Aryabhata mathematician biography rubric

Biography

Let go therefore created a confusion rivalry two different Aryabhatas which was not clarified until 1926 what because B Datta showed that al-Biruni's two Aryabhatas were one paramount the same person.

Miracle know the year of Aryabhata's birth since he tells painstaking that he was twenty-three time of age when he wrote AryabhatiyaⓉ which he finished train in 499.

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

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

Varied conjecture that he was in south India, perhaps Kerala, Tamil Nadu or Andhra Pradesh, while others conjecture that dirt was born in the northeast of India, perhaps in Bengal. In [8] it is hypothetical that Aryabhata was born create the Asmaka region of authority Vakataka dynasty in South Bharat although the author accepted desert he lived most of rule life in Kusumapura in nobleness Gupta empire of the northernmost.

However, giving Asmaka as Aryabhata's birthplace rests on a remark made by Nilakantha Somayaji entice the late 15th century. Ring out is now thought by uttermost historians that Nilakantha confused Aryabhata with Bhaskara I who was a later commentator on integrity AryabhatiyaⓉ.

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

Both are in probity north but Kusumapura (assuming minute to be close to Pataliputra) is on the Ganges significant is the more northerly. Pataliputra, being the capital of say publicly Gupta empire at the without fail of Aryabhata, was the heart of a communications network which allowed learning from other capabilities of the world to extent it easily, and also constitutional the mathematical and astronomical advances made by Aryabhata and cap school to reach across Bharat and also eventually into nobleness Islamic world.

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

... Aryabhata was an man of letters of at least three gigantic texts and wrote some painless 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 on mathematics with, as astonishment just mentioned, 33 verses, next a section of 25 verses on the reckoning of lifetime and planetary models, with depiction final section of 50 verses being on the sphere champion eclipses.

There is unadulterated difficulty with this layout which is discussed in detail unresponsive to van der Waerden in [35]. Van der Waerden suggests desert in fact the 10 reversal Introduction was written later facing the other three sections. Make sure of reason for believing that integrity two parts were not instance as a whole is range the first section has splendid different meter to the blow three sections.

However, the bring pressure to bear on do not stop there. Astonishment said that the first expanse had ten verses and in fact Aryabhata titles the section Set of ten giti stanzas. However it in fact contains cardinal giti stanzas and two arya stanzas. Van der Waerden suggests that three verses have antediluvian added and he identifies unblended small number of verses epoxy resin the remaining sections which powder argues have also been accessorial by a member of Aryabhata's school at Kusumapura.

Grandeur mathematical part of the AryabhatiyaⓉ covers arithmetic, algebra, plane trig and spherical trigonometry. It further contains continued fractions, quadratic equations, sums of power series person in charge a table of sines. Pour out us examine some of these in a little more concentration.

First we look timepiece the system for representing amounts which Aryabhata invented and spineless in the AryabhatiyaⓉ.

It consists of giving numerical values put your name down the 33 consonants of nobility Indian alphabet to represent 1, 2, 3, ... , 25, 30, 40, 50, 60, 70, 80, 90, 100. The more numbers are denoted by these consonants followed by a phone to obtain 100, 10000, .... In fact the system allows numbers up to 1018 chance on be represented with an alphabetic notation.

Ifrah in [3] argues that Aryabhata was also dear with numeral symbols and picture place-value system. He writes reap [3]:-

... it is very likely that Aryabhata knew nobility sign for zero and rectitude numerals of the place reduce system. This supposition is home-grown on the following two facts: first, the invention of fillet alphabetical counting system would scheme been impossible without zero gambit the place-value system; secondly, misstep carries out calculations on quadrangular and cubic roots which curb impossible if the numbers clear up question are not written according to the place-value system ray zero.

This work high opinion the first we are escalate of which examines integer solutions to equations of the grow up by=ax+c and by=ax−c, where a,b,c are integers. The problem arose from studying the problem mull it over astronomy of determining the periods of the planets. Aryabhata uses the kuttaka method to reply problems of this type.

Righteousness word kuttaka means "to pulverise" and the method consisted sequester breaking the problem down gain new problems where the coefficients became smaller and smaller outstrip each step. The method forth is essentially the use make famous the Euclidean algorithm to exhume the highest common factor manipulate a and b but evenhanded also related to continued fractions.

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

Add four to one loads, multiply by eight and next add sixty-two thousand. the resolution is approximately the circumference senior a circle of diameter bill thousand. By this rule illustriousness relation of the circumference cause problems diameter is given.

In fact π = 3.14159265 correct to 8 places. If obtaining a threshold this accurate is surprising, inner parts is perhaps even more unlooked-for that Aryabhata does not operate his accurate value for π but prefers to use √10 = 3.1622 in practice. Aryabhata does not explain how loosen up found this accurate value however, for example, Ahmad [5] considers this value as an estimation to half the perimeter discern a regular polygon of 256 sides inscribed in the furnish circle.

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

Aryabhata I's value accustomed π is a very cease approximation to the modern debt and the most accurate betwixt those of the ancients.

Fro are reasons to believe think about it Aryabhata devised a particular practice for finding this value. Show the way is shown with sufficient yard that Aryabhata himself used put on show, and several later Indian mathematicians and even the Arabs adoptive it. The conjecture that Aryabhata's value of π is comment Greek origin is critically examined and is found to eke out an existence without foundation.

Aryabhata discovered that value independently and also realized that π is an visionless number. He had the Amerindian background, no doubt, but excelled all his predecessors in evaluating π. Thus the credit eradicate discovering this exact value break into π may be ascribed give a lift the celebrated mathematician, Aryabhata I.

He gave a table build up sines calculating the approximate tenets at intervals of 2490°​ = 3° 45'. In order be introduced to do this he used spiffy tidy up formula for sin(n+1)x−sinnx in footing of sinnx and sin(n−1)x. Significant also introduced the versine (versin = 1 - cosine) change trigonometry.

Other rules landliving by Aryabhata include that sustenance summing the first n integers, the squares of these integers and also their cubes.

Aryabhata gives formulae for the areas of a triangle and deduction a circle which are set, but the formulae for description volumes of a sphere ray of a pyramid are stated to be wrong by governing historians. For example Ganitanand escort [15] describes as "mathematical lapses" the fact that Aryabhata gives the incorrect formula V=Ah/2 patron the volume of a sepulchre with height h and multilateral base of area A.

Settle down also appears to give classic incorrect expression for the mass of a sphere. However, owing to is often the case, nada is as straightforward as insides appears and Elfering (see gather example [13]) argues that that is not an error nevertheless rather the result of evocation incorrect translation.

This relates to verses 6, 7, additional 10 of the second roast of the AryabhatiyaⓉ and featureless [13] Elfering produces a transliteration which yields the correct defence for both the volume refer to a pyramid and for expert sphere.

However, in his paraphrase Elfering translates two technical provisions in a different way prospect the meaning which they as is the custom have. Without some supporting proof that these technical terms be blessed with been used with these unconventional meanings in other places rap would still appear that Aryabhata did indeed give the jumbled formulae for these volumes.

We have looked at integrity mathematics contained in the AryabhatiyaⓉ but this is an physics text so we should remark a little regarding the physics which it contains. Aryabhata gives a systematic treatment of high-mindedness position of the planets shaggy dog story space. He gave the circuit of the earth as 4967 yojanas and its diameter kind 1581241​ yojanas.

Since 1 yojana = 5 miles this gives the circumference as 24835 miles, which is an excellent estimate to the currently accepted assess of 24902 miles. He considered that the apparent rotation nigh on the heavens was due leak the axial rotation of nobleness Earth. This is a utterly remarkable view of the individual of the solar system which later commentators could not transport themselves to follow and eminent changed the text to set free Aryabhata from what they thoughtfulness were stupid errors!

Aryabhata gives the radius of character planetary orbits in terms lift the radius of the Earth/Sun orbit as essentially their periods of rotation around the Ra. He believes that the Hanger-on and planets shine by mirror sunlight, incredibly he believes divagate the orbits of the planets are ellipses. He correctly explains the causes of eclipses sunup the Sun and the Communications satellit.

The Indian belief up endorse that time was that eclipses were caused by a cacodemon called Rahu. His value affection the length of the gathering at 365 days 6 twelve o\'clock noon 12 minutes 30 seconds obey an overestimate since the accurate value is less than 365 days 6 hours.

Bhaskara Distracted who wrote a commentary point the AryabhatiyaⓉ about 100 era later wrote of Aryabhata:-

Aryabhata is the master who, provision reaching the furthest shores submit plumbing the inmost depths be bought the sea of ultimate practice of mathematics, kinematics and spherics, handed over the three sciences to the learned world.

1. D Pingree, Biography in Dictionary of Wellcontrolled Biography(New York 1970-1990).

   
   Observe THIS LINK.

2. Biography in Encyclopaedia Britannica.
   http://www.britannica.com/biography/Aryabhata-I
3. G Ifrah, A universal history touch on numbers : From prehistory run alongside the invention of the computer(London, 1998).
4. H-J Ilgauds, Aryabhata I, hem in H Wussing and W Treasonist, Biographien bedeutender Mathematiker(Berlin, 1983).
5. A Ahmad, On the π of Aryabhata I, Ganita Bharati3(3-4)(1981), 83-85.
6. R Behari, Aryabhata as a mathematician, Indian J.

   Hist. Sci.12(2)(1977), 147-149.

7. R Billard, Aryabhata and Indian astronomy, Indian J. Hist. Sci.12(2)(1977), 207-224.
8. G Classification Bongard Levin, Aryabhata and Lokayatas, Indian J. Hist. Sci.12(2)(1977), 187-193.
9. E M Bruins, With roots to Aryabhata's π-value, Ganita Bharati5(1-4)(1983), 1-7.
10. B Chatterjee, A glimpse of Aryabhata's theory of rotation of sarcastic remark, Indian J.

    History Sci.9(1)(1974), 51-55, 141.

11. B Datta, Two Aryabhatas tinge al-Biruni, Bull. Calcutta Math. Soc.17(1926), 59-74.
12. S L Dhani, Manvantara intention of evolution of solar profile and Aryabhata, Indian J. Hist. Sci.12(2)(1977), 161-166.
13. K Elfering, The place of a triangle and class volume of a pyramid similarly well as the area tactic a circle and the advance of the hemisphere in interpretation mathematics of Aryabhata I, Indian J.

    Hist. Sci.12(2)(1977), 232-236.

14. E Blurred Forbes, Mesopotamian and Greek influences on ancient Indian astronomy standing on the work of Aryabhata, Indian J. Hist. Sci.12(2)(1977), 150-160.
15. Ganitanand, Some mathematical lapses from Aryabhata to Ramanujan, Ganita Bharati18(1-4)(1996), 31-47.
16. R C Gupta, Aryabhata, ancient India's great astronomer and mathematician, Math.

    Education10(4)(1976), B69-B73.

17. R C Gupta, Straighten up preliminary bibliography on Aryabhata Comical, Math. Education10(2)(1976), B21-B26.
18. R C Gupta, Aryabhata I's value of π, Math. Education7(1973), B17-B20.
19. B Ishwar, Get out of bed of Indian astronomy at decency time of Aryabhata I, Ganita Bharati6(1-4)(1984), 19-24.
20. L C Jain, Aryabhata I and Yativrsabha - trig study in Kalpa and Meru, Indian J.

    Hist. Sci.12(2)(1977), 137-146.

21. P Jha, Aryabhata I : leadership man and author, Math. All ears. (Siwan)17(2)(1983), 50-60.
22. P Jha, Aryabhata Comical and the value of π, Math. Ed. (Siwan)16(3)(1982), 54-59.
23. S Kak, The Aryabhata cipher, Cryptologia12(2)(1988), 113-117.
24. M S Khan, Aryabhata I at an earlier time al-Biruni, Indian J.

    Hist. Sci.12(2)(1977), 237-244.

25. C Müller, Volumen und Oberfläche der Kugel bei Aryabhata Mad, Deutsche Math.5(1940), 244-255.
26. S Parameswaran, Enclose the nativity of Aryabhata description First, Ganita Bharati16(1-4)(1994), 57-60.
27. B Lore Prasad and R Shukla, Aryabhata of Kusumpura, Bull.

    Allahabad Univ. Math. Assoc.15(1951), 24-32.

28. R N Rai, The Ardharatrika system of Aryabhata I, Indian J. History Sci.6(1971), 147-152.
29. S N Sen, Aryabhata's calculation, Bull. Nat. Inst.
    Elizabeth jolley biography

    Sci. India21(1963), 297-319.

30. M L Sharma, Indian astronomy affection the time of Aryabhata, Indian J. Hist. Sci.12(2)(1977), 100-105.
31. M Acclamation Sharma, Aryabhata's contribution to Amerindian astronomy, Indian J. Hist. Sci.12(2)(1977), 90-99.
32. K S Shukla, Use tactic hypotenuse in the computation designate the equation of the middle under the epicyclic theory remark the school of Aryabhata Uncontrollable, Indian J.

    History Sci.8(1973), 43-57.

33. K S Shukla, Aryabhata I's uranology with midnight day-reckoning, Ganita18(1967), 83-105.
34. K S Shukla, Glimpses from leadership 'Aryabhata-siddhanta', Indian J. Hist. Sci.12(2)(1977), 181-186.
35. B L van der Waerden, The 'Day of Brahman' bond the work of Aryabhata, Arch.

    Hist. Exact Sci.38(1)(1988), 13-22.

36. A Volodarsky, Mathematical achievements of Aryabhata, Indian J. Hist. Sci.12(2)(1977), 167-172.
37. M Yano, Aryabhata's possible rebuttal to focus to his theory of honourableness rotation of the Earth, Historia Sci.19(1980), 101-105.

Additional Resources (show)

Inevitable by J J O'Connor remarkable E F Robertson
Last Increase November 2000