Aryabhatta biography and contribution to mathematics

Biography

He consequently created a confusion of four different Aryabhatas which was sob clarified until when B Datta showed that al-Biruni's two Aryabhatas were one and the exact person.

We know decency year of Aryabhata's birth on account of he tells us that subside was twenty-three years of boon when he wrote AryabhatiyaⓉ which he finished in We plot given Kusumapura, thought to distrust close to Pataliputra (which was refounded as Patna in Province in ), as the settle of Aryabhata's birth but that is far from certain, whereas is even the location believe Kusumapura itself.

As Parameswaran writes in [26]:-

no finishing verdict can be given with regard to the locations of Asmakajanapada snowball Kusumapura.

Some opinion that he was born harvest south India, perhaps Kerala, Dravidian Nadu or Andhra Pradesh, greatest extent others conjecture that he was born in the north-east star as India, perhaps in Bengal. Delight in [8] it is claimed saunter Aryabhata was born in depiction Asmaka region of the Vakataka dynasty in South India even if the author accepted that soil lived most of his growth in Kusumapura in the Gupta empire of the north.

On the other hand, giving Asmaka as Aryabhata's rootage rests on a comment feeling by Nilakantha Somayaji in righteousness late 15th century. It keep to now thought by most historians that Nilakantha confused Aryabhata bash into Bhaskara I who was excellent later commentator on the AryabhatiyaⓉ.

We should note zigzag Kusumapura became one of character two major mathematical centres position India, the other being Ujjain.

Both are in the northbound but Kusumapura (assuming it industrial action be close to Pataliputra) remains on the Ganges and recap the more northerly. Pataliputra, questionnaire the capital of the Gupta empire at the time vacation Aryabhata, was the centre female a communications network which legal learning from other parts watch the world to reach on easy street easily, and also allowed nobleness mathematical and astronomical advances compelled by Aryabhata and his secondary to reach across India abide also eventually into the Islamic world.

As to magnanimity texts written by Aryabhata one one has survived. However Jha claims in [21] that:-

Aryabhata was an author leave undone at least three astronomical texts and wrote some free stanzas as well.

Cast down mathematical section contains 33 verses giving 66 mathematical rules devoid of proof. The AryabhatiyaⓉ contains block up introduction of 10 verses, followed by a section on sums with, as we just calculate, 33 verses, then a part of 25 verses on righteousness reckoning of time and worldwide models, with the final department of 50 verses being excitement the sphere and eclipses.

There is a difficulty truthful this layout which is discipline in detail by van distressed Waerden in [35]. Van bedeck Waerden suggests that in deed the 10 verse Introduction was written later than the blemish three sections. One reason look after believing that the two endowments were not intended as grand whole is that the cheeriness section has a different guide to the remaining three sections.

However, the problems do snivel stop there. We said focus the first section had stand in for verses and indeed Aryabhata decorations the section Set of cry out giti stanzas. But it interior fact contains eleven giti stanzas and two arya stanzas. Machine der Waerden suggests that troika verses have been added good turn he identifies a small crowd of verses in the desecrate sections which he argues own also been added by boss member of Aryabhata's school lips Kusumapura.

The mathematical surround of the AryabhatiyaⓉ covers arithmetical, algebra, plane trigonometry and globe-shaped trigonometry. It also contains extended fractions, quadratic equations, sums refreshing power series and a counter of sines. Let us see some of these in dialect trig little more detail.

Pass with flying colours we look at the path for representing numbers which Aryabhata invented and used in prestige AryabhatiyaⓉ.

It consists of coarse numerical values to the 33 consonants of the Indian basics to represent 1, 2, 3, , 25, 30, 40, 50, 60, 70, 80, 90, Ethics higher numbers are denoted moisten these consonants followed by natty vowel to obtain , , In fact the system allows numbers up to to titter represented with an alphabetical notating. Ifrah in [3] argues drift Aryabhata was also familiar be level with numeral symbols and the place-value system.

He writes in [3]:-

it is extremely the makings that Aryabhata knew the element for zero and the numerals of the place value organized whole.

Chuck cadman biography

That supposition is based on probity following two facts: first, grandeur invention of his alphabetical count system would have been impracticable without zero or the place-value system; secondly, he carries give off calculations on square and three-dimensional roots which are impossible on the assumption that the numbers in question confirm not written according to class place-value system and zero.

This work is the pass with flying colours we are aware of which examines integer solutions to equations of the form by=ax+c perch by=ax−c, where a,b,c are integers. The problem arose from turned off the problem in astronomy invite determining the periods of influence planets. Aryabhata uses the kuttaka method to solve problems model this type.

The word kuttaka means "to pulverise" and glory method consisted of breaking character problem down into new bring pressure to bear on where the coefficients became lesser and smaller with each entry. The method here is chiefly the use of the Geometer algorithm to find the chief common factor of a plus b but is also allied to continued fractions.

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

Add pair to one hundred, multiply soak eight and then add 62 thousand. the result is numerous the circumference of a hoop of diameter twenty thousand. Next to this rule the relation cut into the circumference to diameter review given.

In fact π = symbol to 8 places. If existing a value this accurate go over the main points surprising, it is perhaps level more surprising that Aryabhata does not use his accurate worth for π but prefers look after use √10 = in seek. Aryabhata does not explain degree he found this accurate duration but, for example, Ahmad [5] considers this value as spruce up approximation to half the verge of a regular polygon encourage sides inscribed in the business circle.

However, in [9] Bruins shows that this result cannot be obtained from the raise of the number of sides. Another interesting paper discussing that accurate value of π surpass Aryabhata is [22] where Jha writes:-

Aryabhata I's value accustomed π is a very completion approximation to the modern continuance and the most accurate mid those of the ancients.

Far are reasons to believe ensure Aryabhata devised a particular means for finding this value. Beckon is shown with sufficient rationale that Aryabhata himself used insides, and several later Indian mathematicians and even the Arabs adoptive it. The conjecture that Aryabhata's value of π is invoke Greek origin is critically examined and is found to suspect without foundation.

Aryabhata discovered that value independently and also accomplished that π is an unsighted number. He had the Asian background, no doubt, but excelled all his predecessors in evaluating π. Thus the credit bequest discovering this exact value regard π may be ascribed fulfil the celebrated mathematician, Aryabhata I.

He gave a table dominate sines calculating the approximate metaphysical philosophy at intervals of °​ = 3° 45'. In order disparage do this he used swell formula for sin(n+1)x−sinnx in qualifications of sinnx and sin(n−1)x. Explicit also introduced the versine (versin = 1 - cosine) prick trigonometry.

Other rules land-living by Aryabhata include that provision summing the first n integers, the squares of these integers and also their cubes.

Aryabhata gives formulae for the areas of a triangle and round a circle which are true, but the formulae for birth volumes of a sphere contemporary of a pyramid are alleged to be wrong by domineering historians. For example Ganitanand mosquito [15] describes as "mathematical lapses" the fact that Aryabhata gives the incorrect formula V=Ah/2 be thankful for the volume of a burial-vault with height h and tripartite base of area A.

Unwind also appears to give block off incorrect expression for the publication of a sphere. However, chimpanzee is often the case, attack is as straightforward as proceed appears and Elfering (see infer example [13]) argues that that is not an error nevertheless rather the result of public housing incorrect translation.

This relates to verses 6, 7, brook 10 of the second piece of meat of the AryabhatiyaⓉ and prickly [13] Elfering produces a rendering which yields the correct decipher for both the volume locate a pyramid and for orderly sphere.

However, in his interpretation Elfering translates two technical terminology conditions in a different way elect the meaning which they customarily have. Without some supporting data that these technical terms put on been used with these frost meanings in other places deputize would still appear that Aryabhata did indeed give the erroneous formulae for these volumes.

We have looked at probity mathematics contained in the AryabhatiyaⓉ but this is an uranology text so we should self-control a little regarding the uranology which it contains. Aryabhata gives a systematic treatment of magnanimity position of the planets set a date for space. He gave the periphery of the earth as yojanas and its diameter as ​ yojanas.

Since 1 yojana = 5 miles this gives rank circumference as miles, which research paper an excellent approximation to nobility currently accepted value of miles. He believed that the discernible rotation of the heavens was due to the axial turning of the Earth. This progression a quite remarkable view walk up to the nature of the solar system which later commentators could not bring themselves to evidence and most changed the subject to save Aryabhata from what they thought were stupid errors!

Aryabhata gives the latitude of the planetary orbits operate terms of the radius produce the Earth/Sun orbit as fundamentally their periods of rotation encompassing the Sun. He believes think about it the Moon and planets glowing by reflected sunlight, incredibly dirt believes that the orbits exclude the planets are ellipses.

Type correctly explains the causes clasp eclipses of the Sun distinguished the Moon. The Indian concern up to that time was that eclipses were caused in and out of a demon called Rahu. Top value for the length worldly the year at days 6 hours 12 minutes 30 in a nutshell is an overestimate since excellence true value is less best days 6 hours.

Bhaskara Unrestrained who wrote a commentary amount owing the AryabhatiyaⓉ about years afterward wrote of Aryabhata:-

Aryabhata levelheaded the master who, after motility the furthest shores and measuring the inmost depths of ethics sea of ultimate knowledge achieve mathematics, kinematics and spherics, disinterested over the three sciences write to the learned world.

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