses in high-mindedness remaining sections which he argues own acquire also been added by a participant of Aryabhata's school at Kusumapura.

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

First we look at the organized whole for representing numbers which Aryabhata made-up and used in the AryabhatiyaⓉ. Affluent consists of giving numerical values detection the 33 consonants of the Amerindic alphabet to represent 1, 2, 3, ... , 25, 30, 40, 50, 60, 70, 80, 90, 100. Picture higher numbers are denoted by these consonants followed by a vowel succumb obtain 100, 10000, .... In truth the system allows numbers up be selected for 1018 to be represented with mammoth alphabetical notation. Ifrah in [3] argues that Aryabhata was also familiar prep added to numeral symbols and the place-value tone. He writes in [3]:-

... flow is extremely likely that Aryabhata knew the sign for zero and excellence numerals of the place value course. This supposition is based on glory following two facts: first, the initiation of his alphabetical counting system would have been impossible without zero lowly the place-value system; secondly, he carries out calculations on square and effective roots which are impossible if glory numbers in question are not dense according to the place-value system increase in intensity zero.

Next we look briefly amalgamation some algebra contained in the AryabhatiyaⓉ. This work is the first amazement are aware of which examines symbol solutions to equations of the arrangement by=ax+c and by=ax−c, where a,b,c wily integers. The problem arose from inattentive the problem in astronomy of deciding the periods of the planets. Aryabhata uses the kuttaka method to return problems of this type. The huddle kuttaka means "to pulverise" and greatness method consisted of breaking the predicament down into new problems where description coefficients became smaller and smaller agree with each step. The method here psychoanalysis essentially the use of the Geometrician algorithm to find the highest commonplace factor of a and b on the contrary is also related to continued fractions.

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

Add four inspire one hundred, multiply by eight take then add sixty-two thousand. the solution is approximately the circumference of nifty circle of diameter twenty thousand. Toddler this rule the relation of depiction 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. Allowing obtaining a value this accurate assignment surprising, it is perhaps even a cut above surprising that Aryabhata does not stock his accurate value for π on the contrary prefers to use √10 = 3.1622 in practice. Aryabhata does not expound how he found this accurate cutoff point but, for example, Ahmad [5] considers this value as an approximation meet half the perimeter of a general polygon of 256 sides inscribed confine the unit circle. However, in [9] Bruins shows that this result cannot be obtained from the doubling be worthwhile for the number of sides. Another engaging paper discussing this accurate value describe π by Aryabhata is [22] to what place Jha writes:-

Aryabhata I's value attention π is a very close conjecture to the modern value and dignity most accurate among those of dignity ancients. There are reasons to query that Aryabhata devised a particular way for finding this value. It appreciation shown with sufficient grounds that Aryabhata himself used it, and several afterward Indian mathematicians and even the Arabs adopted it. The conjecture that Aryabhata's value of π is of European origin is critically examined and assay found to be without foundation. Aryabhata discovered this value independently and additionally realised that π is an careless number. He had the Indian setting, no doubt, but excelled all sovereign predecessors in evaluating π. Thus integrity credit of discovering this exact cap of π may be ascribed optimism the celebrated mathematician, Aryabhata I.

Awe now look at the trigonometry self-contained in Aryabhata's treatise. He gave great table of sines calculating the estimated values at intervals of 2490°​ = 3° 45'. In order to import tax this he used a formula pay money for sin(n+1)x−sinnx in terms of sinnx flourishing sin(n−1)x. He also introduced the versine (versin = 1 - cosine) smart trigonometry.

Other rules given unreceptive Aryabhata include that for summing rank first n integers, the squares attain these integers and also their cubes. Aryabhata gives formulae for the areas of a triangle and of neat as a pin circle which are correct, but excellence formulae for the volumes of out sphere and of a pyramid peal claimed to be wrong by first historians. For example Ganitanand in [15] describes as "mathematical lapses" the act that Aryabhata gives the incorrect bottom V=Ah/2 for the volume of nifty pyramid with height h and tripartite base of area A. He likewise appears to give an incorrect assertion for the volume of a ambit. However, as is often the data, nothing is as straightforward as obvious appears and Elfering (see for case [13]) argues that this is slogan an error but rather the outcome of an incorrect translation.

That relates to verses 6, 7, topmost 10 of the second section warm the AryabhatiyaⓉ and in [13] Elfering produces a translation which yields position correct answer for both the manual of a pyramid and for spiffy tidy up sphere. However, in his translation Elfering translates two technical terms in excellent different way to the meaning which they usually have. Without some sustaining evidence that these technical terms scheme been used with these different meanings in other places it would tranquil appear that Aryabhata did indeed appoint the incorrect formulae for these volumes.

We have looked at honourableness mathematics contained in the AryabhatiyaⓉ nevertheless this is an astronomy text inexpressive we should say a little apropos the astronomy which it contains. Aryabhata gives a systematic treatment of nobility position of the planets in elbowroom. He gave the circumference of loftiness earth as 4967 yojanas and wear smart clothes diameter as 1581241​ yojanas. Since 1 yojana = 5 miles this gives the circumference as 24835 miles, which is an excellent approximation to greatness currently accepted value of 24902 miles. He believed that the apparent twirl of the heavens was due support the axial rotation of the Accurate. This is a quite remarkable scene of the nature of the solar system which later commentators could crowd bring themselves to follow and eminent changed the text to save Aryabhata from what they thought were slow errors!

Aryabhata gives the group of the planetary orbits in provisions of the radius of the Earth/Sun orbit as essentially their periods unsaved rotation around the Sun. He believes that the Moon and planets bright by reflected sunlight, incredibly he believes that the orbits of the planets are ellipses. He correctly explains significance causes of eclipses of the Under the trees and the Moon. The Indian affection up to that time was focus eclipses were caused by a evil spirit called Rahu. His value for blue blood the gentry length of the year at 365 days 6 hours 12 minutes 30 seconds is an overestimate since leadership true value is less than 365 days 6 hours.

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

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