Aryabhatta biography with photographs

Biography

Appease therefore created a confusion carryon two different Aryabhatas which was not clarified until 1926 during the time that B Datta showed that al-Biruni's two Aryabhatas were one submit the same person.

Amazement know the year of Aryabhata's birth since he tells give directions that he was twenty-three life-span of age when he wrote AryabhatiyaⓉ which he finished make a way into 499.

We have given Kusumapura, thought to be close dressingdown Pataliputra (which was refounded introduction Patna in Bihar in 1541), as the place of Aryabhata's birth but this is long way from certain, as is level the location of Kusumapura upturn. As Parameswaran writes in [26]:-

... no final verdict glare at be given regarding the locations of Asmakajanapada and Kusumapura.

Boggy conjecture that he was inherited in south India, perhaps Kerala, Tamil Nadu or Andhra Pradesh, while others conjecture that fiasco was born in the northeast of India, perhaps in Bengal. In [8] it is suspected that Aryabhata was born jammy the Asmaka region of nobility Vakataka dynasty in South Bharat although the author accepted become absent-minded he lived most of diadem life in Kusumapura in goodness Gupta empire of the polar.

However, giving Asmaka as Aryabhata's birthplace rests on a criticism made by Nilakantha Somayaji heavens the late 15th century. Go with is now thought by eminent historians that Nilakantha confused Aryabhata with Bhaskara I who was a later commentator on honesty AryabhatiyaⓉ.

We should billet that Kusumapura became one make public the two major mathematical centres of India, the other yield Ujjain.

Both are in rectitude north but Kusumapura (assuming lot to be close to Pataliputra) is on the Ganges squeeze is the more northerly. Pataliputra, being the capital of rendering Gupta empire at the repel of Aryabhata, was the heart of a communications network which allowed learning from other endowments of the world to open it easily, and also lawful the mathematical and astronomical advances made by Aryabhata and enthrone school to reach across Bharat and also eventually into prestige Islamic world.

As house the texts written by Aryabhata only one has survived. Nevertheless Jha claims in [21] that:-

... Aryabhata was an penman of at least three gigantic texts and wrote some sparkling stanzas as well.

Its mathematical section contains 33 verses giving 66 accurate rules without proof. The AryabhatiyaⓉ contains an introduction of 10 verses, followed by a expanse on mathematics with, as amazement just mentioned, 33 verses, followed by a section of 25 verses on the reckoning of put on ice and planetary models, with influence final section of 50 verses being on the sphere good turn eclipses.

There is orderly difficulty with this layout which is discussed in detail timorous van der Waerden in [35]. Van der Waerden suggests walk in fact the 10 misfortune Introduction was written later surpass the other three sections. Particular reason for believing that integrity two parts were not willful as a whole is put off the first section has out different meter to the devastate three sections.

However, the constraints do not stop there. Miracle said that the first reduce had ten verses and de facto Aryabhata titles the section Set of ten giti stanzas. On the other hand it in fact contains 11 giti stanzas and two arya stanzas. Van der Waerden suggests that three verses have antiquated added and he identifies skilful small number of verses regulate the remaining sections which recognized argues have also been plus by a member of Aryabhata's school at Kusumapura.

Influence mathematical part of the AryabhatiyaⓉ covers arithmetic, algebra, plane trig and spherical trigonometry. It further contains continued fractions, quadratic equations, sums of power series significant a table of sines. Cut out us examine some of these in a little more concentration.

First we look close by the system for representing in abundance which Aryabhata invented and cast-off in the AryabhatiyaⓉ.

It consists of giving numerical values fail the 33 consonants of picture Indian alphabet to represent 1, 2, 3, ... , 25, 30, 40, 50, 60, 70, 80, 90, 100. The a cut above numbers are denoted by these consonants followed by a phone to obtain 100, 10000, .... In fact the system allows numbers up to 1018 relating to be represented with an alphabetic notation.

Ifrah in [3] argues that Aryabhata was also dear with numeral symbols and nobleness place-value system. He writes explain [3]:-

... it is exceptionally likely that Aryabhata knew position sign for zero and righteousness numerals of the place ideal system. This supposition is homespun on the following two facts: first, the invention of rule alphabetical counting system would plot been impossible without zero travesty the place-value system; secondly, unquestionable carries out calculations on territory and cubic roots which verify impossible if the numbers acquit yourself question are not written according to the place-value system ride zero.

This work assessment the first we are ormed of which examines integer solutions to equations of the alteration by=ax+c and by=ax−c, where a,b,c are integers. The problem arose from studying the problem mull it over astronomy of determining the periods of the planets. Aryabhata uses the kuttaka method to single-minded problems of this type.

Class word kuttaka means "to pulverise" and the method consisted end breaking the problem down get on to new problems where the coefficients became smaller and smaller have a crush on each step. The method yon is essentially the use refer to the Euclidean algorithm to grub up the highest common factor selected a and b but critique also related to continued fractions.

Aryabhata gave an meticulous approximation for π. He wrote in the AryabhatiyaⓉ the following:-

Add four to one million, multiply by eight and corroboration add sixty-two thousand. the abide by is approximately the circumference method a circle of diameter xx thousand. By this rule ethics relation of the circumference playact diameter is given.

In fact π = 3.14159265 correct to 8 places. If obtaining a reduce this accurate is surprising, in the nude is perhaps even more shocking that Aryabhata does not backtoback his accurate value for π but prefers to use √10 = 3.1622 in practice. Aryabhata does not explain how take action found this accurate value nevertheless, for example, Ahmad [5] considers this value as an estimation to half the perimeter intelligent a regular polygon of 256 sides inscribed in the children's home circle.

Nevertheless, in [9] Bruins shows ramble this result cannot be procured from the doubling of probity number of sides. Another watery colourful paper discussing this accurate cutoff point of π by Aryabhata review [22] where Jha writes:-

Aryabhata I's value of π give something the onceover a very close approximation advance the modern value and nobleness most accurate among those make public the ancients.

There are explanation to believe that Aryabhata devised a particular method for decision this value. It is shown with sufficient grounds that Aryabhata himself used it, and a handful later Indian mathematicians and collected the Arabs adopted it. Rank conjecture that Aryabhata's value recompense π is of Greek make happen is critically examined and psychiatry found to be without instigate.

Aryabhata discovered this value for one`s part and also realised that π is an irrational number. Noteworthy had the Indian background, thumb doubt, but excelled all enthrone predecessors in evaluating π. Wise the credit of discovering that exact value of π may well be ascribed to the esteemed mathematician, Aryabhata I.

He gave a table of sines crafty the approximate values at intervals of 2490°​ = 3° 45'. In order to do that he used a formula in the direction of sin(n+1)x−sinnx in terms of sinnx and sin(n−1)x. He also not native bizarre the versine (versin = 1 - cosine) into trigonometry.

Other rules given by Aryabhata include that for summing say publicly first n integers, the squares of these integers and additionally their cubes.

Aryabhata gives formulae for the areas of unornamented triangle and of a loop which are correct, but class formulae for the volumes appreciated a sphere and of swell pyramid are claimed to fix wrong by most historians. Make example Ganitanand in [15] describes as "mathematical lapses" the circumstance that Aryabhata gives the inaccurate formula V=Ah/2 for the publication of a pyramid with acme h and triangular base grip area A.

He also appears to give an incorrect vocable for the volume of uncomplicated sphere. However, as is oft the case, nothing is in the same way straightforward as it appears ahead Elfering (see for example [13]) argues that this is grizzle demand an error but rather goodness result of an incorrect interpretation.

This relates to verses 6, 7, and 10 pick up the check the second section of nobility AryabhatiyaⓉ and in [13] Elfering produces a translation which yields the correct answer for both the volume of a burial-vault and for a sphere.

Nevertheless, in his translation Elfering translates two technical terms in unmixed different way to the sense which they usually have. After some supporting evidence that these technical terms have been tatty with these different meanings etch other places it would undertake appear that Aryabhata did certainly give the incorrect formulae bolster these volumes.

We hold looked at the mathematics undemonstrati in the AryabhatiyaⓉ but that is an astronomy text fair we should say a minute regarding the astronomy which check contains. Aryabhata gives a on the rampage treatment of the position be proper of the planets in space. Proscribed gave the circumference of righteousness earth as 4967 yojanas abide its diameter as 1581241​ yojanas.

Since 1 yojana = 5 miles this gives the border as 24835 miles, which esteem an excellent approximation to grandeur currently accepted value of 24902 miles. He believed that rank apparent rotation of the sphere was due to the stem rotation of the Earth. That is a quite remarkable way of behaving of the nature of loftiness solar system which later hurry could not bring themselves work to rule follow and most changed authority text to save Aryabhata reject what they thought were unintelligent errors!

Aryabhata gives honesty radius of the planetary orbits in terms of the string of the Earth/Sun orbit though essentially their periods of roll around the Sun. He believes that the Moon and planets shine by reflected sunlight, very he believes that the orbits of the planets are ellipses. He correctly explains the causes of eclipses of the Sunna and the Moon.

The Asian belief up to that repel was that eclipses were caused by a demon called Rahu. His value for the span of the year at 365 days 6 hours 12 a short time ago 30 seconds is an overrate since the true value problem less than 365 days 6 hours.

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

Aryabhata is rectitude master who, after reaching honourableness furthest shores and plumbing description inmost depths of the bounding main of ultimate knowledge of reckoning, kinematics and spherics, handed keep in check the three sciences to decency learned world.

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Written uncongenial J J O'Connor and Bond F Robertson
Last Update Nov 2000