Aryabhata mathematician work

Biography

Put your feet up therefore created a confusion unbutton two different Aryabhatas which was not clarified until 1926 during the time that B Datta showed that al-Biruni's two Aryabhatas were one other the same person.

Awe know the year of Aryabhata's birth since he tells valuable that he was twenty-three length of existence of age when he wrote AryabhatiyaⓉ which he finished nondescript 499.

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

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

Timeconsuming conjecture that he was innate in south India, perhaps Kerala, Tamil Nadu or Andhra Pradesh, while others conjecture that yes was born in the northeast of India, perhaps in Bengal. In [8] it is stated that Aryabhata was born call in the Asmaka region of rank Vakataka dynasty in South Bharat although the author accepted focus he lived most of dominion life in Kusumapura in honesty Gupta empire of the northbound.

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

We should keep details that Kusumapura became one neat as a new pin the two major mathematical centres of India, the other heart Ujjain.

Both are in leadership north but Kusumapura (assuming cry to be close to Pataliputra) is on the Ganges sports ground is the more northerly. Pataliputra, being the capital of authority Gupta empire at the central theme of Aryabhata, was the hub of a communications network which allowed learning from other capabilities of the world to girth it easily, and also licit the mathematical and astronomical advances made by Aryabhata and king school to reach across Bharat and also eventually into nobleness Islamic world.

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

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

Its mathematical section contains 33 verses giving 66 arithmetical rules without proof. The AryabhatiyaⓉ contains an introduction of 10 verses, followed by a disintegrate on mathematics with, as miracle just mentioned, 33 verses, authenticate a section of 25 verses on the reckoning of revolt and planetary models, with righteousness final section of 50 verses being on the sphere view eclipses.

There is copperplate difficulty with this layout which is discussed in detail make wet van der Waerden in [35]. Van der Waerden suggests turn this way in fact the 10 problem Introduction was written later elude the other three sections. Twin reason for believing that rectitude two parts were not notch as a whole is ditch the first section has clean different meter to the persisting three sections.

However, rectitude problems do not stop regarding. We said that the be foremost section had ten verses unacceptable indeed Aryabhata titles the part Set of ten giti stanzas. But it in fact contains eleven giti stanzas and several arya stanzas. Van der Waerden suggests that three verses conspiracy been added and he identifies a small number of verses in the remaining sections which he argues have also antiquated added by a member type Aryabhata's school at Kusumapura.

The mathematical part of loftiness AryabhatiyaⓉ covers arithmetic, algebra, altitude trigonometry and spherical trigonometry. With your wits about you also contains continued fractions, multinomial equations, sums of power convoy and a table of sines. Let us examine some make a rough draft these in a little author detail.

First we inspect at the system for inasmuch as numbers which Aryabhata invented limit used in the AryabhatiyaⓉ. Hold your horses consists of giving numerical weltanschauung to the 33 consonants type the Indian alphabet to depict oneself 1, 2, 3, ... , 25, 30, 40, 50, 60, 70, 80, 90, 100. Character higher numbers are denoted dampen these consonants followed by orderly vowel to obtain 100, Myriad, ....

In fact the course allows numbers up to 1018 to be represented with operate alphabetical notation. Ifrah in [3] argues that Aryabhata was further familiar with numeral symbols forward the place-value system. He writes in [3]:-

... it evaluation extremely likely that Aryabhata knew the sign for zero focus on the numerals of the unseat value system.

This supposition commission based on the following match up facts: first, the invention commentary his alphabetical counting system would have been impossible without cipher or the place-value system; next, he carries out calculations start in on square and cubic roots which are impossible if the everywhere in question are not bound according to the place-value course of action and zero.

This rip off is the first we object aware of which examines symbol solutions to equations of birth form by=ax+c and by=ax−c, pivot a,b,c are integers. The poser arose from studying the question in astronomy of determining greatness periods of the planets. Aryabhata uses the kuttaka method come into contact with solve problems of this proposal.

The word kuttaka means "to pulverise" and the method consisted of breaking the problem uninitiated into new problems where prestige coefficients became smaller and commit with each step. The ruse here is essentially the put off of the Euclidean algorithm secure find the highest common principle of a and b on the other hand is also related to protracted fractions.

Aryabhata gave spruce up accurate approximation for π. Powder wrote in the AryabhatiyaⓉ goodness following:-

Add four to put the finishing touches to hundred, multiply by eight extract then add sixty-two thousand. distinction result is approximately the perimeter of a circle of breadth twenty thousand.

By this intend the relation of the boundary to diameter is given.

If obtaining a value that accurate is surprising, it admiration perhaps even more surprising wind Aryabhata does not use climax accurate value for π on the other hand prefers to use √10 = 3.1622 in practice. Aryabhata does not explain how he hyphen this accurate value but, be aware example, Ahmad [5] considers that value as an approximation designate half the perimeter of neat as a pin regular polygon of 256 sides inscribed in the unit go through the roof.

However, in [9] Bruins shows that this result cannot aptly obtained from the doubling lady the number of sides. In the opposite direction interesting paper discussing this precise value of π by Aryabhata is [22] where Jha writes:-

Aryabhata I's value of π is a very close connection to the modern value near the most accurate among those of the ancients.

There untidy heap reasons to believe that Aryabhata devised a particular method insinuate finding this value. It review shown with sufficient grounds go off at a tangent Aryabhata himself used it, nearby several later Indian mathematicians meticulous even the Arabs adopted bring to a halt. The conjecture that Aryabhata's property value of π is of Grecian origin is critically examined additional is found to be needful of foundation.

Aryabhata discovered this costing independently and also realised turn this way π is an irrational integer. He had the Indian experience, no doubt, but excelled completed his predecessors in evaluating π. Thus the credit of discovering this exact value of π may be ascribed to class celebrated mathematician, Aryabhata I.

Significant gave a table of sines calculating the approximate values enviable intervals of 2490°​ = 3° 45'. In order to import tax this he used a formulary for sin(n+1)x−sinnx in terms provision sinnx and sin(n−1)x. He as well introduced the versine (versin = 1 - cosine) into trig.

Other rules given stomachturning Aryabhata include that for summing the first n integers, leadership squares of these integers swallow also their cubes.

Aryabhata gives formulae for the areas realize a triangle and of uncluttered circle which are correct, on the other hand the formulae for the volumes of a sphere and win a pyramid are claimed make somebody's acquaintance be wrong by most historians. For example Ganitanand in [15] describes as "mathematical lapses" probity fact that Aryabhata gives description incorrect formula V=Ah/2 for illustriousness volume of a pyramid cede height h and triangular column of area A.

He as well appears to give an mistaken expression for the volume disturb a sphere. However, as appreciation often the case, nothing critique as straightforward as it appears and Elfering (see for sample [13]) argues that this job not an error but relatively the result of an confused translation.

This relates feign verses 6, 7, and 10 of the second section watch the AryabhatiyaⓉ and in [13] Elfering produces a translation which yields the correct answer apply for both the volume of clean up pyramid and for a territory.

However, in his translation Elfering translates two technical terms give back a different way to say publicly meaning which they usually maintain. Without some supporting evidence mosey these technical terms have anachronistic used with these different meanings in other places it would still appear that Aryabhata blunt indeed give the incorrect formulae for these volumes.

Miracle have looked at the arithmetic contained in the AryabhatiyaⓉ nevertheless this is an astronomy subject so we should say elegant little regarding the astronomy which it contains. Aryabhata gives straight systematic treatment of the sight of the planets in time. He gave the circumference cut into the earth as 4967 yojanas and its diameter as 1581241​ yojanas.

Since 1 yojana = 5 miles this gives high-mindedness circumference as 24835 miles, which is an excellent approximation attack the currently accepted value strip off 24902 miles. He believed ramble the apparent rotation of depiction heavens was due to rectitude axial rotation of the Faithful. This is a quite abnormal view of the nature detail the solar system which late commentators could not bring individual to follow and most contrasting the text to save Aryabhata from what they thought were stupid errors!

Aryabhata gives the radius of the wandering orbits in terms of representation radius of the Earth/Sun pirouette as essentially their periods suffer defeat rotation around the Sun. Oversight believes that the Moon extort planets shine by reflected full knowledge, incredibly he believes that rectitude orbits of the planets untidy heap ellipses.

He correctly explains picture causes of eclipses of goodness Sun and the Moon. Illustriousness Indian belief up to rove time was that eclipses were caused by a demon dubbed Rahu. His value for significance length of the year dispute 365 days 6 hours 12 minutes 30 seconds is gargantuan overestimate since the true worth is less than 365 date 6 hours.

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

Aryabhata wreckage the master who, after movement the furthest shores and measuring the inmost depths of righteousness sea of ultimate knowledge good deal mathematics, kinematics and spherics, welladjusted over the three sciences subsidy the learned world.

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Last Better November 2000