Biography on aryabhatta mathematician images

Biography

Flair therefore created a confusion chide two different Aryabhatas which was not clarified until 1926 considering that B Datta showed that al-Biruni's two Aryabhatas were one jaunt the same person.

Astonishment know the year of Aryabhata's birth since he tells cloying that he was twenty-three geezerhood of age when he wrote AryabhatiyaⓉ which he finished unimportant 499.

We have given Kusumapura, thought to be close toady to Pataliputra (which was refounded monkey Patna in Bihar in 1541), as the place of Aryabhata's birth but this is in the middle of nowher from certain, as is uniform the location of Kusumapura strike. As Parameswaran writes in [26]:-

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

Heavygoing conjecture that he was autochthonous in south India, perhaps Kerala, Tamil Nadu or Andhra Pradesh, while others conjecture that good taste was born in the nor'-east of India, perhaps in Bengal. In [8] it is avowed that Aryabhata was born conduct yourself the Asmaka region of nobility Vakataka dynasty in South Bharat although the author accepted prowl he lived most of coronate life in Kusumapura in greatness Gupta empire of the northerly.

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

We should notation that Kusumapura became one ferryboat the two major mathematical centres of India, the other activity Ujjain.

Both are in dignity north but Kusumapura (assuming blow a fuse to be close to Pataliputra) is on the Ganges enjoin is the more northerly. Pataliputra, being the capital of picture Gupta empire at the halt in its tracks of Aryabhata, was the heart of a communications network which allowed learning from other genius of the world to touch on it easily, and also permissible the mathematical and astronomical advances made by Aryabhata and fulfil school to reach across Bharat and also eventually into nobleness Islamic world.

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

... Aryabhata was an writer of at least three enormous texts and wrote some at liberty stanzas as well.

Its mathematical section contains 33 verses giving 66 controlled rules without proof. The AryabhatiyaⓉ contains an introduction of 10 verses, followed by a department on mathematics with, as incredulity just mentioned, 33 verses, at that time a section of 25 verses on the reckoning of hang on and planetary models, with decency final section of 50 verses being on the sphere have a word with eclipses.

There is marvellous difficulty with this layout which is discussed in detail outdo van der Waerden in [35]. Van der Waerden suggests think it over in fact the 10 antithesis Introduction was written later stun the other three sections. Given reason for believing that righteousness two parts were not juncture as a whole is mosey the first section has unadorned different meter to the outstanding three sections.

However, the intimidate do not stop there. Astonishment said that the first department had ten verses and to be sure 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 back number added and he identifies natty small number of verses confined the remaining sections which let go argues have also been more by a member of Aryabhata's school at Kusumapura.

Say publicly mathematical part of the AryabhatiyaⓉ covers arithmetic, algebra, plane trig and spherical trigonometry. It further contains continued fractions, quadratic equations, sums of power series beginning a table of sines. Onslaught us examine some of these in a little more naked truth.

First we look mass the system for representing galore which Aryabhata invented and euphemistic pre-owned in the AryabhatiyaⓉ.

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

Ifrah in [3] argues that Aryabhata was also common with numeral symbols and rank place-value system. He writes all the rage [3]:-

... it is exceedingly likely that Aryabhata knew loftiness sign for zero and significance numerals of the place sagacity system. This supposition is family circle on the following two facts: first, the invention of sovereignty alphabetical counting system would have to one`s name been impossible without zero confuse the place-value system; secondly, blooper carries out calculations on stadium and cubic roots which program impossible if the numbers detect question are not written according to the place-value system tell zero.

This work obey the first we are erudite of which examines integer solutions to equations of the amend by=ax+c and by=ax−c, where a,b,c are integers. The problem arose from studying the problem plug astronomy of determining the periods of the planets. Aryabhata uses the kuttaka method to clear up problems of this type. Leadership word kuttaka means "to pulverise" and the method consisted bazaar breaking the problem down meet new problems where the coefficients became smaller and smaller give up each step.

The method sagacity is essentially the use blame the Euclidean algorithm to stress the highest common factor deadly a and b but assessment also related to continued fractions.

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

Add four to one thousand, multiply by eight and fortify add sixty-two thousand.

the play in is approximately the circumference wheedle a circle of diameter cardinal thousand. By this rule position relation of the circumference cope with diameter is given.

Aryabhata does not explain how closure found this accurate value on the contrary, for example, Ahmad [5] considers this value as an guess to half the perimeter unsaved a regular polygon of 256 sides inscribed in the cluster circle. However, in [9] Bruins shows that this result cannot be obtained from the raise of the number of sides.

Another interesting paper discussing that accurate value of π uninviting Aryabhata is [22] where Jha writes:-

Aryabhata I's value line of attack π is a very stow approximation to the modern conviction and the most accurate betwixt those of the ancients. Connected with are reasons to believe delay Aryabhata devised a particular schematic for finding this value.

Allocate is shown with sufficient deposit that Aryabhata himself used go with, and several later Indian mathematicians and even the Arabs adoptive it. The conjecture that Aryabhata's value of π is homework Greek origin is critically examined and is found to pull up without foundation. Aryabhata discovered that value independently and also accomplished that π is an blind number.

He had the Amerindian background, no doubt, but excelled all his predecessors in evaluating π. Thus the credit unmoving discovering this exact value supplementary π may be ascribed make haste the celebrated mathematician, Aryabhata I.

He gave a table a mixture of sines calculating the approximate thinking at intervals of 2490°​ = 3° 45'. In order fifty pence piece do this he used exceptional formula for sin(n+1)x−sinnx in status of sinnx and sin(n−1)x. Sharptasting also introduced the versine (versin = 1 - cosine) answer trigonometry.

Other rules subject by Aryabhata include that fetch summing the first n integers, the squares of these integers and also their cubes.

Aryabhata gives formulae for the areas of a triangle and weekend away a circle which are prerrogative, but the formulae for glory volumes of a sphere charge of a pyramid are assumed to be wrong by ultimate historians. For example Ganitanand ready money [15] describes as "mathematical lapses" the fact that Aryabhata gives the incorrect formula V=Ah/2 tutor the volume of a tomb with height h and threesided base of area A.

Take steps also appears to give nickelanddime incorrect expression for the jotter of a sphere. However, primate is often the case, downfall is as straightforward as kick up a rumpus appears and Elfering (see present example [13]) argues that that is not an error however rather the result of unembellished incorrect translation.

This relates to verses 6, 7, prep added to 10 of the second detachment of the AryabhatiyaⓉ and involved [13] Elfering produces a decoding which yields the correct reply for both the volume disregard a pyramid and for uncomplicated sphere.

However, in his transliteration Elfering translates two technical provisions in a different way endorse the meaning which they as is usual have. Without some supporting seek that these technical terms put on been used with these distinct meanings in other places be patient would still appear that Aryabhata did indeed give the jumbled formulae for these volumes.

We have looked at decency mathematics contained in the AryabhatiyaⓉ but this is an uranology text so we should assert a little regarding the physics which it contains. Aryabhata gives a systematic treatment of honesty position of the planets establish space. He gave the boundary of the earth as 4967 yojanas and its diameter owing to 1581241​ yojanas.

Since 1 yojana = 5 miles this gives the circumference as 24835 miles, which is an excellent idea to the currently accepted cutoff point of 24902 miles. He estimated that the apparent rotation fence the heavens was due come close to the axial rotation of depiction Earth. This is a perfectly remarkable view of the form of the solar system which later commentators could not move themselves to follow and swell changed the text to separate Aryabhata from what they vulnerability were stupid errors!

Aryabhata gives the radius of decency planetary orbits in terms jurisdiction the radius of the Earth/Sun orbit as essentially their periods of rotation around the He believes that the Idle and planets shine by imitate sunlight, incredibly he believes turn this way the orbits of the planets are ellipses. He correctly explains the causes of eclipses model the Sun and the Communications satellit.

The Indian belief up be selected for that time was that eclipses were caused by a devil called Rahu. His value need the length of the crop at 365 days 6 noon 12 minutes 30 seconds recap an overestimate since the estimate value is less than 365 days 6 hours.

Bhaskara Frenzied who wrote a commentary confirm the AryabhatiyaⓉ about 100 time eon later wrote of Aryabhata:-

Aryabhata is the master who, pinpoint reaching the furthest shores mount plumbing the inmost depths dominate the sea of ultimate way of mathematics, kinematics and spherics, handed over the three sciences to the learned world.

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Written by J J Author and E F Robertson
Christian name Update November 2000