# Ghosts of mathematicians past: Bharati Krishna and Gabriel Cramer.

Introduction

Jagadguru Shankaracharya Swami Bharati Krishna Tirtha (commonly abbreviated to Bharati Krishna) was a scholar who studied ancient Indian eda texts and between 1911 and 1918 (vedicmaths.org, n.d.) and wrote a collection of 16 major rules and a number of minor rules which have collectively become known as the sutras of Vedic mathematics. The numbering of the sutras in this article has been adopted from Williams and Gaskell (2010) which matches the numbering from the online references.

Some of the rules described by the Vedic Sutras can be used to improve the efficiency in a number of calculations (which makes them interesting to students and teachers of mathematics) and as theorems, they can be proven algebraically (which makes them interesting to students and teachers of mathematics).

In some cases, the processes of calculation are remarkably similar to other historical gems: for example, Cramer's rule (Katz, 2004, pp. 378-379) when applied to 2 by 2 systems of linear equations is quite similar to the solution process obtained using the Paravartya Yojayet Sutra (Babajee, 2012) which translates roughly as "transpose and apply" (Williams & Gaskell, 2010).

This article aims to introduce some of the Vedic techniques, give an example of how each technique may be used and how it may be proven algebraically, with the big-picture question of where and when such ideas could be utilised within the secondary mathematics curriculum.

Sutra six: Anurupye sunayamanyat

If one is in ratio, the other is zero (Williams & Gaskell, 2010).

To explain this sutra, consider a general system of two linear equations in two unknowns:

ax + by = c dx + ey = f

This sutra states that 1/d = c/f [??] y = 0 and also that b/e = c/f [??] x = 0.

Example

To solve the system of equations

3x - 2y = 6 x + 3y = 2

notice that the ratio of coefficients of x are equal to the ratio of constant terms. Hence, according to the sutra, y = 0. From there, it is simple to conclude that x = 2.

Proof

Proving this theorem should not be beyond the capabilities of a student in Year 10 or above (provided they are familiar with the elimination method for

solving simultaneous linear equations, of course):

ax + by = c

dx + ey = f

Then we make the coefficients of y equal in order to eliminate the variable y:

aex + bey = ec

bdx + bey = bf

Then we subtract

aex - bdx = ec - bf

Solving for x gives

x = ec - bf/ae - bd

However, if we assume that

b/e = c/f

then bf = ec which in turn implies that ec - bf = 0 and hence

x = ec- bf/ae-bd = 0

as required.

QED

Sutra seven: Sankalana vyavakalanabhyam

This sutra describes a process for solving simultaneous linear equations in two variables where the coefficients of the unknowns in one equation are the same numbers as the coefficients of the unknowns in the second equation except in the reverse order.

Example

Solve the system of equations

4x + 7y = 5 7x + 4y = 17

for x and y.

By the elimination method, our solution would look like this:

4x + 7y = 5 (1)

7x + 4y = 17 (2)

7 x (1): 28x + 49y = 35 (3)

4 x (2): 28x + 16y = 68 (4)

(3) - (4): 33y = -33

y = 1

4x + 7 x -1 =5

4x = 12

x = 3

This is approximately six steps and requires multiplying by 7 (which is not difficult in itself, but see the second example.)

However, if we first add the two equations and then subtract one from the other, the following results are obtained:

4x + 7y = 5

7x + 4y = 17

Addition gives 11x + 11y = 22. Dividing by 11 then gives x + y = 2.

Subtraction gives -3x + 3y = -12. Dividing by -3 gives x - y = 4.

Adding these two new equations gives 2x = 6 [??] x = 3.

Subtracting these new equations gives 2y = -2 [??] y = -1.

This Vedic method required four steps and required no multiplication (and only very simple division to remove a common factor).

Consider now this second example which would be considered grossly unfair to ask a year 10 student under normal circumstances:

Example

Solve the following system of equations for x and y:

279x + 321y = 405 321 x + 279y = 195

One can only imagine the reaction that a teacher would get putting this on a test!

However, if the Vedic sutra is applied, the solution may look like this:

279x + 321y = 405 (1)

321x + 279y = 195 (2)

(1) + (2): 600x + 600y = 600

x + y = 1 (3)

Initially it may appear difficult to subtract 321 from 279. However, it should be noted that the result is just the negative of subtracting 279 from 321:

(1) - (2): -42 x + 42y = 210

x - y = -5 (4)

Then we add and subtract again:

(3) + (4): 2x = -4

x = -2

(3) - (4): 2y = 6

y = 3

And this (very) difficult looking problem has been solved (without a calculator) in four steps with only simple addition, subtraction and division to remove common factors.

Before moving to a proof, we need to formally state a theorem.

Theorem

For a system of linear equations

ax + by = c bx + ay = d

the unique solution for x and y can be obtained through two additions, two subtractions and division.

In this general case, we only need to show that a unique solution for x and y exists. Simplifying the final expressions in terms of a, b, c and d may be an interesting exercise in itself, but it is not a necessary ingredient in the proof.

Proof

ax + by = c (1) bx + ay = d

(1) + (2): (a + b)x + (a + b)y = c + d x + y = c + d/ a + b (3)

(1) - (2): (a - b)x + (b - a)y = c - d (a - b)x - (a - b)y = c - d x + y = c - d/ a - b (4)

(3) + (4): 2x = c+d/a+b + c-d/a-b x = 1/2(c+d/a+b + c-d/a-b)

(3) + (4): 2y = c + d/a + b + c - d/a - b y = 1/2(c+d/a+b - c-d/a-b)

QED

Sutra four: Paravartya yojayet

Transpose and apply (Williams & Gaskell 2010).

An excellent description of this sutra by Babajee (2012) makes a comparison to Cramer's rule which is an interesting side note in itself. (The proof of Cramer's rule, in the 2 by 2 case at least, should not be beyond the ability of any student of mathematics who has studied matrix determinants.) The Vedic sutra itself can be proven using algebra without matrices although without matrices, the general case is a little less succinct. It is worth noting that in the case of two unknowns, the Vedic sutra and Cramer's rule are essentially the same process, whereas for three or more unknowns, the two processes are visibly different (and hence worth exploring).

Whereas sutras six and seven cover specific cases of simultaneous linear equations, sutra four can be used to solve any system of simultaneous linear equations (although for a high school discussion it may not be realistic to go beyond three unknowns).

Consider the general case of two linear equations in two unknowns:

ax + by = c

dx + ey = f

The Vedic sutra, as interpreted by Babajee (2012), then instructs the solver to calculate a denominator, D = ae - bd as well as two numerators:

[N.sub.x] = ce - bf, [N.sub.y] = af - cd

From here, the solutions to the equation are calculated:

x = [N.sub.x]/D, y = [N.sub.y]/D

Example 2x + 3y = 21

Solve the system of equations

4x - y = 7

Proof

Starting with the general system and applying the elimination method:

D = 2 x -1 - 3 x 4 = -14

[N.sub.x] = -1 x 21 - 7 x 3 = -42

[N.sub.y] = 2 x 7 - 4 x 21 = -70

x = -42/-14 = 3, y = -70/-14 = 5

Earlier, we defined [N.sub.y] = af - cd and D = ae - bd.

Then our solution from the elimination method:

y = cd - af/bd - ae = af - cd/ae - bd = [N.sub.y]/D

as required.

A similar argument can now be made for x:

ax + by = c

dx + ey = f

aex + bey = ce (3)

bdx + bey = bf (4)

(3) - (4) (ae - bd)x = ce - bf

x = ce - bf/ ae - bd

And because we earlier defined [N.sub.x] = ce - bf and D = ae - bd:

x = ce - bf/ ae - bd = [N.sub.x]/D

as required which completes the proof.

QED

Cramer's rule

Gabriel Cramer was a Swiss mathematician from the early part of the 18 th century (University of St Andrews, 2000) who devised a process for solving systems of linear equations using the determinants of matrices. In the case of two unknowns, Cramer's rule could be stated thus:

ax + by = c dx + ey = f

Matrices M, X and Y are then defined in terms of the coefficients of x and y and the constant terms (the coefficients of x are replaced by the constant terms in matrix X and the coefficients of y are replaced by the constant terms in matrix Y):

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The solution to the system of equations can then be calculated using the

determinants of these matrices:

x = y = det(X)/det(M) y = det(Y)/ det(M)

Example

Solve the system of equations

2x + 3y = 21 4x - y = 7

This is the same example we solved using the Vedic sutra earlier. The two solutions can be compared for efficiency and similarity of method (this comparison will be made again in the case of three unknowns later).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then solving for x and y:

x = -42/-14 = 3 y = -70/-14 = 5

Proof of Cramer's rule

Earlier it was shown by the elimination method that the solution to

ax + by = c was x = y = ce - bf/ae - bd y = af - cd/ae - bd

dx + ey = J

Now, by Cramer's rule,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then, calculating the determinants:

det(M) = ae - bd, det(X) = ce - bf, det(F) = af - cd

And then the solution according to Cramer's rule is:

x = ce - bf/ae - bd,y = af - cd/ae - bd

which is the same as the result obtained by the elimination method.

QED

Three equations, three unknowns

Babajee (2012) demonstrates that Vedic sutra four can be used to solve a system of equations with any number of unknowns by generalising the technique for a system with three unknowns. It is here that the Vedic method differs noticeably from Cramer's rule.

The following system of equations will be solved using the Vedic sutra number four and also by Cramer's rule from which a discussion about the relative efficiency can arise.

Example

Solve the following system of linear equations for x, y and z:

x + y + z = 4 (1)

x - y + 2z = 9 (2)

3x + 2y - z = 1 (3)

Start by re-arranging equations (1) and (2) to have the z component on the right-hand side:

x + y = 4 - z (1*)

x - y = 9 - 2z (2*)

Then D = 1 x -1 - 1 x 1 = -2

[N.sub.x] = (4 - z) x - 1 - (9 - 2z) x 1

[N.sub.y] = 1 x (9 - 2z) - 1 x (4 - z)

which simplifies to D = -2, [N.sub.x] = 3z - 13, [N.sub.y] = 5 - z.

The solution for x and y is then

x = 3z -13/-2, y = 5 - z/-2

Substituting these solutions into equation (3) gives an equation in z only:

3 (3z-13/-2)+2(5-z/-2)-z = 1

which has solution z = 3.

Substituting into the equations for x and y then gives:

x = 3x3-13/-2 = 2, y = 5-3/-2 = -1

So the complete solution is x = 2, y = -1, z = 3.

By comparison, Cramer's rule would proceed as follows:

x + y + z = 4 x - y + 2z = 9 3x + 2y - z = 1

Then matrices M, X, Y and Z are defined as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

There are a number of ways of calculating the determinant of a 3 x 3 matrix. Using the multiplication of diagonals method gives the following working:

det(M) = (1x-1-x1) + (1x2x3) + (1x1x2)-(3x-1x1)-(1x1x-1)-(1x2x2)=9

det(X) = (4x-1x-1) + (1x2x1) + (1x9x2)-(1x-1x1)-(9x1x-1)-(4x2x2)=18

det(Y) = (1x9x-1) + (4x2x3) + (1x1x1)-(3x9x1)-(1x4x-1)-(1x1x2)=-9

det(Z) = (1x-1x1) + (1x9x3) + (4x1x2)-(3x-1x4)-(1x1x1)-(1x2x9)=27

Then the solutions are x = 18/9 = 2, y = -9/9 = -1, z = 27/9 = 3 as expected.

At a first inspection, the two techniques may appear to be similar in terms of time and mental arithmetic. However, unless a quick method for calculating matrix determinants is known, Cramer's rule requires a significantly more work than the Vedic approach. In addition, using the Vedic technique, once the value of z has been determined, the values of x and y are calculated in one step each by substitution

Four equations with four unknowns

Because the Vedic sutra does not rely on matrix determinants and because the calculation of the determinant of a 4 x 4 matrix could (if there are no zero elements) mean calculating 16 determinants of 3 x 3 matrices (with four unknowns this process would be repeated five times, for a total of eighty 3 x 3 determinants calculated which is roughly 480 calculations by the diagonal multiplication method which is one of the quicker ways to calculate the determinant of a 3 x 3 matrix) it stands to reason that with four equations and four unknowns, the Vedic technique could be superior to Cramer's rule.

Consider the following example of a system of four equations in four unknowns:

w + x + 2y + z = -2 (1)

2w - x + 2y - z = -4 (2)

-w + x + y - 3z = 6 (3)

3w + 2x - y + 2z = -6 (4)

Since the sutra works with systems of two equations, we will separate these four equations into two pairs and transpose them accordingly.

w + x = -2 - 2y - z (1*)

2w - x = -4 - 2y + z (2*)

Then we calculate

[D.sub.1] = (1 x -1) - (2 x 1) = -3 (5)

[N.sub.w1] = ((-2 -2y - z) x -1) - ((-4 - 2y + z) x 1) = 6 + 4y (6)

[N.sub.x1] = (1 x (-4 - 2y + z)) - 2 x (-2 - 2y - z) = 2y + 3z (7)

Which gives w = 6+4y/-3, x = 2y+3z/-3 (8)

Now taking equations (3) and (4) and applying the same process:

-w + x = 6 - y + 3z (3*)

3w + 2x = -6 + y - 2z (4*)

Then, calculating a new D value and new N values:

[D.sub.2] = (-1 x 2) - (3 x 1) = -5 (9)

[N.sub.w2] = ((6 - y + 3z) x 2) - ((-6 + y - 2z) x 1) = 18 - 3y + 8z (10)

[N.sub.x2] = (-1 x (-6 + y - 2z)) - (3 x (6 - y + 3z)) = -12 + 2y - 7z (11)

This gives

Now, equating w and x from equations (8) and (12) gives

w = 6 + 4y/-3, x = 2y + 3z/-3

These can be arranged into two linear equations:

29y - 24z = 24 4y + 36z = -36

Which can, in turn, be solved using the fourth Vedic sutra:

D = 29 x 36 - 4 x -24 = 948 [N.sub.y] = 24 x 36 - (-36 x -24) = 0 [N.sub.z] = 29 x -36 - 4 x 24 = -948

So the solutions for y and z are:

y = 0/948 = 0, z = -948/948 = -1

Substituting into (12) gives

w = 18-8/-5 = -2, x = -12+7/-5 = 1

So the final solution is x = -2, x = 1, y = 0, z = -1

Note again how quickly the last unknowns were found once the first unknown had been evaluated.

It should be noted that the numbers used in calculations in this case of four unknowns become rather large three digit numbers (even when the coefficients in the original equations were all single digit numbers). The question that therefore arises is whether the Vedic technique is more efficient than row-reduction for solving a system of four or more equations.

This is therefore a topic for further investigation.

Conclusion

There is no doubting that the methods for solving systems of linear equations in the Vedic sutras numbers four six and seven do work and are reasonably efficient methods of solution.

Sutras six and seven apply to only specific cases of systems of equations, and as such their usefulness in the regular curriculum could be questioned. Their usefulness as an extension or investigation exercise, however, makes them undeniably interesting and useful in the right setting.

The fourth sutra is perhaps the most interesting in a more general sense as it works for all simultaneous systems of linear equations. In systems of two equations with two unknowns, the elimination method (which is currently prescribed in the Australian Senior Mathematics curriculum) is comparably efficient to the Vedic sutra, in the case of three equations and three unknowns, the sutra is comparable in efficiency to Cramer's rule and Cramer's rule may have the edge here as the statement of the theorem is more succinct. In the case of four or more unknowns, where computing matrix determinants becomes inefficient, the question (for the moment unanswered) is whether row-reduction techniques are superior or not in terms of efficiency.

Because the Vedic sutras appear (based on this admittedly limited exploration) to be more useful in specific cases of systems of simultaneous linear equations rather than general cases, there is not a strong case to argue including them as part of the regular curriculum.

That said, the theorems within the sutras can be proven algebraically, add a unique (and non-European) sense of history into the mathematics classroom and in the specific cases where the sutras are more efficient than other techniques they are amazingly efficient.

For this reason, there is a strong argument to be made in favour of teaching the Vedic sutras to high school mathematics teachers with the intention of seeing them more readily used in investigation tasks.

References

Babajee, D. K. R. (August 2012). Solving systems of linear equations using the Paravartya rule in Vedic mathematics. Retrieved 13 December 2015 from https://docs.google.com/viewerng/ viewer?url=http://wwwvedicmaths.org/images/PDFs/VM_Journal/Cross+Multiplication+ Method+in+Vedic+Mathematics.pdf

Katz, V. (2004). A history of mathematics (brief ed.). Boston, MA: Pearson Education.

University of St Andrews (2000). Gabriel Cramer. Retrieved 13 December 2015 from http:// www-history.mcs.st-and.ac.uk/Biographies/Cramer.html vedicmaths.org (n.d.). What is Vedic mathematics ? Retrieved 13 December 2015 from http:// wwwvedicmaths.org/introduction/what-is-vedic-mathematics Williams, K. & Gaskell, M. (2010). The cosmic calculator: A Vedic mathematics course for schools. UK: Inspiration Books.

John Fitzherbert

Ivanhoe Girls' Grammar School, Vic.

jfitzherbert@ivanhoegirls.vic.edu.au
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