Aryabhata s Root Extraction Methods. Abhishek Parakh Louisiana State University Aug 31 st 2006


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1 Aryabhata s Root Extracton Methods Abhshek Parakh Lousana State Unversty Aug 1 st 1 Introducton Ths artcle presents an analyss of the root extracton algorthms of Aryabhata gven n hs book Āryabhatīya [1, ] Aryabhata, who was born n 4, occupes an mportant place n the hstory of mathematcs and astronomy Ths year s RSA Conference on cryptology honored hm, and several studes of hs algorthm to solve lnear ndetermnate equatons, whch has applcatons n computer scence, have appeared recently [,4,5] Aryabhata s book presents hs astronomcal and mathematcal theores He took the earth to rotate on ts axs and he gave planet perods wth respect to the sun The books by Datta and Sngh [] and Srnvasengar [] are a good source for a quck hstory of Indan computng algorthms For combnatorc and astronomcal motvatons of Indan mathematcs, see the recent papers by Kak [11], and for an assessment of Indan computatonal methods n a wder context, see the book by Joseph [1] and the essay by Pearce [1] In ths paper we analyze Aryabhata s root extracton methods gven n the mathematcal secton of the Aryabhatya, whch s called the Gantapada In hs earler analyss [1], Klntberg showed that Aryabhata s methods were dfferent from that of Greek mathematcs Here we go beyond Klntberg s analyss and dscuss the larger queston of the theory behnd Aryabhata s algorthms Our analyss shows that Aryabhata was well aware of the place value system of numbers We also look at the computatonal complexty of these methods and observe that the methods taught today n schools for root extracton are essentally an extenson of Aryabhata s methods The Cube Root extracton method Aryabhata presents the followng concse verse of the descrpton of cube root extracton method: 1
2 Translaton []: (Havng subtracted the greatest possble cube from the last cube place and then havng wrtten down the cube root of the number subtracted n the lne of the cube root), dvde the second noncube place (standng on the rght of the last cube place) by thrce the square of the cube root (already obtaned); (then) subtract form the frst noncube place (standng on the rght of the second noncube place) the square of the quotent multpled by thrce the prevous (cuberoot); and (then subtract) the cube (of the quotent) from the cube place (standng on the rght of the frst noncube place) (and wrte down the quotent on the rght of the prevous cube root n the lne of the cube root, and treat ths as the new cube root Repeat the process f there s stll dgts on the rght) Algorthm: Represent the gven number as a seres of ndexed dgts, e, d nd n 1 KK d d1d, where d s the dgt n unts place, d1s the dgt n tens place, d s the dgt n hundreds place and so on Let the fnal root be R and n s the ndex of the left most dgt n the gven number Then the algorthm s as follows: 1 Pck the dgt d such that such that, + k = nteger, k nteger and >, k =1,,, n 1 p =
3 Let k = 1 d d d f d + does not exst, let d = + ; f d +1 does not exst, let d = +1 4 Choose A such that A k and k A s mnmum 5 S = k A R = A l S + d = 1 1 S = l mod ( R ) m S + d 1 = 1 l B = R 11 S m ( R B ) = 1 R = 1 R + B 1 n S + d = 1 14 S = n B 15 = 1 p = p 1 1 If p then go to, else qut Example [1]: Before dscussng the theory behnd the algorthm, let us frst look at ts workng wth the help of an example Let the gven number be x = 4,5, We have to estmate the value of x Step 1: Above each dgt, mark out the cubc places (wth a dash), and noncubc places (wth a hat), startng from rght The unts place s always consdered as a cubc place: 4ˆ 5ˆ ˆ
4 Step : Locate the last cubc place (e the second dgt from the left) and fnd the number that holds ths poston (=4): 4ˆ 5ˆ ˆ Step : Now, fnd the number s nearest lesser (or equal) cube (= e ) and subtract t from 4 Put the cube root of ths lesser cube () n the root result area The root result area now contans the most sgnfcant dgt of the root result: 4ˆ Root Result = 5ˆ ˆ Step 4: Locate the next place to the rght and move ts dgt () down to the rght of the result of the prevous subtracton to form Snce we are now on a second noncubc place, we dvde by three tmes the square of the current root and evaluate the quotent, e = Multply ths quotent wth three tmes the square of current assembled root, e = 54 and subtract t from Ths s equvalent to dong a mod operaton n the step of our algorthm 4ˆ LL 5 4 5ˆ ˆ Root Result = 5 Step 5: Now, brng down the next dgt n the number to form 5 and subtract three tmes the assembled root tmes the square of the last quotent, e = And then place ths quotent as the next dgt of the assembled root result, e 1 + = 4ˆ LL 5 5ˆ ˆ Root Result = 4
5 Step : Agan, brng down the next dgt to the rght of the prevous subtracton to form 5 and subtract the cube of the last quotent, e to get 1 = 4ˆ 5ˆ LL 5 ˆ Root Result = 1 Step : Repeat the steps 4, 5 and untl the remander s zero The resultng calculaton s as follows: 4ˆ 5ˆ ˆ Root Result = We see that the procedure s very smple and the ntermedate numbers generated are easy to handle computatonally Hence, a large problem s broken down nto a seres of small steps Theory: If we look at the bnomal expanson ( A + B) = A + A B + A B + B, the smlarty of Aryabhata s method wth ths expanson s easly notced The gven number s consdered to be ( and our am s to estmate the value of A + B A + B) ( ) 5
6 Aryabhata starts out wth frst subtractng A from the gven number and the correspondng cube root A becomes the frst dgt of our fnal result The value A s determned by tral and error If k s the number from whch A s to be subtracted, then t s observed n the algorthm that the maxmum value of k can be Hence, estmaton of A to determne k A s very easy and equvalent to one look up table complexty In the next step Aryabhata subtracts tmes the assembled root, whch n frst teraton s A Then he estmates the value of B and subtracts tmes the assembled root tmes the square of the estmate of B In the fnal step he subtracts the cube of the estmate of B Thus, our frst approxmaton of cube root becomes1 A + B After the frst teraton of algorthm s done, our frst estmate of cube root s obtaned In other words what we have effectvely acheved s to subtract the closest cube, n our example =, from the number formed by last 5 dgts, that s45 In each teraton, we mprove our knowledge of estmate of cube root by B When we have subtracted exactly ( A + B), the fnal remander wll be zero and the answer we seek wll be ( A + B) The Square Root extracton method Aryabhata presents hs square root extracton method n the followng verse: Translaton []: (Havng subtracted the greatest possble square from the last odd place and then havng wrtten down the square root of the number subtracted n the lne of the square root) always dvde the even place (standng on the rght) by twce the square root Then, havng subtracted the square (of the quotent) from the odd place (standng on the rght), set down the quotent at the next place (e, on the rght of the number already wrtten n the lne of the square root) Ths s the square root (Repeat the process f there are stll dgts on the rght) Algorthm: Represent the gven number as a seres of ndexed dgts, e d nd n 1 KK d d1d, where d s the dgt n unts place, d1s the dgt n tens place, d s
7 the dgt n hundreds place and so on Let the fnal root be R and n s the ndex of the left most dgt n the gven number 1 Pck the dgt d such that = nteger, Where, + k nteger and k >, k =1,,, n 1 p = If d +1 exsts then a + = 1 d +1 d, else d a = 4 Choose A such that A a and a A s mnmum 5 S = a A R = A y = 1 S + d 1 S = y mod ( R) y B = R 1 R = 1 R + B 11 c = 1 S + d 1 S = c B 1 14 = p = p 1 15 If p then go to, else qut Example [1]: Let us look at the workng of the algorthm We perform the steps d escrbed above usng an example: x = 114 Our am s to fnd the value of x
8 Step 1: Mark out the odd places (wth a hat) and the even places (wth a dash), startng from the rght hand sde (unts place), whch s always consdered as an odd place: 1 1ˆ 4ˆ ˆ ˆ Step : Locate the last odd place (n ths case, the second dgt from the left) and fnd the number that holds the poston (=11): 1 1ˆ 4ˆ ˆ ˆ Step : Fnd the number s nearest lesser square (n ths case = ) and subtract from the same poston Put ts square root (= ) asde as the most sgnfcant dgt of the fnal square root 1 1ˆ 4ˆ ˆ ˆ Root result: Step 4: Locate next place to the rght and move the dgt down next to the result of prevous subtracton (= ) to form Snce we are n the even place, we dvde by twce the current root (= ) And then put the quotent of, e 4 ( ) = to the ( ) rght of the current root as the second dgt Now, multply 4 wth (x) gvng 4 and subtract t from 1 1ˆ 4ˆ LL 4 ˆ ˆ Root result: 4 5 Step 5: Move the next dgt down next to the result of prevous subtracton to form 54 Snce, we are at an odd place; we take the prevous quotent and subtract the square of t from 54
9 1 1ˆ 4ˆ LLL ˆ ˆ Root result: 4 Step : We repeat the steps 4 and 5 n tll we reach the frst odd place 1 1ˆ 4ˆ ˆ ˆ Root result: 4 5 Theory: Agan t s notced that bnomal expanson ( A + B) = A + A B + B s smlar to the steps performed n Aryabhata s method The gven number s consdered to be ( A + Our am s to fnd ( A + B) B) Aryabhata starts out wth frst subtractng A from the gven number and the correspondng square root A becomes the frst dgt of our root result The value of A s determned by tralanderror However, ths step s not computatonal ntensve as there are only numbers wth two dgt squares We can set up a lookup table for ths operaton
10 In the next step we subtract tmes the current assembled root (n ths case A ) and estmate B and then subsequently subtract B Once we have done our frst round of subtractons and we are not yet at the last odd poston, then we repeat the algorthm agan After the frst round of subtractons, we have effectvely subtracted ( A + B) from the gven number Ths s actually the closest lesser square subtracted from the number formed by the last four most sgnfcant dgts of the number (n ths case, 114) and our estmated ( A + B) = 4 and ( A + B) =115 Hence, the present value of the assembled root s taken as A for the next round subtractons At every step our knowledge of the root ncreases by one dgt B When we have reached the fnal odd place and performed all the subtractons then f the number s a perfect A + B square, we wll have a null remander Ths s equvalent to subtractng ( ) 4 A Note on Computatonal Complexty The algorthms are observed to be teratve n nature We frst look at the computatonal complexty of the Cube Root extracton algorthm: step (1) of the algorthm nvolves one look up table operaton and one subtracton; step () nvolves four multplcatons, one addton, one dvson and one subtracton; step () agan nvolves four multplcaton, one addton, one dvson and one subtracton; and step (4) agan nvolves the same number of calculatons as step () and () In addton to ths, every teraton uses one multplcaton and one addton n order to accumulate the root result If we represent multplcatons by M, addtons by A, dvsons by D and subtractons by S, then f the number of dgts n the gven number s N, the number of teratons wll N be The computatonal complexty of the cube rootng algorthm turns out to be: N ( 1M + A + D + 4S ) + 1 look up table operaton In a smlar manner the computatonal complexty of the Square Root extracton algorthm turns out to be: N ( 5M + A + D + S ) + 1 look up table operaton 1
11 5 Concluson We nvestgated the theory behnd Aryabhata s algorthms and found that the methods taught n today s school for root extracton are essentally the same as those presented by Aryabhata centures ago Also, the methods are based on the place value system of numbers The computatonal complexty of the algorthms was presented Reference 1 BO Klntberg, Were Aryabhata s Square and Cube Root Methods Orgnally from the Greeks? MSc Dssertaton n Phlosophy and Scence, London School of Economcs and Poltcal Scences, London, 1 KS Shukla and DV Sarma, Aryabhatya of Aryabhata Indan Natonal Scence Academy, 1 S Kak, Computatonal Aspects of the Aryabhata Algorthm, Indan Journal of Hstory of Scence, 1: 1, 1 4 TRN Rao and CH Yang, Aryabhata remander theorem: relevance to cryptoalgorthms, Crcuts, Systems, and Sgnal Processng, 5: 115, 5 S Vuppala, The Aryabhata algorthm usng least absolute remanders arxv: cscr/41 B Datta and AN Sngh, Hstory of Hndu Mathematcs, A Source Book, Parts 1 and, (sngle volume) Asa Publshng House, Bombay, 1 CN Srnvasengar, The Hstory of Ancent Indan Mathematcs The World Press Prvate, Calcutta, 1 S Kak, Arstotle and Gautama on logc and physcs arxv: physcs/551 S Kak, The golden mean and the physcs of aesthetcs arxv: physcs/ S Kak, Indan Physcs: Outlne of Early Hstory arxv: physcs/11 11 S Kak, Brth and Early Development of Indan Astronomy In Astronomy Across Cultures: The Hstory of NonWestern Astronomy, Helane Seln (ed), Kluwer, pp  4, arxv: physcs/11 1 G G Joseph, The Crest of the Peacock, NonEuropean Roots of Mathematcs Prnceton Unversty Press, 1 IG Pearce, Indan mathematcs: redressng the balance 11
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