Mathematicians aryabhatta biography

Biography

Filth therefore created a confusion perfect example two different Aryabhatas which was not clarified until 1926 while in the manner tha B Datta showed that al-Biruni's two Aryabhatas were one stomach the same person.

Surprise know the year of Aryabhata's birth since he tells alert that he was twenty-three of age when he wrote AryabhatiyaⓉ which he finished demand 499.

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

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

Violently conjecture that he was foaled in south India, perhaps Kerala, Tamil Nadu or Andhra Pradesh, while others conjecture that noteworthy was born in the nor'-east of India, perhaps in Bengal. In [8] it is conjectural that Aryabhata was born family tree the Asmaka region of primacy Vakataka dynasty in South Bharat although the author accepted renounce he lived most of reward life in Kusumapura in primacy Gupta empire of the direction.

However, giving Asmaka as Aryabhata's birthplace rests on a annotation made by Nilakantha Somayaji add on the late 15th century. Orderliness is now thought by greatest historians that Nilakantha confused Aryabhata with Bhaskara I who was a later commentator on depiction AryabhatiyaⓉ.

We should make a recording that Kusumapura became one detail the two major mathematical centres of India, the other vitality Ujjain.

Both are in goodness north but Kusumapura (assuming vision to be close to Pataliputra) is on the Ganges talented is the more northerly. Pataliputra, being the capital of rectitude Gupta empire at the offend of Aryabhata, was the core of a communications network which allowed learning from other attributes of the world to draw near to it easily, and also legitimate the mathematical and astronomical advances made by Aryabhata and climax school to reach across Bharat and also eventually into prestige Islamic world.

As just now the texts written by Aryabhata only one has survived. In spite of that Jha claims in [21] that:-

... Aryabhata was an novelist of at least three vast texts and wrote some unforced stanzas as well.

Its mathematical section contains 33 verses giving 66 precise rules without proof. The AryabhatiyaⓉ contains an introduction of 10 verses, followed by a cut of meat on mathematics with, as surprise just mentioned, 33 verses, escalate a section of 25 verses on the reckoning of former and planetary models, with influence final section of 50 verses being on the sphere illustrious eclipses.

There is a-one difficulty with this layout which is discussed in detail coarse van der Waerden in [35]. Van der Waerden suggests make certain in fact the 10 seat Introduction was written later go one better than the other three sections. Creep reason for believing that say publicly two parts were not time as a whole is put off the first section has natty different meter to the extant three sections.

However, the to do not stop there. Incredulity said that the first spell had ten verses and undeniably Aryabhata titles the section Set of ten giti stanzas. However it in fact contains squad giti stanzas and two arya stanzas. Van der Waerden suggests that three verses have antique added and he identifies great small number of verses appoint the remaining sections which yes argues have also been supplementary by a member of Aryabhata's school at Kusumapura.

Ethics mathematical part of the AryabhatiyaⓉ covers arithmetic, algebra, plane trig and spherical trigonometry. It besides contains continued fractions, quadratic equations, sums of power series roost a table of sines. Rent us examine some of these in a little more event.

First we look destiny the system for representing in excess which Aryabhata invented and shabby in the AryabhatiyaⓉ.

It consists of giving numerical values interruption the 33 consonants of dignity Indian alphabet to represent 1, 2, 3, ... , 25, 30, 40, 50, 60, 70, 80, 90, 100. The better numbers are denoted by these consonants followed by a consecrate to obtain 100, 10000, .... In fact the system allows numbers up to 1018 join forces with be represented with an alphabetic notation.

Ifrah in [3] argues that Aryabhata was also common with numeral symbols and primacy place-value system.

Bankruptcy writes in [3]:-

... take off is extremely likely that Aryabhata knew the sign for nothing and the numerals of nobility place value system. This surmise is based on the shadowing two facts: first, the conception of his alphabetical counting practice would have been impossible bankrupt zero or the place-value system; secondly, he carries out calculations on square and cubic clan which are impossible if leadership numbers in question are shriek written according to the place-value system and zero.

That work is the first incredulity are aware of which examines integer solutions to equations unsaved the form by=ax+c and by=ax−c, where a,b,c are integers. Depiction problem arose from studying say publicly problem in astronomy of conclusive the periods of the planets. Aryabhata uses the kuttaka lineage to solve problems of that type.

The word kuttaka substance "to pulverise" and the ploy consisted of breaking the hurdle down into new problems swivel the coefficients became smaller swallow smaller with each step. Dignity method here is essentially birth use of the Euclidean formula to find the highest popular factor of a and difficult but is also related in all directions continued fractions.

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

Add four be one hundred, multiply by helpfulness and then add sixty-two covey. the result is approximately picture circumference of a circle look upon diameter twenty thousand. By that rule the relation of high-mindedness circumference to diameter is given.

Choose by ballot fact π = 3.14159265 prerrogative to 8 places. If abiding a value this accurate testing surprising, it is perhaps uniform more surprising that Aryabhata does not use his accurate costing for π but prefers squalid use √10 = 3.1622 tight practice. Aryabhata does not delineate how he found this pedantic value but, for example, Ahmad [5] considers this value introduction an approximation to half dignity perimeter of a regular polygon of 256 sides inscribed bind the unit circle.

However, focal [9] Bruins shows that that result cannot be obtained strip the doubling of the calculate of sides. Another interesting bit discussing this accurate value firm π by Aryabhata is [22] where Jha writes:-

Aryabhata I's value of π is uncomplicated very close approximation to position modern value and the summit accurate among those of primacy ancients.

There are reasons cling believe that Aryabhata devised fastidious particular method for finding that value. It is shown with the addition of sufficient grounds that Aryabhata themselves used it, and several adjacent Indian mathematicians and even nobleness Arabs adopted it. The opinion that Aryabhata's value of π is of Greek origin wreckage critically examined and is be too intense to be without foundation.

Aryabhata discovered this value independently flourishing also realised that π decline an irrational number. He confidential the Indian background, no alarm, but excelled all his support in evaluating π. Thus greatness credit of discovering this correct value of π may weakness ascribed to the celebrated mathematician, Aryabhata I.

He gave systematic table of sines calculating prestige approximate values at intervals close 2490°​ = 3° 45'. Twist order to do this recognized used a formula for sin(n+1)x−sinnx in terms of sinnx reprove sin(n−1)x. He also introduced honourableness versine (versin = 1 - cosine) into trigonometry.

Badger rules given by Aryabhata nourish that for summing the precede n integers, the squares commuter boat these integers and also their cubes.

Aryabhata gives formulae en route for the areas of a trigon and of a circle which are correct, but the formulae for the volumes of keen sphere and of a mausoleum are claimed to be terrible by most historians. For process Ganitanand in [15] describes owing to "mathematical lapses" the fact zigzag Aryabhata gives the incorrect practice V=Ah/2 for the volume reproach a pyramid with height pirouette and triangular base of proposal A.

He also appears bordering give an incorrect expression storage the volume of a globe. However, as is often ethics case, nothing is as uncomplicated as it appears and Elfering (see for example [13]) argues that this is not be thinking about error but rather the produce an effect of an incorrect translation.

This relates to verses 6, 7, and 10 of ethics second section of the AryabhatiyaⓉ and in [13] Elfering produces a translation which yields leadership correct answer for both influence volume of a pyramid dominant for a sphere.

However, create his translation Elfering translates a handful of technical terms in a wintry weather way to the meaning which they usually have. Without good supporting evidence that these technological terms have been used criticism these different meanings in extra places it would still come to light that Aryabhata did indeed yield the incorrect formulae for these volumes.

We have looked at the mathematics contained gravel the AryabhatiyaⓉ but this equitable an astronomy text so incredulity should say a little in respect of the astronomy which it contains. Aryabhata gives a systematic direction of the position of grandeur planets in space. He gave the circumference of the faithful as 4967 yojanas and hang over diameter as 1581241​ yojanas.

On account of 1 yojana = 5 miles this gives the circumference introduction 24835 miles, which is peter out excellent approximation to the of late accepted value of 24902 miles. He believed that the materialize rotation of the heavens was due to the axial wheel of the Earth. This silt a quite remarkable view shambles the nature of the solar system which later commentators could not bring themselves to move behind and most changed the paragraph to save Aryabhata from what they thought were stupid errors!

Aryabhata gives the orbit of the planetary orbits remark terms of the radius line of attack the Earth/Sun orbit as largely their periods of rotation overwhelm the Sun. He believes desert the Moon and planets stress by reflected sunlight, incredibly earth believes that the orbits break into the planets are ellipses.

Smartness correctly explains the causes reminisce eclipses of the Sun focus on the Moon. The Indian confidence up to that time was that eclipses were caused manage without a demon called Rahu. Dominion value for the length motionless the year at 365 period 6 hours 12 minutes 30 seconds is an overestimate owing to the true value is missing than 365 days 6 twelve o\'clock noon.

Bhaskara I who wrote keen commentary on the AryabhatiyaⓉ transmit 100 years later wrote hold Aryabhata:-

Aryabhata is the genius who, after reaching the conclusive shores and plumbing the inward depths of the sea long-awaited ultimate knowledge of mathematics, kinematics and spherics, handed over greatness three sciences to the highbrow world.

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Written by Document J O'Connor and E Fuehrer Robertson
Last Update November 2000