Ramanujan, The greatest ever Indian mathematician.

Only recently I read a biography “The Man Who Knew Infinity”, on Ramanujan, the greatest ever Indian mathematician, . It was amazing and poignant at the same time.
Some of the problems mentioned in the book, revved up my aging mind. I tried to prove one of his equations. I am sure this being one of the simpler equations of Ramanujan, many would have solved it during the last century itself. Still I feel elated that I was able to share a very small part of his genious. Ofcourse, unlike Ramanujan, I had a formal education upto master’s level in engineering in one of the IITs. If only we had IITs in those days ……..?
I am attaching  two proofs for Ramanujan’s equation, one of which I call as Derivation. I feel very happy to share the same with my readers.
Happy to be associated with great minds of India,
regards to all,
L V Nagarajan

 

Solution for the Problem by Sri Ramanujan

 

1.      To Prove

(x + n + a) =  √[ax +(n+a)² +x√[a(x+n) +(n+a)² + (x+n)√[a(x+2n) +(n+a)² + (x+2n) √etc  ….

Proof

Let A_x = √[ax +(n+a)² +x√[a(x+n) +(n+a)² + (x+n)√[a(x+2n) +(n+a)² + (x+2n) √etc  ….

Then, We may write

A_x = √[ax +(n+a)² +x A_{x+ n}]

 

i.e.       A_x² = [ax +(n+a)² +x A_{x+ n}]

 

 

i.e.       A_x² – (n+a)²  = [ax + x A_{x+ n}] = [ a + A_{x+ n} ] x

 

 

i.e.       [A_x - (n+a)] . [A_x + (n+a)] = [ a + A_{x+ n} ] x

 

 

i.e.       [A_x - (n+a)] / x = [ a + A_{x+ n} ] / [A_x + (n+a)] = k (say) —– (1)

 

Hence,             A_x = kx + n + a;   And so, A_{x+ n} =  k(x + n) + n + a ———(2)

 

Substituting in the second part of eqn(1) above,

[a + k(x + n) + n + a] / [kx +n +a + n+ a] = k

i.e        xk² + (2n + 2a – x – n) k – (n + 2a) = 0

i.e.       xk² + (n + 2a – x) k – (n + 2a) = 0

i.e.       (xk + n + 2a) (k – 1) = 0, giving the values, k =1 or kx = – n – 2a

Substituting in (2)

            A_x = x + n + a , or A_x = – a, (not admissible)

Hence proved

 

 

2.      The above is a proof. Let us call the following as a derivation.

 

Derivation

(x +n+ a)    =  √ [ (x + n + a)² ]

= √ [ x² + ( n + a)² + 2x(n + a) ]

= √ [ ax + x² + ( n + a)² + x(2n + a) ]

= √ [ ax + ( n + a)² + x(x + n +n + a) ]  ——–(i)

 

Following as per Eqn (i) above, we may write:

(x+n +n + a) = √ [ a(x+n) + ( n + a)² + (x+n)(x + 2n + n + a) ] ——(ii)

 

From Eqns (i) and (ii), we can recursively write:

(x +n +a) = √[ax + ( n + a)² + x√[a(x+n) + ( n + a)² + (x+n)√[a(x + 2n) + (n+a)² + (x+2n)√…

 

Hence Derived

 

 

3.      To Find the value of  √[1+2√[1+3√[1+4√[1+5√[1+ …….

 

(n+1)² = n² + 2n + 1

= 1 + n(n+2)

i.e.

n + 1 = √ [1 + n(n+2]]

Hence we may write,

3 = √ [1 + (2 x 4)] and

4 = √ [1 + (3 x 5)] and

5=  √ [1 + (4 x 6)],  etc

 

Hence,

3= √[1+2√[1+3√[1+4√[1+5√[1+ …….

 

 

 

L V Nagarajan

01 July 2008

 

Once There Were Rivers

Once There Were Rivers.

L V Nagarajan

Rivers are the worst sufferers of human greed. I read somewhere that about 80% of the rivers of the world do not reach the sea at all. They become dry miles before they reach their natural end. River beds are used as illegal sand quarries and later as illegal real estates. What do we do about it? There should be an international law to limit the utilization of river water to, say, 95% and a minimum of 5% of water, as a rule should be discharged into the sea. A norm should be developed for graded utilization of river water all along its route upto the sea, irrespective of political boundaries it passes through. Let us think of our ancient cultures which worshiped rivers, especially its source and its point of collusion with the sea. Let us not pollute the rivers and let us keep their banks and the beds clean and clear. Let us preserve our rivers for our future generations.

 

Sex Education

 

SEX EDUCATION IN SCHOOLS

L V Nagarajan

 

 

1.0 Introduction

It is a good idea to educate the youth of this country on Sex, since it is one of the major aspects of life itself. It is quite dangerous in the present context to leave the youth to learn the aspects of sex on their own, from the society around them. Since such a type of responsible social set-up has long broken down, it is the duty of the society to find an alternative. High Schools are being considered as one of those places where such an education on sex could be imparted. In proceeding to provide such a system of sex education, a lot of thought should be given in commencing, maintaining and improving such a system. We should be careful about four major aspects of this education: they are a) the content, b) the candidate, c) the class and d) the counsellor. These aspects are individually discussed along with my suggestions.

 

 

2.0 The Content:

What should sex education consist of? We should take care to see that it does not become a substitute for pornography. Love and morality should be made the fabric on which the sex education will be scripted. Care should be taken to see that it does not kindle undue passions among the youth undergoing such education. It should respect the institution of marriage and should not encourage sub-normal and abnormal sexual behaviours, such as homo-sexuality. With the above consideration in mind I propose the following broad content to be included in the syllabus for sex education.

 

§         The biology of sex

Coming to age and maturity

Fertility and menstruation

Sperms and Eggs

Genetic aspects of pregnancy

 

§         The physiology of sex

Sexual organs

Sexual urge and union

Sexual Hygiene

The process of pregnancy

Sex determination

Period of pregnancy and child growth

Process of delivery

Lactation and feeding the child

 

§         The psychology of sex

Hormones

Sexual urge, love and lust

Male and female aspects of sex

Impotence, sterility and cure

Erectile Dis-function and Frigidity

Aphrodisiac

Enjoyment and satisfaction

 

§         Sex and moral values

Teenage and sexual care

Normal, subnormal and abnormal Sex

(eg. Man/woman, Self/Masturbation, Homo Sexual – respectively)

One Man / One woman concept

Avoiding homosexual behaviour

Sexual crimes

Sex and society – Ethics and Etiquettes

 

§         Sexual infections and deceases

Evils of promiscuity

Sex on sale

Sexually Transmitted Deceases

Safe sex

 

§         Conception and Contraception

Menstrual Cycle and pregnancy

Oral contraceptives

Day-after Pills

Abortion and health

Vasectomy and tubectomy

IUCDs

Condoms and femidoms

 

The contents of sex education as proposed above may be discussed by a panel of doctors, biologists and social scientists and a final list of contents may be evolved.

 

 

3.0 The Candidate

Normally whatever we learn, there is always an urge to put to practice what we have learnt. In many cultures including ours, premarital sex is not accepted. Hence sex education can only start at an age close to marriageable age. We should also allow our children to remain as children for as long a period of time as possible. In a highly populated and poor country like ours we wish to maintain a marriageable age of 18 years for girls and 21 years for boys. Considering these facts we may say, sex education may start only at the age of 15 for both boys and girls.

The content for sex education, proposed as above, has to be suitably divided into three modules and be prescribed for three years. Starting at a minimum age of 15, the students can complete the three modules at the age of 18. However the age, at which to start the sex education and when to take the successive courses, should be left to the decision of the parents. The classes should be separately held for boys and girls. The students should be monitored for proper understanding of the course material without any mental aberrations. 

 

 

4.0 The Class

Sex education classes should be held separately for boys and girls, at different places and at different times. This will ensure a higher acceptance from the parents of the boys and girls. These classes should be held in such a way that even non school-going children and drop-outs may attend and benefit. The children should be taught to use the books and course material discretely to avoid misuse by other children and other non students.   

  

 

5.0 The Counsellor

The counsellors (or teachers) for sex education should be selected with utmost care. We should have only male teachers for boys and female teachers for girls. The teachers should have impeccable moral values. They should generally be happily married persons. They should have undergone special training in sexology. They should be further trained by a panel of doctors, biologists, psychologists and social scientist on the contents of sex education as proposed above. 

 

6.0 Conclusion

In our country, i.e., India, with an ancient culture and a largely conservative population, we have to approach the subject of sex education with great care. The above are my suggestions which may be considered by all concerned. It may hopefully promote open discussions in public forum.

 

Nagarajan L V

 

 

 

Indian Democracy

Indian Democracy

Nagarajan L V

 

India has achieved a great name for itself as the biggest working democracy in the whole world. Sure the Indian democracy is the biggest with more than 500 million voters exercising their voting rights periodically. Also sure the democracy is working but for a brief aberration in 1971 when ‘national emergency’ was declared. Peaceful changes of governments have occurred on the power of ballot. But is it really a true democracy? Is it reflecting the will of a true majority? Does the democracy exist, in all levels of political activity and organisations, in representation and formation of local and federal governments, in governmental decisions and during change of governments? In all counts it is a big no. Crime and criminals are forming an unacceptably large part of Indian political scene. Corruption has become the major source of funds for political parties for fighting elections and as such corruption in governments has been accepted as normal. Nepotism, hero-worship and sycophancy are at the highest level. Just the conduct of periodic elections of representatives to the legislatures and parliament cannot alone assure a real democracy. The umpteen obstacles are in the way of obtaining a truly representative government, evolved through meaningful debate within and across all political spectrum and governmental bodies. Following are a few suggestions for improving the democratic process in the running of our great country, India.

 

1) Every political party in India should be forced to implement the democratic process within their parties. Periodic elections for organisational posts should be ensured as a precondition for official recognition of the political party.

2) Every political party should be made to submit audited accounts on an annual basis.

(The need for the above two reforms has already been widely accepted. But the political heavy weights across all political parties are turning a blind eye towards implementation of these electoral reforms, less they should loose control of ‘their’ party.)

 

3) There should be a set of minimum qualifications criteria for contesting in various elections to local bodies, legislatures and parliament. Education, experience, service record, moral background will all form part of such criteria.

(The need for minimum qualification, as above, is being scuttled with the lame excuse that it will not give chance for the so called ‘Common Man’ to contest the elections. A graded criteria for candidates will, in fact, encourage the qualified ‘Common Man’ to contest the elections; like, a) elementary education for candidature to Local bodies, b) high School education or membership of local bodies for at least 5-years for candidature to legislature, c) Graduation or membership of legislature for at least 5-years for candidature to Parliament. Good conduct and Social Service record could be common criteria for all candidature. Similarly ministers in State cabinet should be specialists (Post Graduates) in their area (portfolio) or should have been member of legislature for at least 10-years. Ministers in Central cabinet should be specialists (Post Graduates) in their area (portfolio) or should have been members of Lok Sabha or Rajya Sabha for a total of at least 10-years.)

  

4) The candidates for any election should be selected through a democratic process within the political party, instead of the high command deciding the nominations.

(The above suggestion ensures a much wider democracy at the grass root level. Every political party should encourage proposals for candidature from the party members at the constituency level. The party may try to obtain unanimous choice; however it should not shy away from holding party level elections to select the most popular candidate among many aspirants.)

 

5) There should be a limit on number of terms/years for anybody to continue as President, Governor, Prime Minister and Chief Minister.

(This suggestion will see an end to widespread hero-worship, sycophancy and nepotism seen in our democracy.)

 

6) Election expenses by all political parties and candidates should be closely monitored. Any party or candidate spending more than the stipulated amounts should be disqualified.  

7) The Election Commission should subsidise election expenses for all candidates, the amount of subsidy varying depending the size of the constituency. However, after the elections, the candidates placed fourth or lower, on number votes polled, will be asked to repay the amount of subsidy availed.

The candidate, who loses security deposit due to low percentage of votes polled, should also be asked to make good the expenses incurred by Election Commission on account of his candidature, like, providing security, telephone and other such facilities and subsidies. It could even be an ad-hoc amount based on the size of the constituency. This will be in addition to the subsidy of election expenses availed by him from the Election Commission as in item No. 7 above.

(The two suggestions as above will encourage only the serious candidates to contest the elections. Number of contestants per seat will reduce. This will make the election process more meaningful and largely reduce the cost of conducting elections, in terms of simpler voting machines, less security and fewer facilities provided to the candidates, etc.)   

 

9) Elections should not be countermanded for a death of an ‘independent’ candidate who does not officially represent any political party or group.         

10) If an elected member resigns before his term or gets disqualified for any reason during the term, he will be asked to pay the entire election expenses incurred by Election Commission in his constituency. This could also be an ad-hoc amount based on the size of the constituency. This member should also be disqualified from contesting elections for the next 5-years.

11) The term of every elected body should be fixed, be it Local, Legislature or Parliament. Dissolution of any of these bodies should not be permitted, and neither its extension. When the leader of the house loses majority, a new leader should re-elected with majority support. House may be kept in suspension till a new leader is elected.

 

(The three suggestions as above will dispel the need for unnecessary repeated elections and the subsequent expenses and disturbances to public affairs and administration.)

 

12) Voting should be made compulsory for all voters. It should be made a condition for availing any government grants, loans etc. The tax-payers should be asked to present proof of voting in any of the elections (at least once in the preceding 5 Years) to be able to avail tax exemptions. Alternately they should get prior exemption from voting with valid reasons.

13) Electronic voting should be introduced at all levels. The voting machine should include an option for No-Vote.

(The above two suggestions will help attract quality voters to the voting booth. The present trend of ignoring these voters in preference to poorer class is not good for our democracy in the long run.)

 

14) Upper house (Legislative Councils) should be restored in all the states to offer political space for the experts with social objectives.

15) Proportional representation had been discussed umpteen times in India but not considered due to obvious complications. A simple way to obtain the same is to consider membership to Upper Houses (both at State and Central level) based on percentage votes polled, (instead of number of seats won), by the political parties in State and Central elections respectively.

(The above two suggestions will enable future Manmohan Singhs and Abdul Kalams to contribute to our national policy matters and administration.)

 

It is hoped these suggestions will be considered by the election commission along with other electoral reforms already on their cards. These reforms should be widely discussed by the people in all the media including internet, TV channels and print media. Is not that a true democracy?

 

L V Nagarajan

1 Aug 2008

 

I have been closely watching the recent presidential elections in USA. One aspect I liked there was the possibilty for early voting. Early voting opens about one month before actual polling date. The people are encouraged to vote early by Email and postal ballot. This can very easily be adopted in India and it will encourage more middleclass elites to vote. Right now in India this middle class is totally neglected in elctioneering. Nobody bothers for their vote

L V Nagarajan , 6th Nov 2008