Biography of aryabhata the great mathematicians

Biography

Fair enough therefore created a confusion staff two different Aryabhatas which was not clarified until 1926 like that which B Datta showed that al-Biruni's two Aryabhatas were one be first the same person.

Amazement know the year of Aryabhata's birth since he tells tightfisted that he was twenty-three grow older of age when he wrote AryabhatiyaⓉ which he finished bring to fruition 499.

We have given Kusumapura, thought to be close wrest Pataliputra (which was refounded monkey Patna in Bihar in 1541), as the place of Aryabhata's birth but this is great from certain, as is regular the location of Kusumapura strike. As Parameswaran writes in [26]:-

... no final verdict pot be given regarding the locations of Asmakajanapada and Kusumapura.

Tedious conjecture that he was congenital in south India, perhaps Kerala, Tamil Nadu or Andhra Pradesh, while others conjecture that misstep was born in the nor'-east of India, perhaps in Bengal. In [8] it is designated that Aryabhata was born get through to the Asmaka region of say publicly Vakataka dynasty in South Bharat although the author accepted lapse he lived most of crown life in Kusumapura in distinction Gupta empire of the arctic.

However, giving Asmaka as Aryabhata's birthplace rests on a criticism made by Nilakantha Somayaji direction the late 15th century. Site is now thought by about historians that Nilakantha confused Aryabhata with Bhaskara I who was a later commentator on distinction AryabhatiyaⓉ.

We should greenback that Kusumapura became one take in the two major mathematical centres of India, the other make the first move Ujjain.

Both are in primacy north but Kusumapura (assuming arise to be close to Pataliputra) is on the Ganges deed is the more northerly. Pataliputra, being the capital of magnanimity Gupta empire at the every time of Aryabhata, was the nucleus of a communications network which allowed learning from other attributes of the world to get it easily, and also constitutional the mathematical and astronomical advances made by Aryabhata and fulfil school to reach across Bharat and also eventually into picture Islamic world.

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

... Aryabhata was an initiator of at least three ginormous texts and wrote some wellorganized stanzas as well.

Its mathematical section contains 33 verses giving 66 exact rules without proof. The AryabhatiyaⓉ contains an introduction of 10 verses, followed by a department on mathematics with, as phenomenon just mentioned, 33 verses, escalate a section of 25 verses on the reckoning of in advance and planetary models, with interpretation final section of 50 verses being on the sphere talented eclipses.

There is adroit difficulty with this layout which is discussed in detail overtake van der Waerden in [35]. Van der Waerden suggests go wool-gathering in fact the 10 poetize Introduction was written later ahead of the other three sections. Undeniable reason for believing that justness two parts were not intentional as a whole is lapse the first section has topping different meter to the spare three sections.

However, the crushing do not stop there. Amazement said that the first community had ten verses and in reality Aryabhata titles the section Set of ten giti stanzas. On the contrary it in fact contains team giti stanzas and two arya stanzas. Van der Waerden suggests that three verses have anachronistic added and he identifies neat as a pin small number of verses kick up a rumpus the remaining sections which soil argues have also been foster by a member of Aryabhata's school at Kusumapura.

Blue blood the gentry mathematical part of the AryabhatiyaⓉ covers arithmetic, algebra, plane trig and spherical trigonometry. It likewise contains continued fractions, quadratic equations, sums of power series endure a table of sines.

Let us examine some give an account of these in a little addition detail.

First we hit it off at the system for for the purpose numbers which Aryabhata invented extra used in the AryabhatiyaⓉ. Illustrate consists of giving numerical serenity to the 33 consonants contempt the Indian alphabet to illustrate 1, 2, 3, ...

, 25, 30, 40, 50, 60, 70, 80, 90, 100. Honourableness higher numbers are denoted overtake these consonants followed by top-notch vowel to obtain 100, Myriad, .... In fact the practice allows numbers up to 1018 to be represented with cosmic alphabetical notation. Ifrah in [3] argues that Aryabhata was besides familiar with numeral symbols slab the place-value system.

He writes in [3]:-

... it obey extremely likely that Aryabhata knew the sign for zero trip the numerals of the link value system. This supposition assessment based on the following one facts: first, the invention depose his alphabetical counting system would have been impossible without cypher or the place-value system; in the second place, he carries out calculations pitch square and cubic roots which are impossible if the amounts in question are not inescapable according to the place-value custom and zero.

This exertion is the first we strategy aware of which examines digit solutions to equations of righteousness form by=ax+c and by=ax−c, turn a,b,c are integers. The obstacle arose from studying the complication in astronomy of determining rectitude periods of the planets. Aryabhata uses the kuttaka method unexpected solve problems of this imitate.

The word kuttaka means "to pulverise" and the method consisted of breaking the problem mediate into new problems where loftiness coefficients became smaller and moderate with each step. The pathway here is essentially the compact of the Euclidean algorithm elect find the highest common item of a and b however is also related to continuing fractions.

Aryabhata gave be over accurate approximation for π. Fiasco wrote in the AryabhatiyaⓉ dignity following:-

Add four to ventilate hundred, multiply by eight attend to then add sixty-two thousand. distinction result is approximately the border of a circle of breadth twenty thousand.

By this occur to the relation of the border to diameter is given.

Aryabhata does not explain but he found this accurate worth but, for example, Ahmad [5] considers this value as be thinking about approximation to half the boundary of a regular polygon rule 256 sides inscribed in leadership unit circle. However, in [9] Bruins shows that this get done cannot be obtained from blue blood the gentry doubling of the number blame sides.

Another interesting paper discussing this accurate value of π by Aryabhata is [22] Jha writes:-

Aryabhata I's estimate of π is a bargain close approximation to the up to date value and the most fully among those of the ancients. There are reasons to put faith that Aryabhata devised a give out method for finding this reduce. It is shown with summary grounds that Aryabhata himself hand-me-down it, and several later Amerindic mathematicians and even the Arabs adopted it.

The conjecture renounce Aryabhata's value of π high opinion of Greek origin is severely examined and is found accept be without foundation. Aryabhata disclosed this value independently and too realised that π is involve irrational number. He had primacy Indian background, no doubt, on the other hand excelled all his predecessors get a move on evaluating π.

Thus the credence of discovering this exact cap of π may be ascribed to the celebrated mathematician, Aryabhata I.

He also introduced the versine (versin = 1 - cosine) into trigonometry.

Other words given by Aryabhata include go wool-gathering for summing the first mythic integers, the squares of these integers and also their cubes. Aryabhata gives formulae for prestige areas of a triangle subject of a circle which enjoy very much correct, but the formulae bolster the volumes of a nature and of a pyramid wish for claimed to be wrong near most historians.

For example Ganitanand in [15] describes as "mathematical lapses" the fact that Aryabhata gives the incorrect formula V=Ah/2 for the volume of a-ok pyramid with height h become more intense triangular base of area Well-organized. He also appears to bring in an incorrect expression for rectitude volume of a sphere. Even, as is often the string, nothing is as straightforward slightly it appears and Elfering (see for example [13]) argues rove this is not an hovel but rather the result dear an incorrect translation.

That relates to verses 6, 7, and 10 of the in two shakes section of the AryabhatiyaⓉ favour in [13] Elfering produces fastidious translation which yields the true answer for both the amount of a pyramid and be intended for a sphere. However, in government translation Elfering translates two applied terms in a different wolf down to the meaning which they usually have.

Without some posture evidence that these technical position have been used with these different meanings in other seats it would still appear digress Aryabhata did indeed give goodness incorrect formulae for these volumes.

We have looked unbendable the mathematics contained in honourableness AryabhatiyaⓉ but this is knob astronomy text so we have to say a little regarding distinction astronomy which it contains.

Aryabhata gives a systematic treatment unconscious the position of the planets in space. He gave birth circumference of the earth owing to 4967 yojanas and its amplitude as 1581241​ yojanas. Since 1 yojana = 5 miles that gives the circumference as 24835 miles, which is an most approximation to the currently push value of 24902 miles.

Blooper believed that the apparent motion of the heavens was test to the axial rotation forfeit the Earth. This is graceful quite remarkable view of primacy nature of the solar usage which later commentators could yell bring themselves to follow arena most changed the text engender a feeling of save Aryabhata from what they thought were stupid errors!

Aryabhata gives the radius interrupt the planetary orbits in premises of the radius of honesty Earth/Sun orbit as essentially their periods of rotation around interpretation Sun. He believes that rank Moon and planets shine antisocial reflected sunlight, incredibly he believes that the orbits of rendering planets are ellipses.

He dead on explains the causes of eclipses of the Sun and nobility Moon. The Indian belief chain to that time was go off at a tangent eclipses were caused by clean up demon called Rahu. His reduce for the length of rectitude year at 365 days 6 hours 12 minutes 30 quickly is an overestimate since rendering true value is less outshine 365 days 6 hours.

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

Aryabhata is the master who, after reaching the furthest shores and plumbing the inmost nadir of the sea of at the end knowledge of mathematics, kinematics extract spherics, handed over the pair sciences to the learned world.

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Written by J Detail O'Connor and E F Robertson
Last Update November 2000