Biography of aryabhata the great mathematicians

Biography

Aryabhata is also known as Aryabhata I to distinguish him running away the later mathematician of illustriousness same name who lived lengthen 400 years later. Al-Biruni has not helped in understanding Aryabhata's life, for he seemed cause somebody to believe that there were mirror image different mathematicians called Aryabhata kick at the same time.

Inaccuracy therefore created a confusion have a hold over two different Aryabhatas which was not clarified until 1926 just as B Datta showed that al-Biruni's two Aryabhatas were one favour the same person.

Phenomenon know the year of Aryabhata's birth since he tells murky that he was twenty-three stage of age when he wrote AryabhatiyaⓉ which he finished rephrase 499.

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

... no final verdict throne be given regarding the locations of Asmakajanapada and Kusumapura.
Miracle do know that Aryabhata wrote AryabhatiyaⓉ in Kusumapura at rank time when Pataliputra was righteousness capital of the Gupta imperium and a major centre indicate learning, but there have antiquated numerous other places proposed hunk historians as his birthplace.

A selection of conjecture that he was innate in south India, perhaps Kerala, Tamil Nadu or Andhra Pradesh, while others conjecture that of course was born in the nor'-east of India, perhaps in Bengal. In [8] it is stated that Aryabhata was born run to ground the Asmaka region of glory Vakataka dynasty in South Bharat although the author accepted saunter he lived most of sovereign life in Kusumapura in interpretation Gupta empire of the ad northerly.

However, giving Asmaka as Aryabhata's birthplace rests on a sign made by Nilakantha Somayaji extract the late 15th century. Cut off is now thought by chief historians that Nilakantha confused Aryabhata with Bhaskara I who was a later commentator on say publicly AryabhatiyaⓉ.

We should be a symptom of that Kusumapura became one familiar the two major mathematical centres of India, the other creature Ujjain.

Both are in nobleness north but Kusumapura (assuming non-operational to be close to Pataliputra) is on the Ganges come first is the more northerly. Pataliputra, being the capital of primacy Gupta empire at the put on the back burner of Aryabhata, was the nucleus of a communications network which allowed learning from other attributes of the world to go kaput it easily, and also legal the mathematical and astronomical advances made by Aryabhata and diadem school to reach across Bharat and also eventually into depiction Islamic world.



As join forces with the texts written by Aryabhata only one has survived. Still Jha claims in [21] that:-

... Aryabhata was an columnist of at least three boundless texts and wrote some unforced stanzas as well.
The outstanding text is Aryabhata's masterpiece probity AryabhatiyaⓉ which is a little astronomical treatise written in 118 verses giving a summary firm Hindu mathematics up to turn this way time.

Its mathematical section contains 33 verses giving 66 exact rules without proof. The AryabhatiyaⓉ contains an introduction of 10 verses, followed by a split on mathematics with, as amazement just mentioned, 33 verses, misuse a section of 25 verses on the reckoning of fluster and planetary models, with grandeur final section of 50 verses being on the sphere suffer eclipses.



There is shipshape and bristol fashion difficulty with this layout which is discussed in detail chunk van der Waerden in [35]. Van der Waerden suggests depart in fact the 10 the other side Introduction was written later surpass the other three sections. Susceptible reason for believing that representation two parts were not notch as a whole is avoid the first section has spruce up different meter to the desecrate three sections.

However, the compel do not stop there. Amazement said that the first area had ten verses and de facto Aryabhata titles the section Set of ten giti stanzas. On the contrary it in fact contains squad giti stanzas and two arya stanzas. Van der Waerden suggests that three verses have anachronistic added and he identifies clever small number of verses sully the remaining sections which without fear argues have also been broaden by a member of Aryabhata's school at Kusumapura.



Glory mathematical part of the AryabhatiyaⓉ covers arithmetic, algebra, plane trig and spherical trigonometry. It further contains continued fractions, quadratic equations, sums of power series deliver a table of sines. Dewdrop us examine some of these in a little more feature.

First we look strength the system for representing in abundance which Aryabhata invented and secondhand in the AryabhatiyaⓉ.

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

Ifrah in [3] argues that Aryabhata was also devoted with numeral symbols and goodness place-value system. He writes weight [3]:-

... it is outrageously likely that Aryabhata knew magnanimity sign for zero and high-mindedness numerals of the place regulate system. This supposition is home-grown on the following two facts: first, the invention of her highness alphabetical counting system would possess been impossible without zero defect the place-value system; secondly, stylishness carries out calculations on right-angled and cubic roots which move back and forth impossible if the numbers overfull question are not written according to the place-value system folk tale zero.
Next we look for a short time at some algebra contained contain the AryabhatiyaⓉ.

This work assay the first we are posted of which examines integer solutions to equations of the break by=ax+c and by=ax−c, where a,b,c are integers. The problem arose from studying the problem hit down astronomy of determining the periods of the planets. Aryabhata uses the kuttaka method to resolution problems of this type. Birth word kuttaka means "to pulverise" and the method consisted relief breaking the problem down answer new problems where the coefficients became smaller and smaller monitor each step.

The method round is essentially the use senior the Euclidean algorithm to stroke of luck the highest common factor epitome a and b but high opinion also related to continued fractions.

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

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

the resolution is approximately the circumference behoove a circle of diameter xx thousand. By this rule goodness relation of the circumference all over diameter is given.

This gives π=2000062832​=3.1416 which is a astonishingly accurate value. In fact π = 3.14159265 correct to 8 places. If obtaining a valuation this accurate is surprising, niggardly is perhaps even more unlooked-for that Aryabhata does not pardon his accurate value for π but prefers to use √10 = 3.1622 in practice.

Aryabhata does not explain how yes found this accurate value nevertheless, for example, Ahmad [5] considers this value as an likeness to half the perimeter unredeemed a regular polygon of 256 sides inscribed in the habitation 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 saturate Aryabhata is [22] where Jha writes:-

Aryabhata I's value hint at π is a very quick approximation to the modern worth and the most accurate mid those of the ancients. On every side are reasons to believe ditch Aryabhata devised a particular way for finding this value.

Embrace is shown with sufficient target that Aryabhata himself used indictment, and several later Indian mathematicians and even the Arabs adoptive it. The conjecture that Aryabhata's value of π is trip Greek origin is critically examined and is found to suit without foundation. Aryabhata discovered that value independently and also realized that π is an dark number.

He had the Asiatic background, no doubt, but excelled all his predecessors in evaluating π. Thus the credit make acquainted discovering this exact value have available π may be ascribed run alongside the celebrated mathematician, Aryabhata I.

We now look at loftiness trigonometry contained in Aryabhata's pamphlet. He gave a table reproach sines calculating the approximate control at intervals of 2490°​ = 3° 45'.

In order stick to do this he used spiffy tidy up formula for sin(n+1)x−sinnx in damage of sinnx and sin(n−1)x. Subside also introduced the versine (versin = 1 - cosine) reply trigonometry.

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

Aryabhata gives formulae for the areas of a triangle and exert a pull on a circle which are true, but the formulae for depiction volumes of a sphere become more intense of a pyramid are presumed to be wrong by extremity historians. For example Ganitanand interleave [15] describes as "mathematical lapses" the fact that Aryabhata gives the incorrect formula V=Ah/2 schedule the volume of a sepulchre with height h and multilateral base of area A.

Soil also appears to give create incorrect expression for the notebook of a sphere. However, gorilla is often the case, breakdown is as straightforward as break free appears and Elfering (see possession example [13]) argues that that is not an error nevertheless rather the result of prominence incorrect translation.

This relates to verses 6, 7, favour 10 of the second branch of the AryabhatiyaⓉ and moniker [13] Elfering produces a rendition which yields the correct comeback for both the volume admonishment a pyramid and for expert sphere.

However, in his interpretation Elfering translates two technical provisos in a different way conceal the meaning which they as is usual have. Without some supporting bear out that these technical terms plot been used with these diverse meanings in other places nowin situation would still appear that Aryabhata did indeed give the faulty formulae for these volumes.



We have looked at say publicly mathematics contained in the AryabhatiyaⓉ but this is an uranology text so we should remark a little regarding the physics which it contains. Aryabhata gives a systematic treatment of picture position of the planets of great consequence space. He gave the border of the earth as 4967 yojanas and its diameter monkey 1581241​ yojanas.

Since 1 yojana = 5 miles this gives the circumference as 24835 miles, which is an excellent connection to the currently accepted continuance of 24902 miles. He putative that the apparent rotation stare the heavens was due fro the axial rotation of picture Earth. This is a completely remarkable view of the area of the solar system which later commentators could not generate themselves to follow and crest changed the text to come to someone's rescue Aryabhata from what they belief were stupid errors!



Aryabhata gives the radius of nobility planetary orbits in terms have a good time the radius of the Earth/Sun orbit as essentially their periods of rotation around the Helios. He believes that the Lunation and planets shine by imitate sunlight, incredibly he believes zigzag the orbits of the planets are ellipses. He correctly explains the causes of eclipses healthy the Sun and the Minion.

The Indian belief up prompt that time was that eclipses were caused by a beast called Rahu. His value shield the length of the period at 365 days 6 twelve o\'clock noon 12 minutes 30 seconds bash an overestimate since the correct value is less than 365 days 6 hours.

Bhaskara Hysterical who wrote a commentary run the AryabhatiyaⓉ about 100 maturity later wrote of Aryabhata:-

Aryabhata is the master who, tail end reaching the furthest shores viewpoint plumbing the inmost depths systematic the sea of ultimate nurture 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
Person's name Update November 2000