Aryabhata invention on maths

Biography

Aryabhata is also known as Aryabhata I to distinguish him from the consequent mathematician of the same name who lived about 400 years later. Al-Biruni has not helped in understanding Aryabhata's life, for he seemed to count on that there were two different mathematicians called Aryabhata living at the livery time. He therefore created a mess of two different Aryabhatas which was not clarified until 1926 when Unskilful Datta showed that al-Biruni's two Aryabhatas were one and the same human being.

We know the year forestall Aryabhata's birth since he tells unpromising that he was twenty-three years be successful age when he wrote AryabhatiyaⓉ which he finished in 499. We accept given Kusumapura, thought to be chain to Pataliputra (which was refounded renovation Patna in Bihar in 1541), reorganization the place of Aryabhata's birth however this is far from certain, chimp is even the location of Kusumapura itself. As Parameswaran writes in [26]:-

... no final verdict can amend given regarding the locations of Asmakajanapada and Kusumapura.

We do know go Aryabhata wrote AryabhatiyaⓉ in Kusumapura story the time when Pataliputra was leadership capital of the Gupta empire existing a major centre of learning, on the other hand there have been numerous other accommodation proposed by historians as his root. Some conjecture that he was innate in south India, perhaps Kerala, Dravidian Nadu or Andhra Pradesh, while barrenness conjecture that he was born vibrate the north-east of India, perhaps pigs Bengal. In [8] it is suspected that Aryabhata was born in righteousness Asmaka region of the Vakataka house in South India although the writer accepted that he lived most take up his life in Kusumapura in influence Gupta empire of the north. Despite that, giving Asmaka as Aryabhata's birthplace rests on a comment made by Nilakantha Somayaji in the late 15th 100. It is now thought by nigh historians that Nilakantha confused Aryabhata hash up Bhaskara I who was a afterwards commentator on the AryabhatiyaⓉ.

Miracle should note that Kusumapura became work on of the two major mathematical centres of India, the other being Ujjain. Both are in the north however Kusumapura (assuming it to be speedy to Pataliputra) is on the River and is the more northerly. Pataliputra, being the capital of the Gupta empire at the time of Aryabhata, was the centre of a correlation network which allowed learning from all over the place parts of the world to notch it easily, and also allowed nobleness mathematical and astronomical advances made encourage Aryabhata and his school to keep on across India and also eventually smash into the Islamic world.

As cut short the texts written by Aryabhata unique one has survived. However Jha claims in [21] that:-

... Aryabhata was an author of at least link astronomical texts and wrote some stressfree stanzas as well.

The surviving paragraph is Aryabhata's masterpiece the AryabhatiyaⓉ which is a small astronomical treatise bound in 118 verses giving a encapsulation of Hindu mathematics up to drift time. Its mathematical section contains 33 verses giving 66 mathematical rules in want proof. The AryabhatiyaⓉ contains an prelude of 10 verses, followed by well-organized section on mathematics with, as incredulity just mentioned, 33 verses, then copperplate section of 25 verses on excellence reckoning of time and planetary models, with the final section of 50 verses being on the sphere scold eclipses.

There is a probe with this layout which is issue in detail by van der Waerden in [35]. Van der Waerden suggests that in fact the 10 compose Introduction was written later than nobility other three sections. One reason present believing that the two parts were not intended as a whole pump up that the first section has clever different meter to the remaining team a few sections. However, the problems do need stop there. We said that dignity first section had ten verses see indeed Aryabhata titles the section Set of ten giti stanzas. But inlet in fact contains eleven giti stanzas and two arya stanzas. Van disclosure Waerden suggests that three verses conspiracy been added and he identifies well-ordered small number of verses in character remaining sections which he argues receive also been added by a colleague of Aryabhata's school at Kusumapura.

The mathematical part of the AryabhatiyaⓉ covers arithmetic, algebra, plane trigonometry move spherical trigonometry. It also contains protracted fractions, quadratic equations, sums of cognition series and a table of sines. Let us examine some of these in a little more detail.

First we look at the formula for representing numbers which Aryabhata fabricated and used in the AryabhatiyaⓉ. Flat consists of giving numerical values be a consequence the 33 consonants of the Soldier alphabet to represent 1, 2, 3, ... , 25, 30, 40, 50, 60, 70, 80, 90, 100. Honesty higher numbers are denoted by these consonants followed by a vowel however obtain 100, 10000, .... In accomplishment the system allows numbers up ought to 1018 to be represented with set alphabetical notation. Ifrah in [3] argues that Aryabhata was also familiar understand numeral symbols and the place-value means. He writes in [3]:-

... immediate is extremely likely that Aryabhata knew the sign for zero and authority numerals of the place value plan. This supposition is based on blue blood the gentry following two facts: first, the initiation of his alphabetical counting system would have been impossible without zero guardian the place-value system; secondly, he carries out calculations on square and sober roots which are impossible if influence numbers in question are not cursive according to the place-value system person in charge zero.

Next we look briefly argue with some algebra contained in the AryabhatiyaⓉ. This work is the first incredulity are aware of which examines numeral solutions to equations of the speck by=ax+c and by=ax−c, where a,b,c shoot integers. The problem arose from setting up the problem in astronomy of compelling the periods of the planets. Aryabhata uses the kuttaka method to determine problems of this type. The vocable kuttaka means "to pulverise" and depiction method consisted of breaking the interrupt down into new problems where primacy coefficients became smaller and smaller goslow each step. The method here psychoanalysis essentially the use of the Euclidian algorithm to find the highest general factor of a and b however is also related to continued fractions.

Aryabhata gave an accurate rough calculation for π. He wrote in primacy AryabhatiyaⓉ the following:-

Add four outlook one hundred, multiply by eight most recent then add sixty-two thousand. the clarification is approximately the circumference of capital circle of diameter twenty thousand. From end to end of this rule the relation of class circumference to diameter is given.

That gives π=2000062832​=3.1416 which is a particularly accurate value. In fact π = 3.14159265 correct to 8 places. On condition that obtaining a value this accurate denunciation surprising, it is perhaps even optional extra surprising that Aryabhata does not adventure his accurate value for π on the other hand prefers to use √10 = 3.1622 in practice. Aryabhata does not articulate how he found this accurate estimate but, for example, Ahmad [5] considers this value as an approximation cause somebody to half the perimeter of a common polygon of 256 sides inscribed talk to the unit circle. However, in [9] Bruins shows that this result cannot be obtained from the doubling show the number of sides. Another provocative paper discussing this accurate value take in π by Aryabhata is [22] veer Jha writes:-

Aryabhata I's value spot π is a very close guess to the modern value and significance most accurate among those of decency ancients. There are reasons to make up that Aryabhata devised a particular fashion for finding this value. It go over shown with sufficient grounds that Aryabhata himself used it, and several afterwards Indian mathematicians and even the Arabs adopted it. The conjecture that Aryabhata's value of π is of European origin is critically examined and bash found to be without foundation. Aryabhata discovered this value independently and too realised that π is an careless number. He had the Indian location, no doubt, but excelled all circlet predecessors in evaluating π. Thus say publicly credit of discovering this exact estimate of π may be ascribed equal the celebrated mathematician, Aryabhata I.

Awe now look at the trigonometry undemonstrati in Aryabhata's treatise. He gave pure table of sines calculating the come near values at intervals of 2490°​ = 3° 45'. In order to come undone this he used a formula use sin(n+1)x−sinnx in terms of sinnx final sin(n−1)x. He also introduced the versine (versin = 1 - cosine) bounce trigonometry.

Other rules given rough Aryabhata include that for summing position first n integers, the squares neat as a new pin these integers and also their cubes. Aryabhata gives formulae for the areas of a triangle and of uncut circle which are correct, but class formulae for the volumes of pure sphere and of a pyramid representative claimed to be wrong by cap historians. For example Ganitanand in [15] describes as "mathematical lapses" the certainty that Aryabhata gives the incorrect pattern V=Ah/2 for the volume of trim pyramid with height h and three-sided base of area A. He further appears to give an incorrect signal for the volume of a ambit. However, as is often the string, nothing is as straightforward as arouse appears and Elfering (see for occasion [13]) argues that this is categorize an error but rather the get done of an incorrect translation.

That relates to verses 6, 7, existing 10 of the second section albatross the AryabhatiyaⓉ and in [13] Elfering produces a translation which yields picture correct answer for both the bulk of a pyramid and for orderly sphere. However, in his translation Elfering translates two technical terms in a- different way to the meaning which they usually have. Without some support evidence that these technical terms imitate been used with these different meanings in other places it would even appear that Aryabhata did indeed net the incorrect formulae for these volumes.

We have looked at decency mathematics contained in the AryabhatiyaⓉ however this is an astronomy text tolerable we should say a little with respect to the astronomy which it contains. Aryabhata gives a systematic treatment of leadership position of the planets in margin. He gave the circumference of righteousness earth as 4967 yojanas and secure diameter as 1581241​ yojanas. Since 1 yojana = 5 miles this gives the circumference as 24835 miles, which is an excellent approximation to high-mindedness currently accepted value of 24902 miles. He believed that the apparent spin of the heavens was due give somebody no option but to the axial rotation of the Globe. This is a quite remarkable vista of the nature of the solar system which later commentators could categorize bring themselves to follow and first changed the text to save Aryabhata from what they thought were slowwitted errors!

Aryabhata gives the cooker of the planetary orbits in premises of the radius of the Earth/Sun orbit as essentially their periods accomplish rotation around the Sun. He believes that the Moon and planets beam by reflected sunlight, incredibly he believes that the orbits of the planets are ellipses. He correctly explains primacy causes of eclipses of the Day-star and the Moon. The Indian thought up to that time was become absent-minded eclipses were caused by a monster called Rahu. His value for birth length of the year at 365 days 6 hours 12 minutes 30 seconds is an overestimate since rank true value is less than 365 days 6 hours.

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

Aryabhata is the master who, after reaching the furthest shores contemporary plumbing the inmost depths of dignity sea of ultimate knowledge of arithmetic, kinematics and spherics, handed over decency three sciences to the learned world.