Bijaganita

8/10/2019 Bijaganita

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DEVELOPMENT OF B JAGAITA

M.D.SRINIVASCENTRE FOR POLICY STUDIES

[email protected]


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BJAGAITA OR AVYKTA-GAITA B jagaita or avyakta-gaita, is computation with seeds, or computationwith unmanifest or unknown quantities, which are usually denoted byvaras, colours or symbols.

The following invocatory verse of B jagaita of Bh skar crya II (c.1150)has been interpreted in three different ways by Ka Daivaja (c.1600)

I salute that avyakta (prakti or primordial nature), which the philosophersof the S khya School declare to be the producer of buddhi (theintellectual principle mahat), while it is being directed by the immanentPurua (the Being). It is the sole b ja (seed or the cause) of all that is

manifest.


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BJAGAITA OR AVYKTA-GAITA

I salute that a (the ruling power, Brahman), which the S khyas(those who have realised the Self) declare to be the producer of buddhi(tattvaj na or true knowledge of reality), which arises in adistinguished person (who has accomplished the four-fold s dhanas ofviveka, etc.). It is the sole b ja (seed or the cause) of all that ismanifest.

I salute that avyakta-gaita (computations with unmanifest orindeterminate quantities), which the S khya (who are proficient innumbers) declare to be the producer of buddhi (mathematicalknowledge), which arises in a distinguished person (proficient inmathematics). It is the sole b ja (seed or the cause) of all vyakta-gaita(computations with manifest quantities, such as arithmetic, geometryetc.)


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The notion of a variable quantity, y vat-t vat (as many as), goes back toulvas tras. The K tyyana ulvas tra deals with the problem ofconstructing a square whose area is n-times that of a given square:

[ . ]As many squares as you wish to combine into one, the transverse line willbe one less than that. Twice the side will be one more than that. That willbe the triangle. Its arrow (altitude) will produce that.


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ryabhaiya (c.499) uses the term gulik for the unknown. There, we alsofind the solution of linear and quadratic equations and also the ku akaprocess for the solution of linear indeterminate equations.

Bhskara I (c.629) uses the notion of y vat-t vat in his commentary of ryabha ya .

Brahmagupta has given a detailed exposition of b jagaita in the ChapterXVIII, Kuak dhy ya of his Br hmasphuasiddh nta (c.628). This workhas been commented upon by Pth dakasv mi (c.860)r pati deals with avyakta-gaita in Chapter XIV of his Siddh nta ekhara(c.1050)

Bhskar crya II has written the most detailed available treatise on B jaganita (1150). There, he states that he has only compiled and abridgedfrom the treatises of r dhara (c.750) and Padman bha, which are notavailable. B jaganita has been commented upon by S ryad sa (c.1540)and Ka Daivaja (c.1600).

Nryaa Paita (c.1350) has also composed a treatise, B jagait vatasa , of which only the first few chapters are available.


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KUAK DHY YA OF BR HMASPHUASIDDH NTA (c.628)

The following are the topics dealt with in the Chapter XVIII,Kuak dhy ya , of Br hmasphuasiddh nta .

Solutions of linear indeterminate equations by ku aka process and itsapplications in astronomical problems

Rule of signs and arithmetic of zero Surds (kara ) Operations with unknowns (vara-avidha or avyakta- avidha) Equations with single unknown (ekavara-sam karaa)

Elimination of middle term in quadratic equations (madhyam haraa) Equations with several unknowns (anekavara-sam karaa) Equations with products of unknowns (bh vita) Vargaprakti: Second order indeterminate equation x2 - D y2 = 1.

Bhvan and applications to finding rational and integral solutions. Various problems


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RULE OF SIGNS AND ARITHMETIC OF ZERO

The Br hmasphuasiddh nta is the first available text which discusses thearithmetic of zero ( nya-parikarma) as well as the six operations withpositive and negative numbers (dhanaa-avidha). These are discussed

together in a set of verses at the beginning of the kuak dhy ya .


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KA ON THE NOTION OF NEGATIVE NUMBERS

Ka Daivaja (c.1600), in his commentary B janav kur on B jaganitaof Bh skara II, explains how negativity is to be understood in differentcontexts. He then goes onto show that this physical interpretation of

negativity can be used to demonstrate the rule of signs in algebra indifferent situations.


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KA ON THE NOTION OF NEGATIVE NUMBERS

'Negativity (atva) here is of three types: Spatial, temporal and thatpertaining to objects. In each case, [negativity] is indeed the vaipar tya orthe oppositeness. As has been clearly stated by the crya in L l vat in

the example The bhuj s are ten and seventeen etc. For instance, theother direction in a line is called the opposite direction (vipar ta dik); justas west is the opposite of east... Further, between two stations if one wayof traversing is considered positive then the other is negative... In the sameway past and future time intervals will be mutually negative of eachother... Similarly, when one possesses said objects they would be called

his dhana (wealth). The opposite would be the case when another owns thesame objects... '

The example discussed by Bh skara in L l vat has to with the calculationof the base-intercepts ( bdhas) in a triangle. There, Bh skara explainsthat if the calculation leads to a negative intercept, it should be interpretedas going in the opposite direction. This happens when the foot of the

altitude falls outside the base.


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BH SKARAS EXAMPLE OF NEGATIVE INTERCEPT

[ ]

'In a triangle, which has sides 10, 17 and 9, tell me quickly, Ohmathematician, the base intercepts and the area.'

In his v san, Bh skara gives the solution of this problem as follows

- ,

Here, using the rule, 'The sum of the sides...', we get (the difference of theintercepts to be) 21. We cannot subtract this from the base (9). Hence, thebase has to be subtracted from this (difference) only and the half of theresult is the intercept which is negative, because it is in the oppositedirection. Thus the intercepts are (-6, 15).


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ALGEBRAIC NOTATION

In the ryabha yabh ya of Bh skara (c.629) we find references to thealgebraic notation used in Indian mathematics. Various features of thenotation are more clearly known from the Bakh l manuscript (c. 700).

The system of algebraic notation is explained and fully exemplified in the B jagaita of Bh skara II (c.1050), together with his own auto-commentary v san .

In v san on verse 3 of B jagaita , Bh skara says

Here (in algebra), the initial letters of both the known and unknownquantities should be written as their signs. Similarly those (quantities)which are negative, they have (to be shown with) a dot over them.


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ALGEBRAIC NOTATION

[Takao Hayashi, Algebra in India: B jagaita, in Selin ed. Encyclopaedia of History of Science,Technology and Medicine in Non-Western Cultures , 2008, p.112]


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VARGA PRAKTI

In Chapter 18 of his Br hmasphu asiddh nta (c.628), Brahmagupta

discussed the problem of solving for integral values of X, Y, the equation

X2 - D Y 2 = K

given a non-square integer D > 0, and an integer K.

X is called the jye ha-m la, Y is called the kani ha-m la

D is the prakti, K is the kepa

One motivation for this problem is that of finding rational approximationsto square-root of D. If X, Y are integers such that X 2 - D Y 2 = 1, then,

The ulva-s tra approximation 2 ~ 1+ 1/3 + 1/3.4 - 1/3.4.34 = 577/408

is an example as (577) 2 - 2 (408) 2 = 1.


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BRAHMAGUPTAS BH VAN

If X 12 - D Y 12 = K 1 and X 22- D Y 22 = K 2 then

(X 1 X2 D Y 1 Y 2)2 - D (X 1 Y2 X 2Y1)2 = K 1 K 2

In particular, given X 2 -D Y 2 = K, we get the rational solution

[(X 2 + D Y 2)/K] 2 - D [(2XY)/K] 2 = 1

Also, if one solution of the Equation X 2- D Y 2 = 1 is found, an infinite

number of solutions can be found, via (X, Y) (X2

+ D Y2

, 2XY)


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USE OF BH VAN WHEN K = -1, 2, 4

The bh van principle can be use to obtain a solution of the equation

x2 - D y2 = 1

if we have a solution of the equation

x12 - D y1

2 = K for K = -1, 2, 4


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BHVAN AND RATIONAL APPROXIMATION OF SQUARE-ROOTS

Let us start with ( x, y) such that x2 - D y2 = 1

x1 /y1 = ( x2 + D y2)/ (2 x y) = (2 x2 -1)/ (2 x y) = ( x / y) - 1/ y.2 x

If the solution ( x2, y2) is obtained by bh van of ( x1, y1) with itself, then

x2 /y2 = ( x1 /y1) - 1/ y.2 x1 = ( x / y) - (1/ y.2 x) - [1/ y.2 x.(4 x2-2)]

Thus, we have a series of better approximations which may expressed inthe form

( x r /yr) = ( x / y) - (1/ y.n 1) - (1/ y. n 1 .n 2) - ... - (1/ y. n 1 .n 2... n r)

where n 1 = 2 x and n i = n i-12 - 2 , for i = 2, 3, ..., r .

Example : For D = 2, we start with x = 3 and y = 2. We have

x2 /y2 =( 3/2) - 1/2.6 -1/2.6. (62-2) = (3/2) -1/2.6 -1/2.6.34

By re-grouping the first two terms, the above approximation can be seento be the same as in the ulva-s tras. We can now generate further termsto get the series

2 ~ 1+1/3 + 1/3.4 -1/3.4.34 -1/3.4.34.1154 - 1/3.4.34.1154.1331714 - ...where 1154 = 34 2 -2, 133714 = 1154 2 - 2, and so on.


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CAKRAV LA: THE CYCLIC METHOD

The first known reference to Cakrav la or the cyclic method occurs in a

work of Udayadiv kara (c.1073), who cites the following verses of crya

Jayadeva :


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CAKRAV LA ACCORDING TO JAYADEVA

Given X i , Y i , K i such that X i 2 - D Y i2 = K i

First find P i+1 as follows:

(I) Use kuaka process to solve(Y i P i+1 + X i)/ K i = Y i+1

for integral P i+1 , Y i+1

(II) Of the solutions of the above, choose P i+1 such that

(P i+1 2 - D)/K i has the least value

Then setK i+1 = (P i+1 2 - D)/ K i Y i+1= (Y i P i+1 + X i)/ |K i|

X i+1= (X i P i+1 + DY i)/ |K i|

These satisfy X i+12 - D Y i+12 = K i+1

Iterate the process till K i+1 = 1, 2 or 4, and then solve the equationusing bh van .

Jayadevas verses do not reveal how condition II is to be interpreted.


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CAKRAV LA ACCORDING TO BH SKARA (c. 1150)

We do not have any examples from Jayadeva to illustrate how condition II

is to be interpreted.

In his B jagaita, Bh skar crya gives the following description ofCakrav la:

Bhskara has given the Condition II in the precise form:

(II) Choose P i+1 such that |(P i+12 - D)| has the least value


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CAKRAV LA ACCORDING TO BH SKARA

In 1930, Krishnaswamy Ayyangar showed that the Cakravala algorithmalways leads to a solution of the Varga-Prakti equation with K=1. He alsoshowed that condition (I) is equivalent to the simpler condition

(I) P i + P i+1 is divisible by K i

Thus, we shall now use the Cakrav la algorithm in the following form:

To solve X2 - D Y 2 = 1

Set X 0 = 1, Y 0 = 0, K 0 = 1 and P 0 = 1.

Given X i , Y i , K i such that X i 2 - D Y i2 = K i

First find P i+1 so as to satisfy:(I) P i + P i+1 is divisible by K i (II) P i+12 -D is minimum .

Then setK i+1 = (P i+1 2 - D)/K i Y i+1= (Y i P i+1 + X i)/ |K i| X i+1= (X i P i+1 + DY i)/ |K i|

These satisfy X i +12 - D Y i +1

2 = K i +1

Iterate till K i +1 = 1, 2 or 4, and then use Bh van .


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BHASKARAS EXAMPLES


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BH SKARAS EXAMPLE: X 2 - 61 Y 2 = 1

To find P 1: 0+7, 0+8, 0+9 ... divisible by 1. 8 2 closest to 61. P 1 = 8, K 1 = 3

To find P 2: 8+4, 8+7, 8+10 ... divisible by 3. 72

closest to 61. P 2 =7, K 1= -4

After the second step, we have: 39 2 - 61 x 5 2 = -4

Now, since have reached K=-4, we can use bh van principle to obtain

X = (39 2 +2) [ () (39 2 +1) (39 2 +3) - 1] = 1,766,319,049

Y = () (39 x 5) (39 2 +1) (39 2 +3) = 226,153,980

1766319049 2 - 61. 226153980 2 = 1

I P i K i a i i X i Y i

0 0 1 8 1 1 0

1 8 3 5 -1 8 1

2 7 -4 4 1 39 5

3 9 -5 3 -1 164 21


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BH SKARAS EXAMPLE: X 2 - 61 Y 2 = 1

I P i K i a i i X i Y i

3 9 -5 3 -1 164 2

4

6

5

3 1 453

55 9 4 4 -1 1523 19

6 7 -3 5 1 5639 72

7 8 -1 16 -1 29718 380

8 8 -3 5 -1 469849 6015

9 7 4 4 1 2319527 29698

10 9 5 3 -1 9747957 124809

11 6 -5 3 1 26924344 344730

12 9 -4 4 -1 90520989 1159002

13 7 3 5 1 335159612 4291279

14 8 1 16 -1 1766319049 226153980


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BHASKARAS EXAMPLE: X 2 - 67 Y 2 = 1

I P i K i a i i X i Y i0 0 1 8 1 1 0

1 8 -3 5 1 8 1

2 7 6 2 1 41 5

3 5 -7 2 1 90 11

4 9 -2 9 -1 221 27

5 9 -7 2 -1 1899 232

6 5 6 2 1 3577 437

7 7 -3 5 1 9053 1106

8 8 1 16 1 48842 5967

To find P 1: 0+7, 0+8, 0+9 ... divisible by 1. 8 2 closest to 67. P 1 = 8, K 1 = -3

To find P 2: 8+4, 8+7, 8+10...divisible by 3. 7 2 closest to 67. P 2 = 7, K 2 = 6

To find P 3: 7+5, 7+11, 7+17...divisible by 6. 5 2 closest to 67 P 3 = 5, K 3 =-7

To find P 4: 5+2, 5+9, 5+16...divisible by 7. 92 closest to 67 P 4 = 9, K 3 =-2

Now, since have reached K=2, we can do bh van to find the solution:

48842 2 - 2. 5967 2 = 1


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BH SKARA SEMI-REGULAR CONTINUED FRACTIONS

Krishnaswamy Ayyangar has also showed that the Cakrav la algorithmcorresponds to a semi-regular continued fraction expansion of D.

A simple continued fraction is of the form ( a i are positive integers for i>0)

This is denoted by [ a 0, a 1, a 2, a 3, ... ] or by

Given any real number , to get the continued fraction expansion, takea 0 = [ ]Let 1 = 1/( - []). Then we take a 1 = [ 1]

Let

2 = 1/(

1 - [

]). Then we takea

1 = [

2]. And so onThis will terminate if and only if is rational.


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BH SKARA SEMI-REGULAR CONTINUED FRACTIONS

It is a famous result of Lagrange that every quadratic surd has a periodicsimple continued fraction expansion. This expansion is closely linked withthe Euler-Lagrange algorithm for the solution of the so called Pells

equation [vargaprakti equation of Brahmagupta].Krishnasswamy Ayyangar showed that the Cakrav la method of Bh skaracorresponds to a periodic semi-regular continued function expansion

D = a 0+ 1 / a 1+ 2 / a 2+ 3 / a 3+ ...

where

a i = (P i +P i+1)/ K i and i = (D - P i2

)/ D - P i2

For instance,

67 = 8 + *1/5+ 1/2+ 1/2+ 1/9+ -1/2+ -1/2+ 1/5+ 1/16*+ ...

The simple continued fraction of Euler-Lagrange can also be generated bya Cakrav la type of algorithm if we replace the condition II by

(II) D > P i +1 2 and D - P i +1 2 is minimum


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OPTIMALITY OF CAKRAV LA ALGORITHM

The Cakrav la Algorithm is significantly more optimal than the Euler-Lagrange algorithm as it skips several steps of the latter. It has beenestimated that for large D, Cakrav la skips about 30% of the steps

involved in the Euler-Lagrange algorithm. The use of bh van leads tofurther abridgment of the process when we reach K = -1, 2, 4.

Euler-Lagrange Method for X 2 - 67 Y 2 = 1

I P i K i a i i X i Y i0 0 1 8 1 1 01 8 -3 5 1 8 12 7 6 2 1 41 53 5 -7 1 1 90 114 2 9 1 1 131 165 7 -2 7 1 221 276 7 9 1 1 1678 2057 2 -7 1 1 1899 2328 5 6 2 1 3577 4379 7 -3 5 1 9053 1106

10 8 1 16 1 48842 5967

The steps which are skipped in cakrav la are highlighted

67 = 8 + *1/5+ 1/2+ 1/1+ 1/1+ 1/7+ 1/1+ 1/1+ 1/2+ 1/5+ 1/16*+ ...


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EULER-LAGRANGE METHOD FOR X 2 - 61 Y 2 = 1

I P i K i ai i X i Y i0 0 1 7 1 1 0

1 7 -12 1 1 7 12 5 3 4 1 8 1

3 7 -4 3 1 39 5

4 5 9 1 1 125 165 4 -5 2 1 164 21

6 6 5 2 1 453 58

7 4 -9 1 1 1070 1378 5 4 3 1 1523 195

9 7 -3 4 1 5639 722

10 5 12 1 1 24079 308311 7 -1 14 1 29718 3805



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EULER-LAGRANGE METHOD FOR X 2 - 61 Y 2 = 1 (CONTD)

12 7 12 1 1 440131 56353

13 5 -3 4 1 469849 60158 14 7 4 3 1 2319527 296985

15 5 -9 1 1 7428430 95111316 4 5 2 1 9747967 124809817 6 -5 2 1 26924344 3447309

18 4 9 1 1 63596645 8142716

19

5

-4

3 1 90520989

11590025

20 7 3 4 1 335159612 42912791

21 5 -12 1 1 1431159437 18324118922 7 1 14 1 1766319049 226153980



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CAKRAV LA ACCORDING TO N RYAA (c. 1356)

Nryaa Paita has described the cakrav la process in both of his worksGaitakaumud and B jagait vatasa as follows:

Here, N ryaa seems to formulate the condition (II) somewhat

ambiguously as follows:P i +1 2 may be chosen to be either greater than or lesser than D.


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NRYAAS EXAMPLE: X 2 - 103 Y 2 = 1

At step 4, we can use bh van to obtain the result directly

X = (477 2 + 103 x 47 2)/2 = 455056/2 = 227528

Y = 2 x 477 x 47/2 = 44838/2 = 22419

The sequence of steps here is the same as would follow from Bh skaras

prescription that P i+1 is so chosen that P i+12 -D is minimum.

I P i K i a i i X i Y i0 0 1 10 1 1 0

1 10 -3 7 1 10 1

2 11 -6 3 -1 71 7

3 7 9 2 1 203 20

4 11 2 11 -1 477 47

5 11 9 2 -1 5044 497

6 7 -6 3 1 9611 947

7 11 -3 7 -1 33877 3338

8 10 1 20 1 227528 22419


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THE EQUATION X 2 - D Y 2 = -1

Bhskara states that the equation X 2 - D Y 2 = -1 cannot be solved unless D

is a sum of two squares. Taking D = m2 + n 2 Bh skara gives two rationalsolutions (X, Y) = ( n/m, 1 /m) and (X, Y) = ( m/n, 1 /n).

From these, it is sometimes possible to get integral solutions by bh van ,as Bh skara shows in the case of X 2 - 13 Y 2 = -1. He obtains X=18, Y=5.

While considering the equation X 2 - 8 Y 2 = -1, Bh skara merely gives therational solutions X = 1, Y = 1/2. Here, the commentator Ka Daivajaseems to imply that in this case also we can obtain integral solutions bybhvan .

This is incorrect as it can be shown that X 2 - 8 Y 2 = -1 has no integralsolutions.


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BH SKARAS SOLUTION OF A BIQUADRATIC

Bhskara II has given an example of the method of solution of abiquadratic equation of the special form

x4 + px2+qx+r = 0

by adding to both sides ax 2-qx+b to both sides, choosing a and b such thatboth sides are perfect squares.

This can be done in general, but it could involve a cubic equation. In hisexample, Bh skara seems to have guessed the values of a , b.

Bhskaras example is in the madhyam haraa section of B jagaita :

x4-2 ( x2-200 x) = 10000-1


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The way Bh skara solves this equation is given in his v san commentary:


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Bhskara first obtains the two sides of the equation in the form

x4 - 2x2-400 x = 9999

He then remarks that if we add 400 x+1 to the left side we get a completesquare, but the same thing added to the right hand side will not produceone, and hence proceeding in this way we cannot accomplish anything.Hence, says Bh skara, here one has to apply ones intellect.

If we add 4 x2+ 400 x + 1 to both sides, we get the roots

x2+ 1 = 2 x + 100

This can be solved in the usual way to obtain x = 11

Bhskara remarks that this is how the intelligent should attempt suchproblems.

A.A.Krishnaswami Ayyangar has noted that Bh skaras is indeed the firstsolution of a non-trivial biquadratic equation.

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