What is special about square root of 2?

Mathematics of ancient India can be very useful to understand the motivation behind the development of Mathematics and can help the students understand how applied Mathematics can be. The interesting questions that can be asked what is so special about √2. Why did Boudhayana specifically provide its value? The length of the diagonal of a square is √2 times the side of the square. Why should I know the diagonal of a square? Squares were used for geometrical constructions and had applications in designing sacrificial altars.

For example: Boudhayana's sulbasutra (800 BCE) provides the value of √2

Baudhāyana i.61-2 (elaborated in Āpastamba Sulbasūtra i.6) gives the length of the diagonal of a square in terms of its sides, which is equivalent to a formula for the square root of 2:¹

samasya dvikaraṇī. pramāṇaṃ tṛtīyena vardhayet
tac caturthenātmacatustriṃśonena saviśeṣaḥ

samasya problem or puzzle to be solved.

dvikaraṇī. means “that which produces 2” or diagonal of a square

pramāṇaṃ means proof or first term in a rule of three sum

vardhayet means increase

saviśeṣaḥ one that remains

The last word is the most important part of the sutra because it tells us that Boudhayana knew that the result will be an approximation and not the exact value of √2.

The prescription in sulbasutra can be translated as:

Increase the length [of the side] by its third and this third by its own fourth less the thirty-fourth part of that fourth. The increased length is a small amount in excess²

Thus the above passage from the Sulbasutram gives the approximation:

The symbol ≈ instead of = is because of the word saviśeṣaḥ

In fact this method will in one more step obtain: ³

 ,

Where the only numerical computation needed is 1154 = 2[(34) (17) -1] and, moreover, the method shows that the square of this approximation is less than 2 by exactly

This approximation is accurate to eleven decimal places.

Another thing that I observe is how thin were the compartmentalisation of Mathematics. For example root 2 by Boudhayana uses Geometry. However, value of √2 can be also found using Brahmagupta's varga-prakrti (Brahma-sputa-siddhanata). Brahmagupta (628 CE) uses algebra to find √2 by solving the equation  using “the principle of composition”⁴ or bhavana and finding the ratio  for large values of x & y. 

As can be observed the first positive integer solution for (x, y) is (3, 2) and there are infinitely many solutions to this problem. Although, Brahmagupta couldn’t give the solution to all types of this problem. It sowed the seeds for Acharya Jayadeva and Bhaskara II to design Chakravala process to get a general solution. Another solution to the above equation is (17, 12) which corresponds to Boudhayana’s value of 17/12 if we take first 3 terms  (1+1/3+1/12). Another solution is (99, 70) or (577,408). 

1) Can you figure out how to obtain 577/408 in the Boudhayana’s value of √2? 

2) Can you tell why does x/y gives the approximate value of √2? 

3) Why does the approximation improve for larger values of x & y?

It is needless to say that finding √2 is technically a mundane arithmetic exercise and is not interesting.

Learning this way, we have given an altogether different meaning to √2 and also shown how there can be multiple ways to tackle a same problem in Mathematics. 

Conclusion:

This is just one of the many examples where Ancient Indian Mathematics can be used to motivate higher order thinking among school students and break the barriers of conditional thinking and such inter-linkages will be very useful in contemporary times when knowledge is becoming more and more inter-disciplinary. It is very interesting that the ancient knowledge especially in India was multi-disciplinary. 

References:

1 http://self.gutenberg.org/articles/baudhayana_sulba_sutra#Square_root_of_2

2 This last word is translated by some authors as "The increased length is called savi´e¸a". I follow the translation of "savi´e¸a" given by B. Datta on pp. 196-202 in The Science of the Sulba, University of Calcutta, 1932; see also G. Joseph (The Crest of the Peacock, I.B. Taurus, London, 1991) who translates the word as "a special quantity in excess" and also based on a presentation by Prof Kannan at International Conference on History and Development of Mathematics, Jaipur 2013

3 Square roots in the Sulbasutra – David W Henderson http://www.math.cornell.edu/~dwh/papers/sulba/sulba.html#FOOTNOTE

4 Rationale of the Chakravala process of Jayadeva and Bhaskara II (Historia Mathematica, Page 168)