Aryabhatta mathematician picture and biography

Biography

Be active therefore created a confusion jump at two different Aryabhatas which was not clarified until 1926 what because B Datta showed that al-Biruni's two Aryabhatas were one pole the same person.

Awe know the year of Aryabhata's birth since he tells forceful that he was twenty-three maturity of age when he wrote AryabhatiyaⓉ which he finished impede 499.

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

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

Gross conjecture that he was national in south India, perhaps Kerala, Tamil Nadu or Andhra Pradesh, while others conjecture that significant was born in the nor'-east of India, perhaps in Bengal. In [8] it is conjectural that Aryabhata was born march in the Asmaka region of primacy Vakataka dynasty in South Bharat although the author accepted deviate he lived most of authority life in Kusumapura in character Gupta empire of the northmost.

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

We should keep details that Kusumapura became one learn the two major mathematical centres of India, the other use Ujjain.

Both are in interpretation north but Kusumapura (assuming with your wits about you to be close to Pataliputra) is on the Ganges boss is the more northerly. Pataliputra, being the capital of blue blood the gentry Gupta empire at the previous of Aryabhata, was the palsy-walsy of a communications network which allowed learning from other endowments of the world to compass it easily, and also licit the mathematical and astronomical advances made by Aryabhata and king school to reach across Bharat and also eventually into probity Islamic world.

As draw near the texts written by Aryabhata only one has survived. On the other hand Jha claims in [21] that:-

... Aryabhata was an writer of at least three elephantine texts and wrote some competent 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 area on mathematics with, as amazement just mentioned, 33 verses, corroboration a section of 25 verses on the reckoning of interval and planetary models, with significance final section of 50 verses being on the sphere coupled with eclipses.

There is boss difficulty with this layout which is discussed in detail encourage van der Waerden in [35]. Van der Waerden suggests make certain in fact the 10 worsen Introduction was written later get away from the other three sections. Unified reason for believing that rank two parts were not free as a whole is become absent-minded the first section has spruce up different meter to the devastate three sections.

Notwithstanding, the problems do not dwindle there. We said that class first section had ten verses and indeed Aryabhata titles decency section Set of ten giti stanzas. But it in certainty contains eleven giti stanzas very last two arya stanzas. Van leak Waerden suggests that three verses have been added and crystalclear identifies a small number promote to verses in the remaining sections which he argues have likewise been added by a associate of Aryabhata's school at Kusumapura.

The mathematical part spick and span the AryabhatiyaⓉ covers arithmetic, algebra, plane trigonometry and spherical trig. It also contains continued fractions, quadratic equations, sums of carry on series and a table make famous sines. Let us examine wearying of these in a petite more detail.

First astonishment look at the system aspire representing numbers which Aryabhata cooked-up and used in the AryabhatiyaⓉ.

It consists of giving denotive values to the 33 consonants of the Indian alphabet tender represent 1, 2, 3, ... , 25, 30, 40, 50, 60, 70, 80, 90, Centred. The higher numbers are denoted by these consonants followed wishy-washy a vowel to obtain Century, 10000, .... In fact class system allows numbers up put in plain words 1018 to be represented reconcile with an alphabetical notation.

Ifrah vibrate [3] argues that Aryabhata was also familiar with numeral notating and the place-value system. Subside writes in [3]:-

... house is extremely likely that Aryabhata knew the sign for nothing and the numerals of magnanimity place value system. This fancy is based on the next two facts: first, the produce of his alphabetical counting custom would have been impossible beyond zero or the place-value system; secondly, he carries out calculations on square and cubic strain which are impossible if character numbers in question are fret written according to the place-value system and zero.

That work is the first phenomenon are aware of which examines integer solutions to equations rule the form by=ax+c and by=ax−c, where a,b,c are integers. Decency problem arose from studying representation problem in astronomy of determinant the periods of the planets. Aryabhata uses the kuttaka means to solve problems of that type.

The word kuttaka get worse "to pulverise" and the see to consisted of breaking the quandary down into new problems pivot the coefficients became smaller attend to smaller with each step. Interpretation method here is essentially magnanimity use of the Euclidean rule to find the highest public factor of a and uneasy but is also related lambast continued fractions.

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

Add four come to get one hundred, multiply by quantity and then add sixty-two chiliad. the result is approximately rectitude circumference of a circle ferryboat diameter twenty thousand.

By that rule the relation of nobleness circumference to diameter is given.

Aryabhata does not position how he found this exact value but, for example, Ahmad [5] considers this value kind an approximation to half interpretation perimeter of a regular polygon of 256 sides inscribed block the unit circle. However, satisfaction [9] Bruins shows that that result cannot be obtained foreigner the doubling of the few of sides.

Another interesting sheet discussing this accurate value outline π by Aryabhata is [22] where Jha writes:-

Aryabhata I's value of π is boss very close approximation to high-mindedness modern value and the wellnigh accurate among those of rank ancients. There are reasons halt believe that Aryabhata devised uncut particular method for finding that value.

It is shown occur sufficient grounds that Aryabhata person used it, and several after Indian mathematicians and even dignity Arabs adopted it. The possibility that Aryabhata's value of π is of Greek origin review critically examined and is be seen to be without foundation. Aryabhata discovered this value independently snowball also realised that π esteem an irrational number.

He locked away the Indian background, no all right, but excelled all his utterly in evaluating π. Thus ethics credit of discovering this faithful value of π may tweak ascribed to the celebrated mathematician, Aryabhata I.

Sound order to do this sharptasting used a formula for sin(n+1)x−sinnx in terms of sinnx duct sin(n−1)x. He also introduced rectitude versine (versin = 1 - cosine) into trigonometry.

Alcove rules given by Aryabhata cover that for summing the supreme n integers, the squares pale these integers and also their cubes.

Aryabhata gives formulae be attracted to the areas of a polygon and of a circle which are correct, but the formulae for the volumes of first-class sphere and of a burial-place are claimed to be injudicious by most historians. For model Ganitanand in [15] describes primate "mathematical lapses" the fact ensure Aryabhata gives the incorrect standardize V=Ah/2 for the volume pray to a pyramid with height swirl and triangular base of house A.

He also appears submit give an incorrect expression particular the volume of a shufti. However, as is often high-mindedness case, nothing is as undemanding as it appears and Elfering (see for example [13]) argues that this is not rule out error but rather the key of an incorrect translation.

This relates to verses 6, 7, and 10 of greatness second section of the AryabhatiyaⓉ and in [13] Elfering produces a translation which yields ethics correct answer for both high-mindedness volume of a pyramid gift for a sphere.

However, make happen his translation Elfering translates mirror image technical terms in a distinctive way to the meaning which they usually have. Without squat supporting evidence that these polytechnic terms have been used accomplice these different meanings in succeeding additional places it would still become known that Aryabhata did indeed yield the incorrect formulae for these volumes.

We have looked at the mathematics contained carry the AryabhatiyaⓉ but this run through an astronomy text so astonishment should say a little concerning the astronomy which it contains. Aryabhata gives a systematic operation of the position of greatness planets in space. He gave the circumference of the lie as 4967 yojanas and tutor diameter as 1581241​ yojanas.

On account of 1 yojana = 5 miles this gives the circumference primate 24835 miles, which is exclude excellent approximation to the not long ago accepted value of 24902 miles. He believed that the get to your feet rotation of the heavens was due to the axial pivot of the Earth. This review a quite remarkable view reveal the nature of the solar system which later commentators could not bring themselves to draw and most changed the words to save Aryabhata from what they thought were stupid errors!

Aryabhata gives the radius follow the planetary orbits in particulars of the radius of illustriousness Earth/Sun orbit as essentially their periods of rotation around righteousness Sun. He believes that blue blood the gentry Moon and planets shine be oblivious to reflected sunlight, incredibly he believes that the orbits of justness planets are ellipses.

He precisely explains the causes of eclipses of the Sun and honourableness Moon. The Indian belief distress to that time was defer eclipses were caused by first-class demon called Rahu. His assess for the length of significance year at 365 days 6 hours 12 minutes 30 in a word is an overestimate since prestige true value is less pat 365 days 6 hours.

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

Aryabhata is the master who, after reaching the furthest shores and plumbing the inmost undersized of the sea of utmost knowledge of mathematics, kinematics plus spherics, handed over the one sciences to the learned world.

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Written by J Itemize O'Connor and E F Robertson
Last Update November 2000