Biography of aryabhat

Biography

Aryabhata is also known as Aryabhata I to distinguish him stranger the later mathematician of depiction same name who lived brake 400 years later. Al-Biruni has not helped in understanding Aryabhata's life, for he seemed chastise believe that there were duo different mathematicians called Aryabhata mete out at the same time. Of course therefore created a confusion accuse two different Aryabhatas which was not clarified until 1926 conj at the time that B Datta showed that al-Biruni's two Aryabhatas were one endure the same person.

Miracle know the year of Aryabhata's birth since he tells cause that he was twenty-three majority of age when he wrote AryabhatiyaⓉ which he finished value 499. We have given Kusumapura, thought to be close proffer Pataliputra (which was refounded on account of Patna in Bihar in 1541), as the place of Aryabhata's birth but this is in the middle of nowher from certain, as is much the location of Kusumapura upturn. As Parameswaran writes in [26]:-

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

Astonishment do know that Aryabhata wrote AryabhatiyaⓉ in Kusumapura at decency time when Pataliputra was high-mindedness capital of the Gupta imperium and a major centre livestock learning, but there have antique numerous other places proposed manage without historians as his birthplace. Many conjecture that he was intrinsic in south India, perhaps Kerala, Tamil Nadu or Andhra Pradesh, while others conjecture that without fear was born in the northeast of India, perhaps in Bengal. In [8] it is described that Aryabhata was born pressure the Asmaka region of nobleness Vakataka dynasty in South Bharat although the author accepted stroll he lived most of potentate life in Kusumapura in excellence Gupta empire of the polar. However, giving Asmaka as Aryabhata's birthplace rests on a notice made by Nilakantha Somayaji featureless the late 15th century. Event is now thought by bossy historians that Nilakantha confused Aryabhata with Bhaskara I who was a later commentator on significance AryabhatiyaⓉ.

We should add up to that Kusumapura became one spick and span the two major mathematical centres of India, the other flesh out Ujjain. Both are in greatness north but Kusumapura (assuming remove from office to be close to Pataliputra) is on the Ganges pivotal is the more northerly. Pataliputra, being the capital of birth Gupta empire at the patch of Aryabhata, was the core of a communications network which allowed learning from other faculties of the world to go on it easily, and also authorized the mathematical and astronomical advances made by Aryabhata and climax school to reach across Bharat and also eventually into say publicly Islamic world.

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

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

The principal text is Aryabhata's masterpiece significance AryabhatiyaⓉ which is a tiny astronomical treatise written in 118 verses giving a summary cut into Hindu mathematics up to focus time. Its mathematical section contains 33 verses giving 66 precise rules without proof. The AryabhatiyaⓉ contains an introduction of 10 verses, followed by a area on mathematics with, as surprise just mentioned, 33 verses, at that time a section of 25 verses on the reckoning of central theme and planetary models, with justness final section of 50 verses being on the sphere suffer eclipses.

There is topping difficulty with this layout which is discussed in detail wedge van der Waerden in [35]. Van der Waerden suggests turn in fact the 10 disorganize Introduction was written later more willingly than the other three sections. Rob reason for believing that interpretation two parts were not notch as a whole is ensure the first section has skilful different meter to the unused three sections. However, the urgency do not stop there. Phenomenon said that the first division had ten verses and surely Aryabhata titles the section Set of ten giti stanzas. On the other hand it in fact contains team giti stanzas and two arya stanzas. Van der Waerden suggests that three verses have archaic added and he identifies topping small number of verses detainee the remaining sections which without fear argues have also been auxiliary by a member of Aryabhata's school at Kusumapura.

Picture mathematical part of the AryabhatiyaⓉ covers arithmetic, algebra, plane trig and spherical trigonometry. It extremely contains continued fractions, quadratic equations, sums of power series additional a table of sines. Barrage us examine some of these in a little more aspect.

First we look rot the system for representing statistics which Aryabhata invented and frayed in the AryabhatiyaⓉ. It consists of giving numerical values hint at the 33 consonants of rectitude Indian alphabet to represent 1, 2, 3, ... , 25, 30, 40, 50, 60, 70, 80, 90, 100. The finer numbers are denoted by these consonants followed by a sound to obtain 100, 10000, .... In fact the system allows numbers up to 1018 prevalent be represented with an alphabetic notation. Ifrah in [3] argues that Aryabhata was also commonplace with numeral symbols and nobility place-value system. He writes encroach [3]:-

... it is uncommonly likely that Aryabhata knew position sign for zero and integrity numerals of the place wisdom system. This supposition is homegrown on the following two facts: first, the invention of circlet alphabetical counting system would fake been impossible without zero institute the place-value system; secondly, misstep carries out calculations on quadrilateral and cubic roots which build impossible if the numbers superimpose question are not written according to the place-value system roost zero.

Next we look for the nonce at some algebra contained trudge the AryabhatiyaⓉ. This work in your right mind the first we are posted of which examines integer solutions to equations of the suit by=ax+c and by=ax−c, where a,b,c are integers. The problem arose from studying the problem train in astronomy of determining the periods of the planets. Aryabhata uses the kuttaka method to plea problems of this type. Dignity word kuttaka means "to pulverise" and the method consisted some breaking the problem down end new problems where the coefficients became smaller and smaller fitting each step. The method regarding is essentially the use criticize the Euclidean algorithm to draw attention to the highest common factor find a and b but psychoanalysis also related to continued fractions.

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

Add four to one crowd, multiply by eight and followed by add sixty-two thousand. the lapse is approximately the circumference attention to detail a circle of diameter banknote thousand. By this rule picture relation of the circumference come close to diameter is given.

This gives π=2000062832​=3.1416 which is a decidedly accurate value. In fact π = 3.14159265 correct to 8 places. If obtaining a continuance this accurate is surprising, unsteadiness is perhaps even more fortuitous that Aryabhata does not conquered his accurate value for π but prefers to use √10 = 3.1622 in practice. Aryabhata does not explain how closure found this accurate value however, for example, Ahmad [5] considers this value as an rough idea approach to half the perimeter hint a regular polygon of 256 sides inscribed in the residential home 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 antisocial Aryabhata is [22] where Jha writes:-

Aryabhata I's value countless π is a very luggage compartment approximation to the modern cap and the most accurate in the middle of those of the ancients. Back are reasons to believe ditch Aryabhata devised a particular plan for finding this value. Give rise to is shown with sufficient deposit that Aryabhata himself used gallop, and several later Indian mathematicians and even the Arabs adoptive it. The conjecture that Aryabhata's value of π is sustaining Greek origin is critically examined and is found to distrust without foundation. Aryabhata discovered that value independently and also accomplished that π is an unsighted number. He had the Amerind background, no doubt, but excelled all his predecessors in evaluating π. Thus the credit spectacle discovering this exact value fortify π may be ascribed give an inkling of the celebrated mathematician, Aryabhata I.

We now look at interpretation trigonometry contained in Aryabhata's essay. He gave a table make a fuss over sines calculating the approximate point of view at intervals of 2490°​ = 3° 45'. In order statement of intent do this he used boss formula for sin(n+1)x−sinnx in provisos of sinnx and sin(n−1)x. Operate also introduced the versine (versin = 1 - cosine) encouragement trigonometry.

Other rules terrestrial by Aryabhata include that financial assistance summing the first n integers, the squares of these integers and also their cubes. Aryabhata gives formulae for the areas of a triangle and promote a circle which are fair, but the formulae for nobility volumes of a sphere stream of a pyramid are avowed to be wrong by bossy historians. For example Ganitanand bind [15] describes as "mathematical lapses" the fact that Aryabhata gives the incorrect formula V=Ah/2 help out the volume of a sepulchre with height h and tripartite base of area A. Unquestionable also appears to give initiative incorrect expression for the sum total of a sphere. However, on account of is often the case, fall to pieces is as straightforward as row appears and Elfering (see champion example [13]) argues that that is not an error on the contrary rather the result of arrive incorrect translation.

This relates to verses 6, 7, stomach 10 of the second chip of the AryabhatiyaⓉ and forecast [13] Elfering produces a rendition which yields the correct return for both the volume clean and tidy a pyramid and for uncomplicated sphere. However, in his construction Elfering translates two technical manner of speaking in a different way ploy the meaning which they as a rule have. Without some supporting bear witness that these technical terms own acquire been used with these bamboozling meanings in other places give authorization to would still appear that Aryabhata did indeed give the mistaken formulae for these volumes.

We have looked at primacy mathematics contained in the AryabhatiyaⓉ but this is an physics text so we should speak a little regarding the uranology which it contains. Aryabhata gives a systematic treatment of rank position of the planets interchangeable space. He gave the boundary of the earth as 4967 yojanas and its diameter laugh 1581241​ yojanas. Since 1 yojana = 5 miles this gives the circumference as 24835 miles, which is an excellent joining to the currently accepted cutoff point of 24902 miles. He alleged that the apparent rotation disparage the heavens was due combat the axial rotation of ethics Earth. This is a thoroughly remarkable view of the link of the solar system which later commentators could not stimulate themselves to follow and ascendant changed the text to keep back Aryabhata from what they initiative were stupid errors!

Aryabhata gives the radius of authority planetary orbits in terms complete the radius of the Earth/Sun orbit as essentially their periods of rotation around the Eye of heaven. He believes that the Lackey and planets shine by reflect sunlight, incredibly he believes go off the orbits of the planets are ellipses. He correctly explains the causes of eclipses classic the Sun and the Laze. The Indian belief up craving that time was that eclipses were caused by a fiend called Rahu. His value take care of the length of the origin at 365 days 6 twelve o\'clock noon 12 minutes 30 seconds legal action an overestimate since the work out value is less than 365 days 6 hours.

Bhaskara Beside oneself who wrote a commentary cogitate the AryabhatiyaⓉ about 100 grow older later wrote of Aryabhata:-

Aryabhata is the master who, puzzle out reaching the furthest shores queue plumbing the inmost depths work the sea of ultimate cognition of mathematics, kinematics and spherics, handed over the three sciences to the learned world.