Gupta1966 67
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The document summarizes the Hindu method of solving quadratic equations, which originated in India around the same time that positional numeration developed. It discusses how Hindus were the first to recognize negative numbers and irrational numbers as numbers, whereas Greeks only accepted positive roots. The Hindu method involves completing the square to obtain the solution formula. It also discusses Hindus' recognition of imaginary numbers and multiple roots for quadratic equations.
Gupta1966 67
- 1. I ,' l^t,, -" 1t r ,ia"{', l*r / t'i(( (',2 i/ The Hindu Method of SolvingQuadraticEquations R. C. GUPTA* 'Is it that the Greeks had such a marked Let us apply the rule to solve the coniemptfor applied Science, leaving even following quadraticequation(written in the the instruction of their children to slaves? ancient way) But if so, how is it that the nation that gave ax'*b x : c us geom.etry and carried this science so far I,Iultiplyingboth sidesby 4a, did not create even rudimentary algebra ? 4 a"xt+4 abx:4 ac Is it not equally strange that algebra, that Adding b: to both sides, corner-stone of modern mathematics,also 4 a'x"14 abxf b!:b!f 4 ac originatedin India, and at about the same i.e. (? axf b)':b'*.lac (2) time that positional numerationdid ?' Taking root on each side (Quoted bv JawaharLal Nehru in his 2 ax --b: v b._,_aac "D isco ve ry of In dia " p . 211) Hence x:! b-+? n;---b (3) Actually the modern algebra can be 2a said to begin with the proof by Paolo Ruffini (1799) that the equation of fifth The solution (3) can be rvritten down degree can not be solved in terms ol the by employing the Rule of Brahmagupta radicals of the coefficients*. Betore this (c.628 A.D.) given in his Brahma Sphuta time algebrawas mostly identical with the Siddhanta(XVIII-14) and is equivalentto : solution of equations. The method of "The quadratic: The absolutequantity solving a quadratic equationby completing (i.e. c) multiplied by four times the coeffi- the squareis called the Hindu Method. It cient (i.e.a)of the square of the unknorvn, is basedon Sridhara's Rule. The book of is increased the squareol' the coellicient by Sridhara (circa 750 A. D.) in which this (i.e. b) of the unknorvn; the square root of rule was extant not extinct. But many the result beingdiminishedby the coefficient subsequentauthors quoted the rule and of the unknown,is the root." attributed it to Sridhara. One such author is Gyanraj (cira 1503A.D.) who gives the Now the quantity (2 ax*b) is the diffe- rule, in his algebra,as rential coeff. of the L.H.S. of (1) and the qg{l€d{ri sil ei: ca-{4 gq+( | quantity (btr-4 ac) is called the Discri- minant of the equation(l). The intermediary aE4m '.tg'o) .{d1 aiil {dq tl {ii step (2) can be written down immediately "Multiply bo th tlre sides by a quant ir y by applying the aphorism equal to four times the coeftcient of the EF[€(I4rrl ^ - Itdd -r:-- -.:l t4qq{': -a:- squareol the unknown; add to both sides a quantity equal to the square of the coeft- 'Differential-coemcient-square(equals) cient of the unknown; then take the root',. discriminant'. *Aftet this demonstratiod of thc insolvability of quintic cquation algcbraically,a transcendcotalsolutioo involving ellipuc integrals was given by Hermite in Comptes Rendus (tE5E). 11
- 2. This aphorism was disclosed by greatEuropeanalgebraistshad 'not quite Jagadguru swami Shri Bharti Krishna Tirtha, risen to the views taught bl Hindus'. Sankaracharya Govardhan Math, Puri, of (Ibid p. 233). Hindus recognisedtwo during the lectures his interpretationof on answers for quadratic equation. Thus the Vedas in relation to Mathematics at Bhaskara gave'50 anC-5 as roots of the Banaras Hindu Universityin 1949. x'-45x-250 'And the rnost important advanceis Nature and Number of roots : the theory of quadratic equations made in 'The Hindusearly sawin "oppositionof India is unifying under one rule the three direction" on a line an interpretationof cases Diophantus' of (Ibid p. 102) positiveand negative numbers, In Europe 'The first clear recognition of imagina- full possessionof these ideas was not ries was Mahavira's extremely intelligent acquiredbeforeGirard and Descartes (l7th remark (9th century) that, in the natureof century)' (Cajori p. 233). Hindus were things, a negative number has no square the first to recognisethe existence of root. Cauchl' made same observation in absolute negativenumbers and of irrational 1847'(Bell p. 175). For Mahaviracf :- numbers. (Cajori p. l0l) A Babylonian q{ qqol4r erit qo qqt a,it: sql( | of sufficiently remote time, who gave4 as rt €qqa'r-s qril 4ir€(qle diqqq ll thc root ol x!:x* 12 (Mahavira's Ganita-sara sangraha Chap- had solvedhis equation completell'because ter I, VerseNo.52) negative numberswere not in his number- Same ideas are found in otherHindu system(Bell p. ll). Diophantus Alexan- of Mathematics books. We a_eainquote dria the famousGreek algebraist regarded Cajori-'An advancefar beyondthe Greeks 4x*20:4 is the statement of Bhaskarathat "the as "absurd" (Kramerp. 99) And although square of a positive number, as also of a he solved equations of higher degreesbut negative number,is positive;that the square accepted only positive roots. Becauseof root of a positivenumber is two.fold posi- his lack of clear conception of negative tive and negative. Thereis no square root numbersand algebraicsymbolismhe had to of a negative number,for it is not a square" study the quadratic equation separately 102). 1p. under the the threetypes : The Greeks sharply discriminated ( i ) ax eabx - s between numbers and magnitudes, that the ( ii ) ax':bx*c irrational was not recognised by them (i i i ) ax ' { c : bx as a number. 81' Hindus irrationals were takingthe coetEcients always positive. subjected the same processas ordinary to 'Cardan(1501-1576) callednegative roots of numbersand were indeedregarded them b5, an equation as "fictitious" and positive as numbers. By doing so the1, greatly roots as "rea1" (Cajori p. 227). Vieta aided the progress Mathematics. of (1540-1603) also statcdto have rejected is From the foregoi ng statements ofthe all , e xce pt pos it iv er oots o l a n e q u a ti o n . factsof IJistor-vof Mathematics reader the (Ibid p. 230). In fact, as Cajori writes, will easily agrce with Hankel with whose 'Before lTth century the majoritl' oi the q u o ta ti on cl osethi s smal larri cl e I : 27
- 3. "If oneunderstands algebra the applica- by or irrational numbers or space-magnitudes, tion o[ arithmetical operationsto complex th e n the l earned rahrni ns Il i ndustan are B of magnitudesof all sorts, whether rational the real inventor of Algebra" (Cajorip. 195). General Bibliography 6 . Vedic Mathematics JagadguruSwami by l. The Encyclopedia Americana (1963ed.): B h arti K ri sna" (B .H .U . 1965). article by D.J. Struik, Prof. of Math., Developrnentol l{athematicsb-v E.T. M.I.T., U.S.A. Bel l . 2. The Discovery of India by Jawahar Lal 8 . The Main stream ol Mathematicsby Nehru. (Iudian ed. 1960). Edna E. Kramer. 3. Historyot'Hindu Math. by Datta and (Oxf. Univ. Pressl95l) Singh 9 . A history of Elernentary Mathematicsby 4. qiqa +r tieqrs t Ejo qqq){4 | F. Cajori. deqs ttqr I (M acmi l l an o. 196l ) C 5. Symposium the History of sciencein on rriqrd €K dqa t qatsJil sr4(?d I India (1961). NationalInst. of Sciences. 10. India. ; editedand Translated L.C. Jain. by 'This Revcred Swami used to say that be b a d wr ittcn 16 vol umes, onc of cach of thc sixteen sutras, and thai tbe MSS of thcsc volumcs wcre doposited at the house of one of his disciples. Unfortunately, rbc said maouscripts were lost (Vcdic Math' P. X). 28