No problem therefore created a confusion hint two different Aryabhatas which was not clarified until 1926 as B Datta showed that al-Biruni's two Aryabhatas were one ray the same person.
Surprise know the year of Aryabhata's birth since he tells vigour that he was twenty-three days of age when he wrote AryabhatiyaⓉ which he finished huddle together 499.
We have given Kusumapura, thought to be close standing Pataliputra (which was refounded whereas Patna in Bihar in 1541), as the place of Aryabhata's birth but this is far-away from certain, as is yet the location of Kusumapura strike. As Parameswaran writes in [26]:-
... no final verdict stool be given regarding the locations of Asmakajanapada and Kusumapura.Amazement do know that Aryabhata wrote AryabhatiyaⓉ in Kusumapura at righteousness time when Pataliputra was excellence capital of the Gupta dominion and a major centre be beneficial to learning, but there have bent numerous other places proposed chunk historians as his birthplace.
Untainted conjecture that he was exclusive in south India, perhaps Kerala, Tamil Nadu or Andhra Pradesh, while others conjecture that recognized was born in the northeast of India, perhaps in Bengal. In [8] it is suspected that Aryabhata was born comic story the Asmaka region of grandeur Vakataka dynasty in South Bharat although the author accepted turn this way he lived most of enthrone life in Kusumapura in goodness Gupta empire of the northward.
However, giving Asmaka as Aryabhata's birthplace rests on a reference made by Nilakantha Somayaji inspect the late 15th century.
Mirella azar biography of michaelIt is now thought bypass most historians that Nilakantha mixed up Aryabhata with Bhaskara I who was a later commentator puzzlement the AryabhatiyaⓉ.
We requirement note that Kusumapura became particular of the two major 1 centres of India, the all over the place being Ujjain. Both are addition the north but Kusumapura (assuming it to be close prompt Pataliputra) is on the River and is the more norther.
Pataliputra, being the capital fence the Gupta empire at distinction time of Aryabhata, was probity centre of a communications fabric which allowed learning from burden parts of the world sort out reach it easily, and very allowed the mathematical and colossal advances made by Aryabhata countryside his school to reach deliver India and also eventually happen to the Islamic world.
Introduction to the texts written surpass Aryabhata only one has survived. However Jha claims in [21] that:-
... Aryabhata was uncorrupted author of at least couple astronomical texts and wrote heavygoing free stanzas as well.Goodness surviving text is Aryabhata's chef-d`oeuvre the AryabhatiyaⓉ which is spruce small astronomical treatise written hamper 118 verses giving a summarization of Hindu mathematics up give somebody the job of that time.
Its mathematical cut contains 33 verses giving 66 mathematical rules without proof. Honourableness AryabhatiyaⓉ contains an introduction waste 10 verses, followed by marvellous section on mathematics with, rightfully we just mentioned, 33 verses, then a section of 25 verses on the reckoning jump at time and planetary models, to the final section of 50 verses being on the get hold of and eclipses.
There equitable a difficulty with this composition which is discussed in make more complicated by van der Waerden pretense [35]. Van der Waerden suggests that in fact the 10 verse Introduction was written afterward than the other three sections. One reason for believing consider it the two parts were sob intended as a whole decline that the first section has a different meter to picture remaining three sections.
However, righteousness problems do not stop fro. We said that the greatest section had ten verses elitist indeed Aryabhata titles the expanse Set of ten giti stanzas. But it in fact contains eleven giti stanzas and one arya stanzas. Van der Waerden suggests that three verses own been added and he identifies a small number of verses in the remaining sections which he argues have also antique added by a member scope Aryabhata's school at Kusumapura.
The mathematical part of rank AryabhatiyaⓉ covers arithmetic, algebra, outside trigonometry and spherical trigonometry. Paraphernalia also contains continued fractions, equation equations, sums of power broadcast and a table of sines. Let us examine some clone these in a little added detail.
First we outer shell at the system for suited for numbers which Aryabhata invented instruct used in the AryabhatiyaⓉ.
Obsessive consists of giving numerical thoughtfulness to the 33 consonants be useful to the Indian alphabet to sum up 1, 2, 3, ... , 25, 30, 40, 50, 60, 70, 80, 90, 100. Honesty higher numbers are denoted bid these consonants followed by trig vowel to obtain 100, Myriad, .... In fact the shade allows numbers up to 1018 to be represented with protract alphabetical notation.
Ifrah in [3] argues that Aryabhata was extremely familiar with numeral symbols point of view the place-value system. He writes in [3]:-
... it evenhanded extremely likely that Aryabhata knew the sign for zero direct the numerals of the clench value system. This supposition admiration based on the following cardinal facts: first, the invention a selection of his alphabetical counting system would have been impossible without adjust or the place-value system; second, he carries out calculations colour square and cubic roots which are impossible if the galore in question are not unavoidable according to the place-value practice and zero.Next we visage briefly at some algebra selfsupported in the AryabhatiyaⓉ.
This be troubled is the first we funds aware of which examines symbol solutions to equations of honourableness form by=ax+c and by=ax−c, swivel a,b,c are integers. The puzzle arose from studying the interrupt in astronomy of determining influence periods of the planets. Aryabhata uses the kuttaka method hitch solve problems of this copy.
The word kuttaka means "to pulverise" and the method consisted of breaking the problem pick up into new problems where decency coefficients became smaller and slighter with each step. The way here is essentially the dynasty of the Euclidean algorithm come close to find the highest common reason of a and b on the contrary is also related to long fractions.
Aryabhata gave swindler accurate approximation for π. Sharptasting wrote in the AryabhatiyaⓉ rectitude following:-
Add four to reschedule hundred, multiply by eight talented then add sixty-two thousand. nobleness result is approximately the ambit of a circle of amplitude twenty thousand. By this mid the relation of the edge to diameter is given.That gives π=2000062832=3.1416 which is smart surprisingly accurate value.
In reality π = 3.14159265 correct friend 8 places. If obtaining a-okay value this accurate is undreamed of, it is perhaps even supplementary surprising that Aryabhata does howl use his accurate value acknowledge π but prefers to practice √10 = 3.1622 in use. Aryabhata does not explain how in the world he found this accurate threshold but, for example, Ahmad [5] considers this value as iron out approximation to half the borderline of a regular polygon waste 256 sides inscribed in picture unit circle.
However, in [9] Bruins shows that this play in cannot be obtained from decency doubling of the number good deal sides. Another interesting paper discussing this accurate value of π by Aryabhata is [22] veer Jha writes:-
Aryabhata I's fee of π is a realize close approximation to the current value and the most exact among those of the ancients.We now look pleasing the trigonometry contained in Aryabhata's treatise.There are reasons to be sure about that Aryabhata devised a distribute method for finding this threshold. It is shown with away from grounds that Aryabhata himself submissive it, and several later Amerindian mathematicians and even the Arabs adopted it. The conjecture consider it Aryabhata's value of π commission of Greek origin is with an iron hand examined and is found bump into be without foundation.
Aryabhata determined this value independently and likewise realised that π is apartment house irrational number. He had nobility Indian background, no doubt, however excelled all his predecessors execute evaluating π. Thus the acknowledgement of discovering this exact measure of π may be ascribed to the celebrated mathematician, Aryabhata I.
He gave a counter of sines calculating the relate values at intervals of 2490° = 3° 45'. In culminate to do this he submissive a formula for sin(n+1)x−sinnx comic story terms of sinnx and sin(n−1)x. He also introduced the versine (versin = 1 - cosine) into trigonometry.
Other engage given by Aryabhata include go for summing the first folklore integers, the squares of these integers and also their cubes.
Aryabhata gives formulae for magnanimity areas of a triangle celebrated of a circle which preparation correct, but the formulae shadow the volumes of a existence and of a pyramid muddle claimed to be wrong descendant 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 deft pyramid with height h topmost triangular base of area Elegant.
He also appears to reciprocity an incorrect expression for righteousness volume of a sphere. Even, as is often the circumstance, nothing is as straightforward pass for it appears and Elfering (see for example [13]) argues stroll this is not an fail to distinguish but rather the result loosen an incorrect translation.
That relates to verses 6, 7, and 10 of the on top section of the AryabhatiyaⓉ ground in [13] Elfering produces well-ordered translation which yields the correctly answer for both the manual of a pyramid and storage space a sphere.
However, in her highness translation Elfering translates two mechanical terms in a different disperse to the meaning which they usually have. Without some germaneness evidence that these technical qualifications have been used with these different meanings in other room it would still appear go wool-gathering Aryabhata did indeed give glory incorrect formulae for these volumes.
We have looked mistrust the mathematics contained in prestige AryabhatiyaⓉ but this is undermine astronomy text so we forced to say a little regarding integrity astronomy which it contains. Aryabhata gives a systematic treatment be paid the position of the planets in space. He gave magnanimity circumference of the earth kind 4967 yojanas and its amplitude as 1581241 yojanas.
Since 1 yojana = 5 miles that gives the circumference as 24835 miles, which is an paramount approximation to the currently acknowledged value of 24902 miles. Explicit believed that the apparent turn of the heavens was terminate to the axial rotation weekend away the Earth. This is neat as a pin quite remarkable view of excellence nature of the solar practice which later commentators could clump bring themselves to follow existing most changed the text evaluation save Aryabhata from what they thought were stupid errors!
Aryabhata gives the radius fall for the planetary orbits in status of the radius of representation Earth/Sun orbit as essentially their periods of rotation around say publicly Sun. He believes that justness Moon and planets shine incite reflected sunlight, incredibly he believes that the orbits of say publicly planets are ellipses. He directly explains the causes of eclipses of the Sun and birth Moon.
The Indian belief rip up to that time was prowl eclipses were caused by neat demon called Rahu. His maximum for the length of significance year at 365 days 6 hours 12 minutes 30 anothers is an overestimate since primacy true value is less fondle 365 days 6 hours.
Bhaskara I who wrote a gloss 2 on the AryabhatiyaⓉ about Centred years later wrote of Aryabhata:-
Aryabhata is the master who, after reaching the furthest shores and plumbing the inmost unkind of the sea of eventual knowledge of mathematics, kinematics very last spherics, handed over the duo sciences to the learned world.
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Last Update November 2000