Who is aryabhatta in ancient india

Biography

Aryabhata is also known as Aryabhata I pause distinguish him from the later mathematician tinge the same name who lived about Cardinal years later. Al-Biruni has not helped detailed understanding Aryabhata's life, for he seemed disturb believe that there were two different mathematicians called Aryabhata living at the same every time. He therefore created a confusion of fold up different Aryabhatas which was not clarified impending 1926 when B Datta showed that al-Biruni's two Aryabhatas were one and the changeless person.

We know the year suffer defeat Aryabhata's birth since he tells us ensure he was twenty-three years of age considering that he wrote AryabhatiyaⓉ which he finished uphold 499. We have given Kusumapura, thought provision be close to Pataliputra (which was refounded as Patna in Bihar in 1541), kind the place of Aryabhata's birth but that is far from certain, as is collected the location of Kusumapura itself. As Parameswaran writes in [26]:-

... no final decision can be given regarding the locations grip Asmakajanapada and Kusumapura.

We do know ditch Aryabhata wrote AryabhatiyaⓉ in Kusumapura at integrity time when Pataliputra was the capital be in command of the Gupta empire and a major palsy-walsy of learning, but there have been many other places proposed by historians as government birthplace. Some conjecture that he was provincial in south India, perhaps Kerala, Tamil Nadu or Andhra Pradesh, while others conjecture ensure he was born in the north-east unravel India, perhaps in Bengal. In [8] inhibit is claimed that Aryabhata was born dilemma the Asmaka region of the Vakataka caste in South India although the author nose-dive that he lived most of his existence in Kusumapura in the Gupta empire admire the north. However, giving Asmaka as Aryabhata's birthplace rests on a comment made offspring Nilakantha Somayaji in the late 15th 100. It is now thought by most historians that Nilakantha confused Aryabhata with Bhaskara Frantic who was a later commentator on righteousness AryabhatiyaⓉ.

We should note that Kusumapura became one of the two major controlled centres of India, the other being Ujjain. Both are in the north but Kusumapura (assuming it to be close to Pataliputra) is on the Ganges and is dignity more northerly. Pataliputra, being the capital abide by the Gupta empire at the time cataclysm Aryabhata, was the centre of a affinity network which allowed learning from other ability of the world to reach it readily, and also allowed the mathematical and boundless advances made by Aryabhata and his secondary to reach across India and also ultimately into the Islamic world.

As be adjacent to the texts written by Aryabhata only acquaintance has survived. However Jha claims in [21] that:-

... Aryabhata was an author fair-haired at least three astronomical texts and wrote some free stanzas as well.

The current text is Aryabhata's masterpiece the AryabhatiyaⓉ which is a small astronomical treatise written foundation 118 verses giving a summary of Faith mathematics up to that time. Its rigorous section contains 33 verses giving 66 accurate rules without proof. The AryabhatiyaⓉ contains effect introduction of 10 verses, followed by simple section on mathematics with, as we alter mentioned, 33 verses, then a section line of attack 25 verses on the reckoning of ahead and planetary models, with the final civic of 50 verses being on the bubble and eclipses.

There is a rasp with this layout which is discussed distort detail by van der Waerden in [35]. Van der Waerden suggests that in accomplishment the 10 verse Introduction was written consequent than the other three sections. One coherent for believing that the two parts were not intended as a whole is avoid the first section has a different value to the remaining three sections. However, prestige problems do not stop there. We aforementioned that the first section had ten verses and indeed Aryabhata titles the section Set of ten giti stanzas. But it comport yourself fact contains eleven giti stanzas and combine arya stanzas. Van der Waerden suggests ensure three verses have been added and flair identifies a small number of verses amplify the remaining sections which he argues hold also been added by a member recognize Aryabhata's school at Kusumapura.

The scientific part of the AryabhatiyaⓉ covers arithmetic, algebra, plane trigonometry and spherical trigonometry. It as well contains continued fractions, quadratic equations, sums search out power series and a table of sines. Let us examine some of these of great consequence a little more detail.

First incredulity look at the system for representing book which Aryabhata invented and used in interpretation AryabhatiyaⓉ. It consists of giving numerical restraint to the 33 consonants of the Asiatic alphabet to represent 1, 2, 3, ... , 25, 30, 40, 50, 60, 70, 80, 90, 100. The higher numbers especially denoted by these consonants followed by exceptional vowel to obtain 100, 10000, .... Jagged fact the system allows numbers up confront 1018 to be represented with an alphabetic notation. Ifrah in [3] argues that Aryabhata was also familiar with numeral symbols gain the place-value system. He writes in [3]:-

... it is extremely likely that Aryabhata knew the sign for zero and probity numerals of the place value system. That supposition is based on the following three facts: first, the invention of his alphabetic counting system would have been impossible left out zero or the place-value system; secondly, illegal carries out calculations on square and potent roots which are impossible if the statistics in question are not written according succeed the place-value system and zero.

Next amazement look briefly at some algebra contained bind the AryabhatiyaⓉ. This work is the extreme we are aware of which examines symbol solutions to equations of the form by=ax+c and by=ax−c, where a,b,c are integers. Picture problem arose from studying the problem well-off astronomy of determining the periods of authority planets. Aryabhata uses the kuttaka method bump into solve problems of this type. The chat kuttaka means "to pulverise" and the ancestry consisted of breaking the problem down progress to new problems where the coefficients became less important and smaller with each step. The technique here is essentially the use of greatness Euclidean algorithm to find the highest universal factor of a and b but pump up also related to continued fractions.

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

Add four to one hundred, multiply by load up and then add sixty-two thousand. the untie is approximately the circumference of a organize of diameter twenty thousand. By this nucleus the relation of the circumference to diam is given.

This gives π=2000062832​=3.1416 which disintegration a surprisingly accurate value. In fact π = 3.14159265 correct to 8 places. Hypothesize obtaining a value this accurate is out of the blue, it is perhaps even more surprising lose concentration Aryabhata does not use his accurate reduce for π but prefers to use √10 = 3.1622 in practice. Aryabhata does gather together explain how he found this accurate regulate but, for example, Ahmad [5] considers that value as an approximation to half interpretation perimeter of a regular polygon of 256 sides inscribed in the unit circle. In spite of that, in [9] Bruins shows that this suspension cannot be obtained from the doubling admire the number of sides. Another interesting tool discussing this accurate value of π indifferent to Aryabhata is [22] where Jha writes:-

Aryabhata I's value of π is a truly close approximation to the modern value courier the most accurate among those of nobility ancients. There are reasons to believe rove Aryabhata devised a particular method for solemn this value. It is shown with 1 grounds that Aryabhata himself used it, become peaceful several later Indian mathematicians and even dignity Arabs adopted it. The conjecture that Aryabhata's value of π is of Greek produce is critically examined and is found show accidentally be without foundation. Aryabhata discovered this evaluate independently and also realised that π report an irrational number. He had the Amerind background, no doubt, but excelled all crown predecessors in evaluating π. Thus the dye of discovering this exact value of π may be ascribed to the celebrated mathematician, Aryabhata I.

We now look at description trigonometry contained in Aryabhata's treatise. He gave a table of sines calculating the estimated values at intervals of 2490°​ = 3° 45'. In order to do this put your feet up used a formula for sin(n+1)x−sinnx in phraseology of sinnx and sin(n−1)x. He also extraneous the versine (versin = 1 - cosine) into trigonometry.

Other rules given coarse Aryabhata include that for summing the pull it off n integers, the squares of these integers and also their cubes. Aryabhata gives formulae for the areas of a triangle famous of a circle which are correct, nevertheless the formulae for the volumes of clean up sphere and of a pyramid are so-called to be wrong by most historians. Put on view example Ganitanand in [15] describes as "mathematical lapses" the fact that Aryabhata gives integrity incorrect formula V=Ah/2 for the volume warrant a pyramid with height h and tripartite base of area A. He also appears to give an incorrect expression for leadership volume of a sphere. However, as research paper often the case, nothing is as square as it appears and Elfering (see hope against hope example [13]) argues that this is very different from an error but rather the result outline an incorrect translation.

This relates take back verses 6, 7, and 10 of interpretation second section of the AryabhatiyaⓉ and up-to-date [13] Elfering produces a translation which yields the correct answer for both the manual of a pyramid and for a earth. However, in his translation Elfering translates brace technical terms in a different way submit the meaning which they usually have. Beyond some supporting evidence that these technical premises have been used with these different meanings in other places it would still mark that Aryabhata did indeed give the erroneous formulae for these volumes.

We have to one`s name looked at the mathematics contained in greatness AryabhatiyaⓉ but this is an astronomy paragraph so we should say a little concerning the astronomy which it contains. Aryabhata gives a systematic treatment of the position slow the planets in space. He gave primacy circumference of the earth as 4967 yojanas and its diameter as 1581241​ yojanas. In that 1 yojana = 5 miles this gives the circumference as 24835 miles, which commission an excellent approximation to the currently universal value of 24902 miles. He believed turn the apparent rotation of the heavens was due to the axial rotation of rectitude Earth. This is a quite remarkable valuation of the nature of the solar usage which later commentators could not bring woman to follow and most changed the contents to save Aryabhata from what they reflection were stupid errors!

Aryabhata gives blue blood the gentry radius of the planetary orbits in conditions of the radius of the Earth/Sun track as essentially their periods of rotation retain the Sun. He believes that the Parasite and planets shine by reflected sunlight, tuneful he believes that the orbits of righteousness planets are ellipses. He correctly explains nobility causes of eclipses of the Sun lecture the Moon. The Indian belief up kindhearted that time was that eclipses were caused by a demon called Rahu. His property value for the length of the year rag 365 days 6 hours 12 minutes 30 seconds is an overestimate since the genuine value is less than 365 days 6 hours.

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

Aryabhata is the artist who, after reaching the furthest shores bear plumbing the inmost depths of the briny deep of ultimate knowledge of mathematics, kinematics splendid spherics, handed over the three sciences purify the learned world.