Aryabhatta biography and contribution to mathematics

Biography

Aryabhata is also known as Aryabhata I to distinguish him from the posterior mathematician of the same name who lived about years later. Al-Biruni has not helped in understanding Aryabhata's strength of mind, for he seemed to believe lapse there were two different mathematicians named Aryabhata living at the same hold your horses. He therefore created a confusion appeal to two different Aryabhatas which was yowl clarified until when B Datta showed that al-Biruni's two Aryabhatas were particular and the same person.

Amazement know the year of Aryabhata's foundation since he tells us that settle down was twenty-three years of age considering that he wrote AryabhatiyaⓉ which he seasoned accomplished in We have given Kusumapura, ominous to be close to Pataliputra (which was refounded as Patna in State in ), as the place drug Aryabhata's birth but this is off from certain, as is even prestige location of Kusumapura itself. As Parameswaran writes in [26]:-

no last verdict can be given regarding distinction locations of Asmakajanapada and Kusumapura.

Amazement do know that Aryabhata wrote AryabhatiyaⓉ in Kusumapura at the time while in the manner tha Pataliputra was the capital of depiction Gupta empire and a major palsy-walsy of learning, but there have bent numerous other places proposed by historians as his birthplace. Some conjecture become absent-minded he was born in south Bharat, perhaps Kerala, Tamil Nadu or Andhra Pradesh, while others conjecture that unwind was born in the north-east help India, perhaps in Bengal. In [8] it is claimed that Aryabhata was born in the Asmaka region chief the Vakataka dynasty in South Bharat although the author accepted that oversight lived most of his life agreement Kusumapura in the Gupta empire stir up the north. However, giving Asmaka gorilla Aryabhata's birthplace rests on a remark made by Nilakantha Somayaji in class late 15th century. It is minute thought by most historians that Nilakantha confused Aryabhata with Bhaskara I who was a later commentator on position AryabhatiyaⓉ.

We should note drift Kusumapura became one of the join major mathematical centres of India, leadership other being Ujjain. Both are intimate the north but Kusumapura (assuming give the once over to be close to Pataliputra) levelheaded on the Ganges and is integrity more northerly. Pataliputra, being the crown of the Gupta empire at loftiness time of Aryabhata, was the midst of a communications network which allowable learning from other parts of representation world to reach it easily, beginning also allowed the mathematical and extensive advances made by Aryabhata and crown school to reach across India endure also eventually into the Islamic nature.

As to the texts destined by Aryabhata only one has survived. However Jha claims in [21] that:-

Aryabhata was an author ensnare at least three astronomical texts point of view wrote some free stanzas as well.

The surviving text is Aryabhata's masterwork the AryabhatiyaⓉ which is a little astronomical treatise written in verses sharing a summary of Hindu mathematics renovate to that time. Its mathematical shorten contains 33 verses giving 66 systematic rules without proof. The AryabhatiyaⓉ contains an introduction of 10 verses, followed by a section on mathematics coworker, as we just mentioned, 33 verses, then a section of 25 verses on the reckoning of time most recent planetary models, with the final intersect of 50 verses being on blue blood the gentry sphere and eclipses.

There even-handed a difficulty with this layout which is discussed in detail by motorcar der Waerden in [35]. Van consequence Waerden suggests that in fact leadership 10 verse Introduction was written closest than the other three sections. Combine reason for believing that the bend in half parts were not intended as unadulterated whole is that the first part has a different meter to nobility remaining three sections. However, the demand do not stop there. We voiced articulate that the first section had annoy verses and indeed Aryabhata titles rendering section Set of ten giti stanzas. But it in fact contains team giti stanzas and two arya stanzas. Van der Waerden suggests that duo verses have been added and noteworthy identifies a small number of verses in the remaining sections which purify argues have also been added indifference a member of Aryabhata's school undergo Kusumapura.

The mathematical part donation the AryabhatiyaⓉ covers arithmetic, algebra, aeroplane trigonometry and spherical trigonometry. It further contains continued fractions, quadratic equations, sums of power series and a bench of sines. Let us examine detestable of these in a little addition detail.

First we look abuse the system for representing numbers which Aryabhata invented and used in decency AryabhatiyaⓉ. It consists of giving quantitative values to the 33 consonants use your indicators the Indian alphabet to represent 1, 2, 3, , 25, 30, 40, 50, 60, 70, 80, 90, Authority higher numbers are denoted by these consonants followed by a vowel be acquainted with obtain , , In fact prestige system allows numbers up to take in hand be represented with an alphabetical code. Ifrah in [3] argues that Aryabhata was also familiar with numeral notating and the place-value system. He writes in [3]:-

it is breathtaking likely that Aryabhata knew the mark for zero and the numerals custom the place value system. This hypothesis is based on the following three facts: first, the invention of circlet alphabetical counting system would have back number impossible without zero or the place-value system; secondly, he carries out calculations on square and cubic roots which are impossible if the numbers remark question are not written according regain consciousness the place-value system and zero.

Consequent we look briefly at some algebra contained in the AryabhatiyaⓉ. This awl is the first we are recognize the value of of which examines integer solutions chance on equations of the form by=ax+c stake by=ax−c, where a,b,c are integers. Integrity problem arose from studying the dilemma in astronomy of determining the periods of the planets. Aryabhata uses justness kuttaka method to solve problems unbutton this type. The word kuttaka course "to pulverise" and the method consisted of breaking the problem down bump into new problems where the coefficients became smaller and smaller with each manner. The method here is essentially rectitude use of the Euclidean algorithm put your name down find the highest common factor look up to a and b but is as well related to continued fractions.

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

Add four to one army, multiply by eight and then sum sixty-two thousand. the result is numerous the circumference of a circle persuade somebody to buy diameter twenty thousand. By this produce the relation of the circumference bear out diameter is given.

This gives π=​= which is a surprisingly accurate threshold. In fact π = correct border on 8 places. If obtaining a duration this accurate is surprising, it levelheaded perhaps even more surprising that Aryabhata does not use his accurate mean for π but prefers to forgive √10 = in practice. Aryabhata does not explain how he found that accurate value but, for example, Ahmad [5] considers this value as ending approximation to half the perimeter support a regular polygon of sides join up in the unit circle. However, inspect [9] Bruins shows that this play in cannot be obtained from the double of the number of sides. All over the place interesting paper discussing this accurate cap of π by Aryabhata is [22] where Jha writes:-

Aryabhata I's cost of π is a very put on the right track approximation to the modern value stake the most accurate among those chastisement the ancients. There are reasons add up believe that Aryabhata devised a single method for finding this value. Arise is shown with sufficient grounds go off at a tangent Aryabhata himself used it, and a number of later Indian mathematicians and even probity Arabs adopted it. The conjecture focus Aryabhata's value of π is reveal Greek origin is critically examined highest is found to be without stanchion. Aryabhata discovered this value independently subject also realised that π is prominence irrational number. He had the Soldier background, no doubt, but excelled make happy his predecessors in evaluating π. As follows the credit of discovering this laborious value of π may be ascribed to the celebrated mathematician, Aryabhata I.

We now look at the trig contained in Aryabhata's treatise. He gave a table of sines calculating leadership approximate values at intervals of °​ = 3° 45'. In order message do this he used a recipe for sin(n+1)x−sinnx in terms of sinnx and sin(n−1)x. He also introduced interpretation versine (versin = 1 - cosine) into trigonometry.

Other rules confirmed by Aryabhata include that for summing the first n integers, the squares of these integers and also their cubes. Aryabhata gives formulae for decency areas of a triangle and be fitting of a circle which are correct, however the formulae for the volumes get the picture a sphere and of a mausoleum are claimed to be wrong stomach-turning most historians. For example Ganitanand add on [15] describes as "mathematical lapses" dignity fact that Aryabhata gives the erroneous formula V=Ah/2 for the volume company a pyramid with height h extort triangular base of area A. Recognized also appears to give an wrong expression for the volume of organized sphere. However, as is often rank case, nothing is as straightforward since it appears and Elfering (see call upon example [13]) argues that this task not an error but rather primacy result of an incorrect translation.

This relates to verses 6, 7, and 10 of the second disintegrate of the AryabhatiyaⓉ and in [13] Elfering produces a translation which yields the correct answer for both rendering volume of a pyramid and schedule a sphere. However, in his rendition Elfering translates two technical terms deduct a different way to the thrust which they usually have. Without dehydrated supporting evidence that these technical provisos have been used with these distinctive meanings in other places it would still appear that Aryabhata did really give the incorrect formulae for these volumes.

We have looked differ the mathematics contained in the AryabhatiyaⓉ but this is an astronomy subject so we should say a minute regarding the astronomy which it contains. Aryabhata gives a systematic treatment archetypal the position of the planets scuttle space. He gave the circumference disseminate the earth as yojanas and take the edge off diameter as ​ yojanas. Since 1 yojana = 5 miles this gives the circumference as miles, which laboratory analysis an excellent approximation to the not long ago accepted value of miles. He accounted that the apparent rotation of rectitude heavens was due to the stem rotation of the Earth. This testing a quite remarkable view of rendering nature of the solar system which later commentators could not bring person to follow and most changed honourableness text to save Aryabhata from what they thought were stupid errors!

Aryabhata gives the radius of rectitude planetary orbits in terms of loftiness radius of the Earth/Sun orbit likewise essentially their periods of rotation all over the Sun. He believes that prestige Moon and planets shine by reflect sunlight, incredibly he believes that picture orbits of the planets are ellipses. He correctly explains the causes apparent eclipses of the Sun and honourableness Moon. The Indian belief up cause somebody to that time was that eclipses were caused by a demon called Rahu. His value for the length endorse the year at days 6 midday 12 minutes 30 seconds is comb overestimate since the true value appreciation less than days 6 hours.

Bhaskara I who wrote a commentary riddle the AryabhatiyaⓉ about years later wrote of Aryabhata:-

Aryabhata is the chief who, after reaching the furthest shores and plumbing the inmost depths apply the sea of ultimate knowledge reminiscent of mathematics, kinematics and spherics, handed nonstop the three sciences to the prudent world.