**Chandrasekharan Raman**

The previous article in this series covered the geometric method of calculating square roots as described in the *Sulba Sutras*. By repeatedly applying the geometric method, we could approximate square root of *2* (henceforth denoted as *√2*) to any required precision, thus getting closer to the exact value of *√2*. Of course, it would take infinite number of such approximation steps to write down the exact value of *√2* in the decimal number system, since it is an *irrational number*. Depending on the precision required, one can stop with certain number of steps in the geometric method. For example, in the previous article, we saw successive approximations of the *√2* as

The above sequence represents *√2* as *rational numbers* and the latter numbers in the sequence approximate *√2* with more accuracy. Such techniques are called **Rational Approximation of Irrational Numbers** in the language of Mathematicians.

One disadvantage of the geometric method is that it gets cumbersome after a few steps. What if one needs more precision in approximating *√2*? The geometric method described the method of rational approximation of *√2*. Extending this technique to other irrational numbers is not trivial. Then, how do we get rational approximation for *any* irrational number? Ancient Indian mathematicians had answers for the above questions. In this article, we will see a systematic algebraic method for rational approximation of irrational numbers and how ancient Indian algebraists elegantly handled this problem.

In ancient India, the equation *x ^{2} – Dy^{2} = 1* (

*D*is a positive integer) was called the

*Vargaprakrti*. This equation had attracted the interest of European mathematicians in the 16th century A.D. Indeterminate polynomial equations of this form are called

*Diophantine equations*and this particular form is called the

*Pell’s equation*. Let us turn our attention to

*Vargaprakrti*and assume that

*y*is not zero. Dividing by

*y*on both sides of the equation, we get

^{2}For large values of *y*, the quantity *1/y ^{2}* is very small in comparison to

*D*. So, it does not contribute much to the sum inside the square root and it can be ignored. Therefore, for large values of

*y*, we can write

Equation (4) suggests that if *(x _{1},y_{1})* is a positive integer solution to the

*Vargaprakrti*, then

*x*

_{1}/

*y*

_{1}will “approximate”

*√D*. This was a very smart observation by Brahmagupta, an Indian mathematician and astronomer of the 6th century A.D. If

*√D*were an irrational number, we have a rational approximation to

*√D*!

Let us now apply this to our well-known example of *√2*. The *Vargaprakrti* for *√2* is *x ^{2}–2y^{2} = 1. *We can check that

*(17,12)*is a solution to this equation, since

*17*. Therefore,

^{2}– 2 × 12^{2}= 289 – 288 = 1*17/12*approximates

*√2*. We can go one step further and check that

*(577, 408)*also satisfies this equation, since

*577*=

^{2}– 2 × 408^{2}*332929*

*–*

*2 × 166464*=

*1*. By Brahmagupta’s observation,

*577/408*indeed approximates √2. With more effort, one can check that

*(665857, 470832)*also solves this equation and hence approximates

*√2*. From (2), we know that the square of

*665857/470832*is so close to

*2*that the difference is only as small as

*1/470832*.

^{2}In his legendary work on astronomy called *Brahmasphuta Siddhanta*, Brahmagupta derived a way to generate infinite number of integral solutions to the *Vargaprakrti* if one positive integer solution is known. In other words, he provided a way to generate further fractions in the sequence (1), if one member of the sequence is known. He used a principle called *Bhavana* to show that if *(x _{1},y_{1})* is an integral solution to

*x*, then

^{2}– Dy^{2}= 1is also a solution to the same equation! This was another great observation by Brahmagupta. Thus, if we make an initial guess for a positive integer solution to the *Vargaprakrti* as *(x _{1},y_{1})*, then we can generate infinite number of solutions. We can then approximate

*√D*to any precision! The following

*k*fractions give more and more accurate approximations of

*√D*:

Returning back to our example where *D = 2*, it is easy to check that *(3, 2)* solves the *Vargaprakrti* equation when *D = 2*. So we can make the initial guess of as *(3,2)*. Then,

In fact, we can easily get further rational approximations of *√2* using the *Bhavana* principle of Brahmagupta.

Now, an interesting question arises. For the case of *D = 2*, it is easy to guess *(3,2)* as a solution. But, that may not be the case for all values of *D*. How does one make the initial guess that solves the *Vargaprakrti* for any arbitrary *D*? For example, the simplest non-trivial solution for the *Vargaprakrti* when *D = 13* is *(649, 180)*. This is not easy to guess! Indian Mathematicians had an elegant method called the *Chakravala* method to find the initial guess.

This problem attracted a lot of attention from the western mathematicians and there is an interesting historical story behind this particular equation. During the 16th century A.D., the Royal Mathematical Society was a prestigious society for scientific knowledge. The famous mathematician Fermat once challenged the head of the British Royal Mathematical society, Lord William Brouncker to find integer solutions to equation *x ^{2} – 61 y^{2} = 1*. William Brouncker devised the method of continued fractions to solve this problem but could not come up with the final solution. Later, another mathematician called John Wallis published a solution to this problem. For this reason, equations of this kind were called Brouncker-Wallis-Pell’s equation, even though the mathematician John Pell had nothing to do with it! The solution was further made elegant by the 17th century Swiss mathematician Leonard Euler. But, it was not known to the western mathematicians that the Ancient Indian mathematicians had a very elegant method to solve this problem. The best available method to solve this problem is the

*Chakravala*method of solving quadratic indeterminate equations. For

*D = 61*, the simplest solution is

*(1766319049, 22615390)*. The method uses a cyclic algorithm to solve the

*Vargaprakrti*, and hence it gets the name (

*Chakra*means wheel). The earliest reference to the

*Chakravala*method is given by the 12th century mathematician and astrologer Acharya Jayadeva. The

*Chakravala*method is wrongly attributed to another famous Indian mathematician Bhaskaracharya II, who describes this method elaborately in his mathematical treatise

*Bijaganita*. More details about the

*Chakravala*method can be found in the reference given below.

*Reference*

T. S. Bhanu Murthy, “*A Modern Introduction to Ancient Indian Mathematics*,” New Age International Publishers, New Delhi, 2005.

Could you please have links to the prior posts on ancient Indian math?

Appreciate this post.

Warm regards

Viswanathan

Dear Sri Viswanathan,

Thanks for your comment. Here are the links to previous volumes:

http://www.svbf.org/journal/vol13/PT_Quarter4_2011.pdf

http://www.svbf.org/journal/vol11/Mathematics.pdf

Thanks for highlighting the glory of ancient Indian Mathematics.