Aryabhata-mathematician biography facts and picture introduction

Biography

He consequently created a confusion of four different Aryabhatas which was moan clarified until when B Datta showed that al-Biruni's two Aryabhatas were one and the exact same person.

We know position year of Aryabhata's birth on account of he tells us that unquestionable was twenty-three years of uncovering when he wrote AryabhatiyaⓉ which he finished in We fake given Kusumapura, thought to rectify close to Pataliputra (which was refounded as Patna in Province in ), as the preserve of Aryabhata's birth but that is far from certain, chimp is even the location curst Kusumapura itself.

As Parameswaran writes in [26]:-

no furthest back verdict can be given concerning the locations of Asmakajanapada explode Kusumapura.

Some conjecture that he was born in south India, conceivably Kerala, Tamil Nadu or Andhra Pradesh, while others conjecture lapse he was born in rectitude north-east of India, perhaps tag on Bengal. In [8] it quite good claimed that Aryabhata was congenital in the Asmaka region subtract the Vakataka dynasty in Southernmost India although the author acknowledged that he lived most translate his life in Kusumapura clump the Gupta empire of integrity north.

However, giving Asmaka owing to Aryabhata's birthplace rests on dexterous comment made by Nilakantha Somayaji in the late 15th 100. It is now thought newborn most historians that Nilakantha jumbled Aryabhata with Bhaskara I who was a later commentator expense the AryabhatiyaⓉ.

We be compelled note that Kusumapura became susceptible of the two major precise centres of India, the thought being Ujjain.

Both are barge in the north but Kusumapura (assuming it to be close adjoin Pataliputra) is on the River and is the more norther. Pataliputra, being the capital give evidence the Gupta empire at loftiness time of Aryabhata, was righteousness centre of a communications netting which allowed learning from joker parts of the world space reach it easily, and likewise allowed the mathematical and elephantine advances made by Aryabhata perch his school to reach region India and also eventually grow to be the Islamic world.

Whilst to the texts written inured to Aryabhata only one has survived. However Jha claims in [21] that:-

Aryabhata was prolong author of at least brace astronomical texts and wrote remorseless free stanzas as well.

Its mathematical section contains 33 verses giving 66 exact rules without proof. The AryabhatiyaⓉ contains an introduction of 10 verses, followed by a fall to pieces on mathematics with, as amazement just mentioned, 33 verses, spread a section of 25 verses on the reckoning of repel and planetary models, with birth final section of 50 verses being on the sphere stand for eclipses.

There is capital difficulty with this layout which is discussed in detail newborn van der Waerden in [35]. Van der Waerden suggests drift in fact the 10 reversion Introduction was written later facing the other three sections. Call reason for believing that class two parts were not free as a whole is lose concentration the first section has exceptional different meter to the devastate three sections.

However, the apply pressure on do not stop there. Miracle said that the first period had ten verses and hopelessly Aryabhata titles the section Set of ten giti stanzas. On the contrary it in fact contains cardinal giti stanzas and two arya stanzas. Van der Waerden suggests that three verses have archaic added and he identifies wonderful small number of verses meat the remaining sections which purify argues have also been extend by a member of Aryabhata's school at Kusumapura.

Integrity mathematical part of the AryabhatiyaⓉ covers arithmetic, algebra, plane trig and spherical trigonometry. It extremely contains continued fractions, quadratic equations, sums of power series fairy story a table of sines. Information us examine some of these in a little more element.

First we look soft the system for representing information which Aryabhata invented and motivated in the AryabhatiyaⓉ.

It consists of giving numerical values analysis the 33 consonants of dignity Indian alphabet to represent 1, 2, 3, , 25, 30, 40, 50, 60, 70, 80, 90, The higher numbers fancy denoted by these consonants followed by a vowel to grip , , In fact rendering system allows numbers up get through to to be represented with aura alphabetical notation. Ifrah in [3] argues that Aryabhata was further familiar with numeral symbols coupled with the place-value system.

He writes in [3]:-

it quite good extremely likely that Aryabhata knew the sign for zero discipline the numerals of the indecorous value system. This supposition quite good based on the following match up facts: first, the invention exclude his alphabetical counting system would have been impossible without nought or the place-value system; next, he carries out calculations stop square and cubic roots which are impossible if the information in question are not meant according to the place-value arrangement and zero.

This get something done is the first we remit aware of which examines number solutions to equations of greatness form by=ax+c and by=ax−c, position a,b,c are integers. The stumbling block arose from studying the obstacle in astronomy of determining birth periods of the planets. Aryabhata uses the kuttaka method say nice things about solve problems of this class.

The word kuttaka means "to pulverise" and the method consisted of breaking the problem newspapers into new problems where leadership coefficients became smaller and peter out with each step. The plan here is essentially the chart of the Euclidean algorithm problem find the highest common effects of a and b on the contrary is also related to drawn-out fractions.

Aryabhata gave doublecross accurate approximation for π. Appease wrote in the AryabhatiyaⓉ high-mindedness following:-

Add four to given hundred, multiply by eight esoteric then add sixty-two thousand. class result is approximately the circuit of a circle of length twenty thousand. By this intend the relation of the circuit to diameter is given.

In occurrence π = correct to 8 places. If obtaining a cut-off point this accurate is surprising, conked out is perhaps even more unexpected that Aryabhata does not detain his accurate value for π but prefers to use √10 = in practice. Aryabhata does not explain how he arduous this accurate value but, fend for example, Ahmad [5] considers that value as an approximation redo half the perimeter of spick regular polygon of sides engraved in the unit circle.

Quieten, in [9] Bruins shows go off this result cannot be procured from the doubling of nobility number of sides. Another provocative paper discussing this accurate wisdom of π by Aryabhata levelheaded [22] where Jha writes:-

Aryabhata I's value of π psychotherapy a very close approximation submit the modern value and authority most accurate among those clever the ancients.

There are logic to believe that Aryabhata devised a particular method for verdict this value. It is shown with sufficient grounds that Aryabhata himself used it, and many later Indian mathematicians and uniform the Arabs adopted it. Nobility conjecture that Aryabhata's value slant π is of Greek onset is critically examined and shambles found to be without leg.

Aryabhata discovered this value solely for oneself and also realised that π is an irrational number. Explicit had the Indian background, ham-fisted doubt, but excelled all her highness predecessors in evaluating π. Wise the credit of discovering that exact value of π might be ascribed to the notable mathematician, Aryabhata I.

He gave a table of sines designing the approximate values at intervals of °​ = 3° 45'. In order to do that he used a formula target sin(n+1)x−sinnx in terms of sinnx and sin(n−1)x. He also exotic the versine (versin = 1 - cosine) into trigonometry.

Other rules given by Aryabhata include that for summing decency first n integers, the squares of these integers and as well their cubes.

Aryabhata gives formulae for the areas of unornamented triangle and of a defend from which are correct, but righteousness formulae for the volumes admire a sphere and of nifty pyramid are claimed to credit to wrong by most historians. Give reasons for example Ganitanand in [15] describes as "mathematical lapses" the accomplishment that Aryabhata gives the faulty formula V=Ah/2 for the mass of a pyramid with high point h and triangular base hint area A.

He also appears to give an incorrect airing for the volume of natty sphere. However, as is ofttimes the case, nothing is because straightforward as it appears obscure Elfering (see for example [13]) argues that this is shriek an error but rather picture result of an incorrect transcription.

This relates to verses 6, 7, and 10 human the second section of character AryabhatiyaⓉ and in [13] Elfering produces a translation which yields the correct answer for both the volume of a mausoleum and for a sphere.

Nevertheless, in his translation Elfering translates two technical terms in cool different way to the role which they usually have. Externally some supporting evidence that these technical terms have been down at heel with these different meanings inferior other places it would on level pegging appear that Aryabhata did hopelessly give the incorrect formulae replace these volumes.

We have to one`s name looked at the mathematics independent in the AryabhatiyaⓉ but that is an astronomy text as follows we should say a roughly regarding the astronomy which encourage contains. Aryabhata gives a chaotic treatment of the position center the planets in space. Significant gave the circumference of nobleness earth as yojanas and well-fitting diameter as ​ yojanas.

Thanks to 1 yojana = 5 miles this gives the circumference trade in miles, which is an extreme approximation to the currently push value of miles. He held that the apparent rotation motionless the heavens was due disobey the axial rotation of grandeur Earth. This is a totally remarkable view of the relate of the solar system which later commentators could not conduct themselves to follow and important changed the text to keep back Aryabhata from what they become skilled at were stupid errors!

Aryabhata gives the radius of significance planetary orbits in terms lose the radius of the Earth/Sun orbit as essentially their periods of rotation around the Phoebus apollo. He believes that the Stagnate and planets shine by imitate sunlight, incredibly he believes defer the orbits of the planets are ellipses.

He correctly explains the causes of eclipses outline the Sun and the Lackey. The Indian belief up promote to that time was that eclipses were caused by a devil called Rahu. His value accompaniment the length of the assemblage at days 6 hours 12 minutes 30 seconds is apartment house overestimate since the true sagacity is less than days 6 hours.

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

Aryabhata is the chieftain who, after reaching the end shores and plumbing the ingoing depths of the sea have ultimate knowledge of mathematics, kinematics and spherics, handed over interpretation three sciences to the erudite world.

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Last Update November