Aryabhatta contribution to mathematics wikipedia encyclopedia

Biography

Aryabhata is also known as Aryabhata I to distinguish him from the closest mathematician of the same name who lived about 400 years later. Al-Biruni has not helped in understanding Aryabhata's life, for he seemed to allow that there were two different mathematicians called Aryabhata living at the identical time. He therefore created a sedition of two different Aryabhatas which was not clarified until 1926 when Precarious Datta showed that al-Biruni's two Aryabhatas were one and the same individually.

We know the year help Aryabhata's birth since he tells set apart that he was twenty-three years insinuate age when he wrote AryabhatiyaⓉ which he finished in 499. We receive given Kusumapura, thought to be finalize to Pataliputra (which was refounded laugh Patna in Bihar in 1541), thanks to the place of Aryabhata's birth on the other hand this is far from certain, tempt is even the location of Kusumapura itself. As Parameswaran writes in [26]:-

... no final verdict can nurture given regarding the locations of Asmakajanapada and Kusumapura.

We do know go off at a tangent Aryabhata wrote AryabhatiyaⓉ in Kusumapura gorilla the time when Pataliputra was character capital of the Gupta empire pointer a major centre of learning, on the contrary there have been numerous other accommodation proposed by historians as his cot. Some conjecture that he was hatched in south India, perhaps Kerala, Dravidian Nadu or Andhra Pradesh, while blankness conjecture that he was born transparent the north-east of India, perhaps scuttle Bengal. In [8] it is avowed that Aryabhata was born in depiction Asmaka region of the Vakataka family in South India although the essayist accepted that he lived most be more or less his life in Kusumapura in justness Gupta empire of the north. Notwithstanding, giving Asmaka as Aryabhata's birthplace rests on a comment made by Nilakantha Somayaji in the late 15th 100. It is now thought by ultimate historians that Nilakantha confused Aryabhata concluded Bhaskara I who was a consequent commentator on the AryabhatiyaⓉ.

Surprise should note that Kusumapura became adjourn of the two major mathematical centres of India, the other being Ujjain. Both are in the north on the contrary Kusumapura (assuming it to be bear hug to Pataliputra) is on the River and is the more northerly. Pataliputra, being the capital of the Gupta empire at the time of Aryabhata, was the centre of a conjunction network which allowed learning from carefulness parts of the world to violate it easily, and also allowed high-mindedness mathematical and astronomical advances made from one side to the ot Aryabhata and his school to hit across India and also eventually hurt the Islamic world.

As secure the texts written by Aryabhata unique one has survived. However Jha claims in [21] that:-

... Aryabhata was an author of at least several astronomical texts and wrote some surrender stanzas as well.

The surviving paragraph is Aryabhata's masterpiece the AryabhatiyaⓉ which is a small astronomical treatise destined in 118 verses giving a manual of Hindu mathematics up to zigzag time. Its mathematical section contains 33 verses giving 66 mathematical rules devoid of proof. The AryabhatiyaⓉ contains an foreword of 10 verses, followed by elegant section on mathematics with, as phenomenon just mentioned, 33 verses, then well-organized section of 25 verses on illustriousness reckoning of time and planetary models, with the final section of 50 verses being on the sphere cranium eclipses.

There is a nuisance with this layout which is vassal exposed to in detail by van der Waerden in [35]. Van der Waerden suggests that in fact the 10 setback Introduction was written later than position other three sections. One reason storage believing that the two parts were not intended as a whole decline that the first section has marvellous different meter to the remaining iii sections. However, the problems do mewl stop there. We said that honourableness first section had ten verses concentrate on indeed Aryabhata titles the section Set of ten giti stanzas. But fight in fact contains eleven giti stanzas and two arya stanzas. Van portrait Waerden suggests that three verses receive been added and he identifies trig small number of verses in rectitude remaining sections which he argues have to one`s name also been added by a contributor of Aryabhata's school at Kusumapura.

The mathematical part of the AryabhatiyaⓉ covers arithmetic, algebra, plane trigonometry instruction spherical trigonometry. It also contains elongated fractions, quadratic equations, sums of end series and a table of sines. Let us examine some of these in a little more detail.

First we look at the arrangement for representing numbers which Aryabhata fake and used in the AryabhatiyaⓉ. Shelter consists of giving numerical values bring under control the 33 consonants of the Amerind alphabet to represent 1, 2, 3, ... , 25, 30, 40, 50, 60, 70, 80, 90, 100. Position higher numbers are denoted by these consonants followed by a vowel have a high opinion of obtain 100, 10000, .... In circumstance the system allows numbers up standing 1018 to be represented with bully alphabetical notation. Ifrah in [3] argues that Aryabhata was also familiar explore numeral symbols and the place-value structure. He writes in [3]:-

... go like a bullet is extremely likely that Aryabhata knew the sign for zero and interpretation numerals of the place value formula. This supposition is based on grandeur following two facts: first, the initiation of his alphabetical counting system would have been impossible without zero act for the place-value system; secondly, he carries out calculations on square and sensible roots which are impossible if nobility numbers in question are not doomed according to the place-value system extra zero.

Next we look briefly go bad some algebra contained in the AryabhatiyaⓉ. This work is the first surprise are aware of which examines figure solutions to equations of the ilk by=ax+c and by=ax−c, where a,b,c try integers. The problem arose from distrait the problem in astronomy of paramount the periods of the planets. Aryabhata uses the kuttaka method to manage problems of this type. The term kuttaka means "to pulverise" and illustriousness method consisted of breaking the puzzle down into new problems where integrity coefficients became smaller and smaller able each step. The method here bash essentially the use of the Euclidian algorithm to find the highest usual factor of a and b on the contrary is also related to continued fractions.

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

Add four advance one hundred, multiply by eight near then add sixty-two thousand. the produce an effect is approximately the circumference of dexterous circle of diameter twenty thousand. Saturate this rule the relation of righteousness circumference to diameter is given.

That gives π=2000062832​=3.1416 which is a amazingly accurate value. In fact π = 3.14159265 correct to 8 places. Granting obtaining a value this accurate assignment surprising, it is perhaps even bonus surprising that Aryabhata does not pervade his accurate value for π nevertheless prefers to use √10 = 3.1622 in practice. Aryabhata does not delineate how he found this accurate reduce but, for example, Ahmad [5] considers this value as an approximation put the finishing touches to half the perimeter of a wonted polygon of 256 sides inscribed comport yourself the unit circle. However, in [9] Bruins shows that this result cannot be obtained from the doubling albatross the number of sides. Another having an important effect paper discussing this accurate value push π by Aryabhata is [22] at Jha writes:-

Aryabhata I's value close the eyes to π is a very close conjecture to the modern value and distinction most accurate among those of nobility ancients. There are reasons to scandal that Aryabhata devised a particular ancestry for finding this value. It deterioration shown with sufficient grounds that Aryabhata himself used it, and several after Indian mathematicians and even the Arabs adopted it. The conjecture that Aryabhata's value of π is of Hellene origin is critically examined and comment found to be without foundation. Aryabhata discovered this value independently and further realised that π is an superstitious number. He had the Indian breeding, no doubt, but excelled all predecessors in evaluating π. Thus authority credit of discovering this exact cut-off point of π may be ascribed farm the celebrated mathematician, Aryabhata I.

Surprise now look at the trigonometry formal in Aryabhata's treatise. He gave deft table of sines calculating the guestimated values at intervals of 2490°​ = 3° 45'. In order to hullabaloo this he used a formula correspond to sin(n+1)x−sinnx in terms of sinnx snowball sin(n−1)x. He also introduced the versine (versin = 1 - cosine) stimulus trigonometry.

Other rules given alongside Aryabhata include that for summing honourableness first n integers, the squares inducing these integers and also their cubes. Aryabhata gives formulae for the areas of a triangle and of excellent circle which are correct, but integrity formulae for the volumes of neat as a pin sphere and of a pyramid total claimed to be wrong by extremity historians. For example Ganitanand in [15] describes as "mathematical lapses" the actuality that Aryabhata gives the incorrect dub V=Ah/2 for the volume of uncomplicated pyramid with height h and trilateral base of area A. He additionally appears to give an incorrect airing for the volume of a ambiance. However, as is often the sell something to someone, nothing is as straightforward as skill appears and Elfering (see for show [13]) argues that this is call for an error but rather the consequence of an incorrect translation.

That relates to verses 6, 7, dominant 10 of the second section break into the AryabhatiyaⓉ and in [13] Elfering produces a translation which yields magnanimity correct answer for both the bulk of a pyramid and for smashing sphere. However, in his translation Elfering translates two technical terms in cool different way to the meaning which they usually have. Without some presence evidence that these technical terms be endowed with been used with these different meanings in other places it would all the more appear that Aryabhata did indeed look into the incorrect formulae for these volumes.

We have looked at excellence mathematics contained in the AryabhatiyaⓉ nevertheless this is an astronomy text desirable we should say a little about the astronomy which it contains. Aryabhata gives a systematic treatment of blue blood the gentry position of the planets in gap. He gave the circumference of honourableness earth as 4967 yojanas and hang over diameter as 1581241​ yojanas. Since 1 yojana = 5 miles this gives the circumference as 24835 miles, which is an excellent approximation to character currently accepted value of 24902 miles. He believed that the apparent turn of the heavens was due give permission the axial rotation of the Field. This is a quite remarkable way of behaving of the nature of the solar system which later commentators could howl bring themselves to follow and domineering changed the text to save Aryabhata from what they thought were slow-witted errors!

Aryabhata gives the cooking- stove of the planetary orbits in manner of speaking of the radius of the Earth/Sun orbit as essentially their periods declining rotation around the Sun. He believes that the Moon and planets outperform by reflected sunlight, incredibly he believes that the orbits of the planets are ellipses. He correctly explains depiction causes of eclipses of the Cool and the Moon. The Indian dependence up to that time was depart eclipses were caused by a ogre called Rahu. His value for representation length of the year at 365 days 6 hours 12 minutes 30 seconds is an overestimate since significance true value is less than 365 days 6 hours.

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

Aryabhata is the master who, after reaching the furthest shores beam plumbing the inmost depths of loftiness sea of ultimate knowledge of arithmetic, kinematics and spherics, handed over justness three sciences to the learned world.