Cricket & Family

It was 22nd April 1998, India V Aus played  the 6th Cricket ODI of the series with Newzealand (NZ) as the 3rd team at Sharjah and if India had to qualify for the finals India had to either win or lose and qualify with a better Net run rate than NZ.

Australia batted first and scored 284. India was struggling at 138/4 as VVS Laxman joined Sachin Tendulkar at the crease. This match had a unique disruption due to sand storm and the revised target under Duckworth and Lewis system was 276 in 46 overs. Sachin played a magnificent innings scoring 143 of 131 balls and had a wonderful partnership with Laxman, who made 23 of 34 balls, of 104 runs. When Sachin got out India had scored 242 and qualified for final although lost the match.

This match is celebrated for Sachin's innings but the role played by Laxman is often understated. The support provided by Laxman in the partnership of 102, although he scored only 23, was crucial in helping India reach the final and ultimately winning the series. If you ask an outsider or look at statistics Laxman's innings may not look important in the outcome but if you ask the team or Sachin you will know how instrumental that innings was.

This is how family life is. Every person is important and each one plays a role in the functioning of the same. Equality only undermines this role and makes one think Sachin's knock was more important than Laxman's. Let us not judge and work in the direction of equity and remember the Asian cultures are family based societies and western cultures are based on Max Weber's sociological theories of individual is supreme and hence individualistic societies. Remember outsiders will always tell you how one person is more equal than other but ONLY you know the role each member of the team plays. Let us not blindly follow the west and understand the great virtues & philosophies that makes our culture unique. Our families are the backbone of our culture, our economy and it is important that we protect this fabric because remember OUR FAMILY BASED SOCIETY IS OUR IDENTITY AND ECONOMIC STRENGTH.

What is the similarity between CEOs, politicians and teachers?

Chief Executive Officers (CEOs) are in charge on the company and are equivalent to the captain of a ship. They provide direction to the company and put efforts to accomplish the vision for which the company is formed. There is a tension between achieving short term outcomes and long term outcomes and sometimes achieving these outcomes can be in apparent conflict. In other words, putting efforts to improve long term outcomes may divert CEO's effort from short term outcomes. However, when the benefits will be reaped by the company the current CEO may not be the CEO of the company and also one may not be able to disentangle the effect of outcome to the effort of this CEO or other factors. This means CEOs have incentives to focus on short term outcomes resulting into decisions with a myopic approach. Needless to say this will be sub-optimal in the long run. In order to overcome this situation, CEOs are incentivized by stock options which motivates them to improve the long term health of the organization.

This kind of behavior is not restricted to executives but we find many professionals having a short term approach to their decision making. Let us take the case of politician. A politician is ideally supposed to take decisions which will be socially optimum in the long run but may be tempted to take decisions which will help the politician to stay in power even if it may be detrimental to the society. Politicians are motivated to "show" that they have taken some action to please the voters. If a politician takes some decision which will have effect in the long run, the same politician may not be in power at that time and hence may not be provided due credit for the actions taken which disincentivizes him/her to take decisions in the interest of the society in the long run.

Now if we look at the case of teachers they are required to help the students improve their marks/grades and also improve their thinking skills, higher order skills and educate them in true sense which will help them in the long run. Focusing on grades/marks (short term goals) is necessary but not sufficient. Not improving higher order skills, thinking skills will be suboptimal in the long run and will not create the necessary human capital. The reason again for the myopic behavior observed is because there is a lot of noise in the relationship between efforts taken by the teacher currently and the outcome that will be observed in the future. However, actions taken can be easily mapped into the improvement of grades/marks and such other performance measures.

What can be a possible solution to improve the situation?
I will focus on education because that to me is the most important area that we need to target sooner than later. There is an apparent constraint in the form of time available for teachers to focus on these two aspects. One thing that can be done is segregate these two responsibilities and have two different persons focusing on short term and long term measures of students' objectives. Since the outcome is difficult to measure in the long run, the educational institutions must be a not-for-profit organizations. Structuring educational institutions as for profit will provide incentives to focus on short term outcomes of marks/grades. Mushrooming of coaching institutes to a large extent is due to the myopic outlook of the society as a whole towards education. This problem is aggravated by the lack of education of parents.

Focusing on outcome rather than efforts is not the first best solution because conversion of efforts into outcome is not straightforward. Incentivizing on efforts although ideal is difficult because it is not easy to observe what the person is doing or how the actions are useful because people who are evaluating performance don't have better information than the persons taking actions. Can we allow professionals to improve long term performance even if it means at the foregoing the benefits in the short term? Our position in the years to come will be determined by the answer to this question.

Ps; I have been an instructor in coaching institutes in the past for Chartered Accountancy (CA), middle school, high school, junior college and degree college students. 

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)