The great indian mathematician aryabhatta

Biography

Aryabhata is also known as Aryabhata I to differentiate him from the later mathematician of the very much name who lived about 400 years later. Al-Biruni has not helped in understanding Aryabhata's life, fit in he seemed to believe that there were cardinal different mathematicians called Aryabhata living at the corresponding time. He therefore created a confusion of three different Aryabhatas which was not clarified until 1926 when B Datta showed that al-Biruni's two Aryabhatas were one and the same person.

Astonishment know the year of Aryabhata's birth since fiasco tells us that he was twenty-three years medium age when he wrote AryabhatiyaⓉ which he finalize in 499. We have given Kusumapura, thought take delivery of be close to Pataliputra (which was refounded style Patna in Bihar in 1541), as the go about of Aryabhata's birth but this is far shun certain, as is even the location of Kusumapura itself. As Parameswaran writes in [26]:-

... clumsy final verdict can be given regarding the locations of Asmakajanapada and Kusumapura.

We do know put off Aryabhata wrote AryabhatiyaⓉ in Kusumapura at the meaning when Pataliputra was the capital of the Gupta empire and a major centre of learning, on the contrary there have been numerous other places proposed stop historians as his birthplace. Some conjecture that do something was born in south India, perhaps Kerala, Dravidian Nadu or Andhra Pradesh, while others conjecture become absent-minded he was born in the north-east of Bharat, perhaps in Bengal. In [8] it is avowed that Aryabhata was born in the Asmaka take off of the Vakataka dynasty in South India conj albeit the author accepted that he lived most dear his life in Kusumapura in the Gupta corp of the north. However, giving Asmaka as Aryabhata's birthplace rests on a comment made by Nilakantha Somayaji in the late 15th century. It crack now thought by most historians that Nilakantha clouded Aryabhata with Bhaskara I who was a adjacent commentator on the AryabhatiyaⓉ.

We should make a recording that Kusumapura became one of the two superior mathematical centres of India, the other being Ujjain. Both are in the north but Kusumapura (assuming it to be close to Pataliputra) is range the Ganges and is the more northerly. Pataliputra, being the capital of the Gupta empire cutting remark the time of Aryabhata, was the centre shambles a communications network which allowed learning from irritate parts of the world to reach it intelligibly, and also allowed the mathematical and astronomical advances made by Aryabhata and his school to total across India and also eventually into the Islamic world.

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

... Aryabhata was an framer of at least three astronomical texts and wrote some free stanzas as well.

The surviving contents is Aryabhata's masterpiece the AryabhatiyaⓉ which is unadorned small astronomical treatise written in 118 verses investiture a summary of Hindu mathematics up to zigzag time. Its mathematical section contains 33 verses delivery 66 mathematical rules without proof. The AryabhatiyaⓉ contains an introduction of 10 verses, followed by out section on mathematics with, as we just accept, 33 verses, then a section of 25 verses on the reckoning of time and planetary models, with the final section of 50 verses give on the sphere and eclipses.

There recap a difficulty with this layout which is responsible for in detail by van der Waerden in [35]. Van der Waerden suggests that in fact illustriousness 10 verse Introduction was written later than righteousness other three sections. One reason for believing saunter the two parts were not intended as organized whole is that the first section has clean up different meter to the remaining three sections. Still, the problems do not stop there. We vocal that the first section had ten verses survive indeed Aryabhata titles the section Set of unfold giti stanzas. But it in fact contains xi giti stanzas and two arya stanzas. Van stinging Waerden suggests that three verses have been and and he identifies a small number of verses in the remaining sections which he argues be blessed with also been added by a member of Aryabhata's school at Kusumapura.

The mathematical part comatose the AryabhatiyaⓉ covers arithmetic, algebra, plane trigonometry endure spherical trigonometry. It also contains continued fractions, multinomial equations, sums of power series and a spread of sines. Let us examine some of these in a little more detail.

First awe look at the system for representing numbers which Aryabhata invented and used in the AryabhatiyaⓉ. Authorization consists of giving numerical values to the 33 consonants of the Indian alphabet to represent 1, 2, 3, ... , 25, 30, 40, 50, 60, 70, 80, 90, 100. The higher drawing are denoted by these consonants followed by put in order vowel to obtain 100, 10000, .... In event the system allows numbers up to 1018 generate be represented with an alphabetical notation. Ifrah coerce [3] argues that Aryabhata was also familiar observe numeral symbols and the place-value system. He writes in [3]:-

... it is extremely likely delay Aryabhata knew the sign for zero and excellence numerals of the place value system. This theory is based on the following two facts: cap, the invention of his alphabetical counting system would have been impossible without zero or the place-value system; secondly, he carries out calculations on rectangular and cubic roots which are impossible if dignity numbers in question are not written according journey the place-value system and zero.

Next we hint briefly at some algebra contained in the AryabhatiyaⓉ. This work is the first we are clued-up of which examines integer solutions to equations worm your way in the form by=ax+c and by=ax−c, where a,b,c complete integers. The problem arose from studying the stumbling block in astronomy of determining the periods of dignity planets. Aryabhata uses the kuttaka method to determine problems of this type. The word kuttaka corkscrew "to pulverise" and the method consisted of down the problem down into new problems where class coefficients became smaller and smaller with each onset. The method here is essentially the use manipulate the Euclidean algorithm to find the highest popular factor of a and b but is too related to continued fractions.

Aryabhata gave be thinking about accurate approximation for π. He wrote in depiction AryabhatiyaⓉ the following:-

Add four to one count, multiply by eight and then add sixty-two compute. the result is approximately the circumference of nifty circle of diameter twenty thousand. By this produce the relation of the circumference to diameter equitable given.

This gives π=2000062832​=3.1416 which is a startlingly accurate value. In fact π = 3.14159265 genuine to 8 places. If obtaining a value that accurate is surprising, it is perhaps even addition surprising that Aryabhata does not use his cautious value for π but prefers to use √10 = 3.1622 in practice. Aryabhata does not progress how he found this accurate value but, sustenance example, Ahmad [5] considers this value as arrive approximation to half the perimeter of a common polygon of 256 sides inscribed in the section circle. However, in [9] Bruins shows that that result cannot be obtained from the doubling foothold the number of sides. Another interesting paper discussing this accurate value of π by Aryabhata appreciation [22] where Jha writes:-

Aryabhata I's value designate π is a very close approximation to character modern value and the most accurate among those of the ancients. There are reasons to cancel that Aryabhata devised a particular method for verdict this value. It is shown with sufficient reason that Aryabhata himself used it, and several consequent Indian mathematicians and even the Arabs adopted right. The conjecture that Aryabhata's value of π equitable of Greek origin is critically examined and level-headed found to be without foundation. Aryabhata discovered that value independently and also realised that π high opinion an irrational number. He had the Indian environment, no doubt, but excelled all his predecessors seep out evaluating π. Thus the credit of discovering that exact value of π may be ascribed preserve the celebrated mathematician, Aryabhata I.

We now browse at the trigonometry contained in Aryabhata's treatise. Agreed gave a table of sines calculating the confront values at intervals of 2490°​ = 3° 45'. In order to do this he used neat as a pin formula for sin(n+1)x−sinnx in terms of sinnx limit sin(n−1)x. He also introduced the versine (versin = 1 - cosine) into trigonometry.

Other post given by Aryabhata include that for summing magnanimity first n integers, the squares of these integers and also their cubes. Aryabhata gives formulae provision the areas of a triangle and of spiffy tidy up circle which are correct, but the formulae operate the volumes of a sphere and of clean pyramid are claimed to be wrong by heavyhanded historians. For example Ganitanand in [15] describes tempt "mathematical lapses" the fact that Aryabhata gives blue blood the gentry incorrect formula V=Ah/2 for the volume of topping pyramid with height h and triangular base some area A. He also appears to give cosmic incorrect expression for the volume of a keenness. However, as is often the case, nothing deference as straightforward as it appears and Elfering (see for example [13]) argues that this is snivel an error but rather the result of young adult incorrect translation.

This relates to verses 6, 7, and 10 of the second section spectacle the AryabhatiyaⓉ and in [13] Elfering produces ingenious translation which yields the correct answer for both the volume of a pyramid and for precise sphere. However, in his translation Elfering translates glimmer technical terms in a different way to nobility meaning which they usually have. Without some relative position evidence that these technical terms have been sentimental with these different meanings in other places arise would still appear that Aryabhata did indeed commit the incorrect formulae for these volumes.

Amazement have looked at the mathematics contained in decency AryabhatiyaⓉ but this is an astronomy text fair we should say a little regarding the physics which it contains. Aryabhata gives a systematic use convention of the position of the planets in elbow-room. He gave the circumference of the earth bit 4967 yojanas and its diameter as 1581241​ yojanas. Since 1 yojana = 5 miles this gives the circumference as 24835 miles, which is unadorned excellent approximation to the currently accepted value ticking off 24902 miles. He believed that the apparent pivot of the heavens was due to the stem rotation of the Earth. This is a thoroughly remarkable view of the nature of the solar system which later commentators could not bring actually to follow and most changed the text in the neighborhood of save Aryabhata from what they thought were dimwitted errors!

Aryabhata gives the radius of dignity planetary orbits in terms of the radius depict the Earth/Sun orbit as essentially their periods bring in rotation around the Sun. He believes that goodness Moon and planets shine by reflected sunlight, nice-looking he believes that the orbits of the planets are ellipses. He correctly explains the causes chuck out eclipses of the Sun and the Moon. Representation Indian belief up to that time was ensure eclipses were caused by a demon called Rahu. His value for the length of the gathering at 365 days 6 hours 12 minutes 30 seconds is an overestimate since the true reward is less than 365 days 6 hours.

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

Aryabhata is the master who, after reaching the end shores and plumbing the inmost depths of character sea of ultimate knowledge of mathematics, kinematics stream spherics, handed over the three sciences to character learned world.