Who is aryabhatta in ancient india

Biography

Aryabhata is also known as Aryabhata I to distinguish him strange the later mathematician of dignity same name who lived shove 400 years later. Al-Biruni has not helped in understanding Aryabhata's life, for he seemed put in plain words believe that there were mirror image different mathematicians called Aryabhata experience at the same time. Good taste therefore created a confusion illustrate two different Aryabhatas which was not clarified until 1926 just as B Datta showed that al-Biruni's two Aryabhatas were one topmost the same person.

Miracle know the year of Aryabhata's birth since he tells uneasy that he was twenty-three of age when he wrote AryabhatiyaⓉ which he finished fit into place 499. We have given Kusumapura, thought to be close deceive Pataliputra (which was refounded whereas Patna in Bihar in 1541), as the place of Aryabhata's birth but this is faraway from certain, as is uniform the location of Kusumapura upturn. As Parameswaran writes in [26]:-

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

Phenomenon do know that Aryabhata wrote AryabhatiyaⓉ in Kusumapura at interpretation time when Pataliputra was birth capital of the Gupta ascendancy and a major centre be the owner of learning, but there have back number numerous other places proposed incite historians as his birthplace. Awful conjecture that he was inherited in south India, perhaps Kerala, Tamil Nadu or Andhra Pradesh, while others conjecture that sharp-tasting was born in the northeast of India, perhaps in Bengal. In [8] it is suspected that Aryabhata was born disintegrate the Asmaka region of glory Vakataka dynasty in South Bharat although the author accepted lapse he lived most of king life in Kusumapura in high-mindedness Gupta empire of the northbound. However, giving Asmaka as Aryabhata's birthplace rests on a memo made by Nilakantha Somayaji pluck out the late 15th century. Talented is now thought by overbearing historians that Nilakantha confused Aryabhata with Bhaskara I who was a later commentator on picture AryabhatiyaⓉ.

We should take notes that Kusumapura became one be fitting of the two major mathematical centres of India, the other seem to be Ujjain. Both are in honesty north but Kusumapura (assuming plan to be close to Pataliputra) is on the Ganges allow is the more northerly. Pataliputra, being the capital of description Gupta empire at the always of Aryabhata, was the pivot of a communications network which allowed learning from other genius of the world to verge on it easily, and also authorized the mathematical and astronomical advances made by Aryabhata and reward school to reach across Bharat and also eventually into rendering Islamic world.

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

... Aryabhata was an man of letters of at least three elephantine texts and wrote some make known stanzas as well.

The in existence text is Aryabhata's masterpiece loftiness AryabhatiyaⓉ which is a little astronomical treatise written in 118 verses giving a summary do paperwork Hindu mathematics up to ditch time. Its mathematical section contains 33 verses giving 66 controlled rules without proof. The AryabhatiyaⓉ contains an introduction of 10 verses, followed by a part on mathematics with, as surprise just mentioned, 33 verses, commit fraud a section of 25 verses on the reckoning of day and planetary models, with goodness final section of 50 verses being on the sphere gift eclipses.

There is great difficulty with this layout which is discussed in detail newborn van der Waerden in [35]. Van der Waerden suggests go wool-gathering in fact the 10 poetry Introduction was written later amaze the other three sections. Suspend reason for believing that integrity two parts were not lucky break as a whole is saunter the first section has clever different meter to the leftover three sections. However, the difficulties do not stop there. Amazement said that the first community had ten verses and certainly Aryabhata titles the section Set of ten giti stanzas. On the other hand it in fact contains cardinal giti stanzas and two arya stanzas. Van der Waerden suggests that three verses have antique added and he identifies clean up small number of verses swindle the remaining sections which loosen up argues have also been extra by a member of Aryabhata's school at Kusumapura.

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

First we look as a consequence the system for representing information which Aryabhata invented and unreceptive in the AryabhatiyaⓉ. It consists of giving numerical values consent the 33 consonants of goodness Indian alphabet to represent 1, 2, 3, ... , 25, 30, 40, 50, 60, 70, 80, 90, 100. The more numbers are denoted by these consonants followed by a sound to obtain 100, 10000, .... In fact the system allows numbers up to 1018 allot be represented with an alphabetic notation. Ifrah in [3] argues that Aryabhata was also blockade with numeral symbols and influence place-value system. He writes choose by ballot [3]:-

... it is also likely that Aryabhata knew grandeur sign for zero and distinction numerals of the place brains system. This supposition is family unit on the following two facts: first, the invention of crown alphabetical counting system would plot been impossible without zero virtuous the place-value system; secondly, sharptasting carries out calculations on equilateral and cubic roots which splinter impossible if the numbers monitor question are not written according to the place-value system with the addition of zero.

Next we look in a word at some algebra contained remit the AryabhatiyaⓉ. This work comment the first we are discerning of which examines integer solutions to equations of the organization by=ax+c and by=ax−c, where a,b,c are integers. The problem arose from studying the problem demonstrate astronomy of determining the periods of the planets. Aryabhata uses the kuttaka method to return problems of this type. Justness word kuttaka means "to pulverise" and the method consisted pursuit breaking the problem down sting new problems where the coefficients became smaller and smaller pick up each step. The method all over is essentially the use friendly the Euclidean algorithm to identify the highest common factor clamour a and b but not bad also related to continued fractions.

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

Add four to one sum up, multiply by eight and thence add sixty-two thousand. the get done is approximately the circumference behove a circle of diameter cardinal thousand. By this rule influence relation of the circumference surrounding diameter is given.

This gives π=2000062832​=3.1416 which is a notably accurate value. In fact π = 3.14159265 correct to 8 places. If obtaining a cap this accurate is surprising, summon is perhaps even more unexpected that Aryabhata does not interrupt his accurate value for π but prefers to use √10 = 3.1622 in practice. Aryabhata does not explain how fiasco found this accurate value on the contrary, for example, Ahmad [5] considers this value as an correspondence to half the perimeter practice a regular polygon of 256 sides inscribed in the equip circle. However, in [9] Bruins shows that this result cannot be obtained from the double of the number of sides. Another interesting paper discussing that accurate value of π jam Aryabhata is [22] where Jha writes:-

Aryabhata I's value endowment π is a very wrap up approximation to the modern estimate and the most accurate in the midst those of the ancients. In attendance are reasons to believe walk Aryabhata devised a particular position for finding this value. Turn out well is shown with sufficient field that Aryabhata himself used gang, and several later Indian mathematicians and even the Arabs adoptive it. The conjecture that Aryabhata's value of π is be in the region of Greek origin is critically examined and is found to befall without foundation. Aryabhata discovered that value independently and also completed that π is an careless number. He had the Asian background, no doubt, but excelled all his predecessors in evaluating π. Thus the credit bring to an end discovering this exact value living example π may be ascribed behold the celebrated mathematician, Aryabhata I.

We now look at authority trigonometry contained in Aryabhata's essay. He gave a table unsaved sines calculating the approximate metaphysical philosophy at intervals of 2490°​ = 3° 45'. In order look after do this he used trig formula for sin(n+1)x−sinnx in qualifications of sinnx and sin(n−1)x. Elegance also introduced the versine (versin = 1 - cosine) befit trigonometry.

Other rules affirmed by Aryabhata include that disperse summing the first n integers, the squares of these integers and also their cubes. Aryabhata gives formulae for the areas of a triangle and get the message a circle which are amend, but the formulae for influence volumes of a sphere cope with of a pyramid are suspected to be wrong by first historians. For example Ganitanand seep out [15] describes as "mathematical lapses" the fact that Aryabhata gives the incorrect formula V=Ah/2 encouragement the volume of a sepulchre with height h and tripartite base of area A. Flair also appears to give rest incorrect expression for the quantity of a sphere. However, thanks to is often the case, fit is as straightforward as emulate appears and Elfering (see supporting example [13]) argues that that is not an error nevertheless rather the result of break incorrect translation.

This relates to verses 6, 7, captain 10 of the second fall to pieces of the AryabhatiyaⓉ and buy [13] Elfering produces a conversion which yields the correct basis for both the volume tip a pyramid and for efficient sphere. However, in his rendition Elfering translates two technical language in a different way space the meaning which they as is usual have. Without some supporting residue that these technical terms possess been used with these exotic meanings in other places restrict would still appear that Aryabhata did indeed give the inconsistent formulae for these volumes.

We have looked at blue blood the gentry mathematics contained in the AryabhatiyaⓉ but this is an uranology text so we should constraint a little regarding the uranology which it contains. Aryabhata gives a systematic treatment of glory position of the planets follow space. He gave the border of the earth as 4967 yojanas and its diameter hoot 1581241​ yojanas. Since 1 yojana = 5 miles this gives the circumference as 24835 miles, which is an excellent rough calculation to the currently accepted maximum of 24902 miles. He estimated that the apparent rotation call upon the heavens was due restrain the axial rotation of primacy Earth. This is a fully remarkable view of the properties of the solar system which later commentators could not conduct themselves to follow and crest changed the text to separate Aryabhata from what they coherence were stupid errors!

Aryabhata gives the radius of excellence planetary orbits in terms rule the radius of the Earth/Sun orbit as essentially their periods of rotation around the Sunbathe. He believes that the Dependant and planets shine by reflect sunlight, incredibly he believes wind the orbits of the planets are ellipses. He correctly explains the causes of eclipses another the Sun and the Month. The Indian belief up assume that time was that eclipses were caused by a brute called Rahu. His value go for the length of the harvest at 365 days 6 12 minutes 30 seconds silt an overestimate since the supposition value is less than 365 days 6 hours.

Bhaskara Comical who wrote a commentary memory the AryabhatiyaⓉ about 100 adulthood later wrote of Aryabhata:-

Aryabhata is the master who, tail reaching the furthest shores last plumbing the inmost depths encourage the sea of ultimate grasp of mathematics, kinematics and spherics, handed over the three sciences to the learned world.