It was June, 1971 when I had passed Pre Engineering examination with 9th position in Punjab. Family income was too meagre to manage three square meals a day. Sparing Rupees 200 a month was beyond my father’s capabilities, a fact that was clearly known to me. Often I had considered myself a burden upon my father. Although Rs 60 a month scholarship was awarded to me by Government of India under National Scholarship Scheme, this amount was like few drops upon the mound of dry sand. I was spending upon my clothings, books and college fees and was not contributing to my family income. I had already made up my mind to go to Ludhiana to undertake training in sewing and weaving hosiery and support my family with allowance that was to be paid to me even during training.
But my father was bent upon sending me to Chandigarh, he wanted to see me Engineer which he himself instinctively was. He was expert in metallurgy and often metal traders used to visit him to know the ratio of different metals present in an alloy.
One afternoon after meals, when my father was in house, I mustered up my courage and said, “Will you be able to send me Rs 200 a month as hostel expenses and fees? If you are not, I would be hung up there waiting for your money?” My father laughed at my question and said, “When I am sending you to Engineering College, I will manage the fees also.” His words were more assuring than giving me the entire fees then and there. He kept his words and regularly sent the fees. I do not know where he advanced the loan from and how he repaid it. Nothing can fill up the loss of father. With heavy heart, today, I miss him on Father’s Day. I am what he made me keeping himself hungry and deprived.
Happy Father’s Day!
About Author:
Narinder Kumar Wadhawan is a graduate in Electronics and Electrical Communication Engineering from Punjab Engineering College, Chandigarh. He worked as Scientist in Solid State Physics Lab DELHI then National Fertilizers Ltd , Bhatinda as Instrumentation Engineer, then Ministry of Labour, Employment, Training as Deputy Director. Thereafter, he joined Civil Services in the year 1986, worked in different capacities on administrative posts and retired on September 30, 2013 as Secretary to Government Punjab from Indian Administratve Services. His interests are Astronomy, Physics, History, Music, Law, Spirituality, Administration and writings. He believes in hard work, determination and consistency in efforts. He loves to write on topics related to Astronomy Daily life experience and human sufferings. His favourite writers are Leo Tolstoy, Rabindranath Tagore, Mulk Raj Anand and Munshi Prem Chand.
When I look at sky on a starry night, I find stars uniformly distributed overcast. When I see flowers, I find their shapes follow a certain pattern. When I count their petals, I find these are mostly two, three, five, eight or thirteen, these follow a particular sequence called Fibonacci sequence. When I look at a charming face, I find, its dimensions and other parts follow a certain ratio called Golden Ratio. When I look aesthetically at nature’s processes, I find beauty in its symmetry.
Do isotropic universe, Fibonacci sequence, Golden Ratio, geometric patterns of its objects, vibrant colours and above all symmetry in the universe not assure us systematicity and Dorderliness of the universe?
But there always exceptions. Nature loves disorderliness. Yes, you read it right. We all are disorderly by nature. Have you not seen Law and Order arrangements also called Police Bandoubast (in south India) made on occasions of gatherings, visit of an important person, events of fair, Melas likewise? Have you not seen members exchanging notes in random groups before the meeting starts formally and, then they are asked to take seats arranged in an order? Have you not seen police man posted at ticket counters to maintain orderliness?
No one likes to be in orderly state let to himself , everyone wants to be leisurely located at his own wish and will. “Nature likes disorderliness”, was first stated by Ludwig Boltzmann an Austrian Physicist. Austria is a European country of the size of Punjab but it has produced finest scientists. A military captain George Lill of Austria after retirement, turned a fine mathematician.
But all was not well with Boltzmann who was awarded PhD in mathematical sciences at the young age of 22 years. He was detested by his colleagues in Vienna, it is same Vienna as of Sant Baba who was assassinated in the year 2009 in Gurdwara and that caused furore and vandalism of property in Punjab as aftermath of the incident.
Constantly bullied over his definition of entropy on the basis of disorderliness in nature’s processes, he left Vienna and joined Leipzig University in Germany. There again he was made a laughing stock and persistently hounded. He came back to Vienna which accepted him on the condition that he would not leave the university for four years. Law of entropy also called second law of thermodynamics always distressed him. Among his critics and opponents were physicists Mach and Ostwald. Ludwig Boltzmann lost ground and went under depression. Although married with two daughters and a son, this Austrian Physics took to his heart non recognition of his theory that nature tends to be in disorderly state with maximum entropy and when it is put to orderly state, its entropy deceases.
When he was away on vacation with his wife, he committed suicide by hanging himself. He suffered mental stress now called bipolar disorder.
Truth prevails and can not be kept in a concealed state for long. Boltzmann theory was recognised by quantum physicist Max Planck. Based upon that he propounded entropy is proportional to logarithm of disorderliness. To the constant of proportionality, he gave nomenclature of Boltzmann constant. Based upon theory put forth by Ludwig, a new subject Statistical Mechanics came into being and is now included in modern physics. Boltzmann is now recognised as a pioneer of statistical mechanics but of what use when he is no more, his theory is not only found applicable to physical sciences but also to living science. Behaviour of neurones, body cells and other phenomena are being explained by law of disorderliness. His theory finds mention in quantum mechanics also. Ludwig Boltzmann died so that we may live comfortably, he would never be forgotten in the annals of science. It pains me to highlight his pitiable situation, he could not muster support of those who would have lent him when he needed it the most. My head bows in reverence.
Have you ever thought (2/3) would be the approximation of logarithm of 2, (2/3 + 2/5) would be that of 3 and (2/3 + 2/5 + 2/7) would be that of 4? It is too simple to believe but simplicity leads to ultimate sophistication. This is not what I am saying but has been said, more than five hundred years back, by Leonardo Da Vinci (Leonardo of town Vinci) Italian polymath of the High Renaissance who was active as a painter, draughtsman, engineer, scientist, theorist, sculptor and architect.
In order to reply opening line query, one wants to know in the first instance what this fuss logarithm is. Before explaining, I will narrate a short fable out of many, popularly known ‘Akbar Birbal Ke Kisse’ in common parlance or ‘Fables of Akbar and Birbal’. Akbar was a Mughal Emperor who once ruled India. In a chess game, when Birbal defeated Emperor Akbar, he, enthralled by the exceptional skill, asked Birbal to state whatever he liked as a reward and his wish would be fulfilled. Hesitatingly, Birbal expressed that he might be given a few grains of rice on chessboard itself in a such a way that one grain be put in first square, two grains in second square, four grains in third square so on and all the sixty four squares be filled up in this fashion. To the surprise of the emperor, the entire rice stock of the empire fell short, even when half of the squares could not be filled up. Witty Birbal knew the hugeness of exponentiation, the inverse of which is logarithm.
In mathematics, serial number of the square is analogous to logarithm of the grains kept in that square and multiplying factor 2 that makes 2, 4, 8 grains in second, third, fourth squares, is analogous to the base of the logarithm. If serial number of the square is 11, then logarithm of the grains kept in this eleventh square, is 10. Similarly, we can state that logarithm of grain, kept in first square is 0. Had Birbal wished for putting 1 grain in first square, 10 in second, 100 in third so on, then the base of the logarithm would have been 10. In scientific use, base of logarithm is taken as ‘Euler’s Number’ abbreviated ‘e’ which equals 2.71828. Such logarithm to the base ‘e’ is called natural logarithm and is denoted by ln. For example, logarithm of 100 to the base ‘e’ is written as ln 100.
In what may lead to an important research work in Number Theory, two amateur mathematicians from India, have innovated the simplest ever method of determining natural logarithm of a real number. Briefly stating, for determining natural logarithm of number 2, one will have to find odd numbers in 4 i.e double of 2. These are obviously 1 and 3. Ignoring odd number 1 in all cases, natural logarithm of 2 will be 2/3. In this way, natural logarithm of 5, will be (2/3 + 2/5 +2/7 +2/9). To determine logarithm of a number ‘n,’ odd numbers (ignoring 1), i.e, 3, 5, …, 2n-1 are first written and, then double the sum of reciprocal of these odd numbers except one, called odd harmonic series, yields value of logarithm of n written as ln (n). Mathematically,
ln (n) = 2/3 + 2/5 + 2/7 + …+ 2/(2n-1).
Proceeding in the same manner, natural logarithm of 5/2 is 2/5 + 2/7 +2/9 and in general
These determinations are approximates and to achieve precise value of natural logarithm, the paper “APPROXIMATION OF LOGARITHM, FACTORIAL AND EULER-MASCHERONI CONSTANT USING ODD HARMONIC SERIES, ” prescribes multiplication and division of the real number by a large number. For example, number, ‘n’ should be written as (100n)/100 and the formula modifies to
More the value of multiplier and divider (it is 100 in the example), the better would be the result. “It arises the curiosity to know the theory that culminates into determination of ln (n) by Equation (1). The authors explain in the paper,
“It will not be out of context to state that it bears a strong analogy with human cells. Human cells trillions in number compose the body and NBB’s, in the same way, though limited to (x− 1) in number, compose a number x. To illustrate how NBB’s compose a number, we present the relation between x and NBB’s.
x= (2/1) · (3/2) · (4/3) … {x/(x− 1)}.
It is obvious, NBB’s (2/1), (3/2), (4/3), … , {x/(x− 1)} are (x− 1) in number and when multiplied generate a number x. In normal practice, a number x is envisaged as that which has magnitude equal to what we get when 1 is added x times. This concept of numbers by addition, arises on account of the fact, we are taught mathematics starting with ‘counting of the numbers,’ in kindergarten and basing thereupon, we distinguish one number from the other on account of weight acquired on accumulation of unities in it. We do addition and subtraction, corresponding integers to our fingers that is why fingers are called digits. With these strong prejudices, we are unable to envisage an integer as product of numbers. Thinking out of box, we have considered, in this paper, an integer to be product of NBB’s. Based upon that we give some examples. (101/3) is product of NBB’s (4/3), (5/4), (7/ 8) … , (101/100) and 20 is product of NBB’s (4/3), (4/3) , (5/4), (5/4), (7/ 6),…,(36/35). Using NBB’s, a number say x can be represented in infinite ways. NBB is the abbreviation of number building block. This is the crux of the research highlighted in the paper.”
Euler Mascheroni constant is equal to (1/1+1/2 +1/3+… 1/p)- ln (p) when p extends to infinity. Using number building blocks its, approximate value is calculated as 0.5736309333.
Before parting I would say, when a complicated problem is analysed in basic and simple ways, the success would never be far away. Steve Jobs was not a qualified person, he was not even Graduate and had overwhelming interest in calligraphy of computer writing. But what he had was, probably does not exist in most. He believed in creativity, thinking differently from common lot with unbiased and unprejudiced mind. He never believed, university education makes one creative but was advocate of the fact, it otherwise biases the thinking in particular ways the others have thought and scribed the books. When the mind is unbiased, it has innumerable options to explore but when it is biased, it has only one option that has been studied.
Lisa Brennan Jobs With Her Father Steve Jobs
“They teach you how other people think, during your most productive years,” he said. “It kills creativity. Makes people into bozos.” Lisa Brennan-Jobs daughter of Steve jobs recalls her father in her book and its abstract was published in Times Of India.
Steve Jobs was a believer not only in one religion but also in many. He had kept “Bhagavad Gita” in his home daughter Lisa writes in the book. Inspired by the ideals of Steve Jobs, this article as well as the research paper now published in the journal Mathematical Forum abstracted in Mathematical Reviews (USA), were written.
I am Electronics and Electrical Communication Engineering graduate and was earlier Scientist, then Instrument Maintenance Engineer, then Civil Servant in Indian Administrative Service (IAS). After retirement, he writes on subjects, Astronomy, Mathematics, Yoga, Humanity etc.
Whenever I happen to appreciate nature, be it flowers laden enticing trees in spacious park, be it the distant hills of Kasauli vividly visible from my roof top, I invariably find, all natural objects exist in pairs. If there were a lone object, I search for its complimentary missing part and I am always successful in locating that. This is not my present occupation with parity, quest started when I turned thirty. In the eyes of my comrades, I was the most eligible bachelor for marriage but in my own eyes, I was a doomed single who could not tie the knot although maturity had already set in. Beyond thirty, chances of marriage dims particularly in the part of the world, where I resided. I used to observe whether a bird sitting at the power line is all alone like I was. But soon, I could find, its mate was pecking nearby the pole. It was extremely difficult to have a case of broken parity to console my bruised soul.
When God had believed in duality why I was a freak case. Soon, I got married and parity restored. He restored the parity for Adam and gave him the company of Eve as his helpmate [2]. Self taught English Mathematician George Boole, affirmed this concept of duality when he propounded, a set A always has a complimentary set A bar and A + A bar = U. That means haves and have nots is a universal phenomenon. In other words, it can be said, universality demands duality. That is why matter that constitutes this universe, has its complimentary part energy. Co-existence of particle and wave, endorses the concept of duality [1].
Moot question that arises for a mathematician is whether 2 that manifests intrinsically in duality and parity, can represent the entire spectrum of numbers. Immediately, affirmation flashes in ones mind in the form of binary system of numbers wherein a number can be written by exponents – also called power – of number 2, utilising integers 1 and 0 [6]. With the purpose of illustrating it, I give an example of number 59 which can be written in the following ways in bianry and decimal systems.
We are not interested in traditional representation of numbers, we want a representation that is out of box and trendsetting. Such a representation of numbers, we will discuss, is based upon upon infinite as well as finite nested radicals using number 2[4]. A rsearch paper has also been written on this type of representation and its link is avaiable at serial number 4 of Referebces. One who has fancy for mathematics which is considered the language of science, may go through it. What are radical and nested radical ? For our purpose, a number under a square root sign is a radical and a radical under square roots is a nested radicals. These nested radicals could be finite or infinite, depending upon the magnitude and type of the number being represented. The seed to germinate a nested radical is recursive relation. Recursive relation as the name states, is a relation that reoccurs and is regenerative in nature. It is a relation where input parameter written in left hand side, abbreviated LHS, again appears, in output written in right hand side, abbreviated RHS of an equation. Input parameter that reappears in output in RHS, can again be substituted using the relation thus successive substitution forms a never ending close loop. Characteristic of this close loop is that approximation improves with completion of every single loop. How can a recursive relation be set up?
For establishing a recursive relation, cosine angle double formula will be utilised. Cosine of angle say x, is the ratio of base to hypotenuse in a right angled triangle with angle x between base and hypotenuse. Angle doubling formula is written asNow recurrence takes place on account of the fact cos (pi/3) appears in RHS and it is already present in LHS. Since cos (pi/3) has value ½, therefore, resultant representation will be
In this way, we can express cos {pi/(2n+1)} where n can have any value 1, 2, 3…
Clock Experiment: We have observed , in some cases, recurrence starts early but in other cases, it starts late. To understand it, we will perform a thought experiment. Consider, we have a set of clocks. Clock A has a hand that rotates in steps of five seconds, B in steps of six seconds and C in steps of seven seconds so on. Also consider all clocks start when their hands are at reading 12 marked at the dial. Hand of Clock A rotates in steps of five seconds, that is, it jumps to 1 marked at a gap of 5 seconds, to 2 in second step so on. In twelfth step, the hand will reach 12 which was its starting point. We assume that when it reaches reading 12, recurrence takes place.
Now consider clock C. In first step, it will cover 7 seconds, in second step, 14 seconds, in eighth step, 56 seconds and in ninth step, 63 secons. It could not come exactly to reading 12 in first round of 60 seconds. It will start the second round, but on completion of that round also, it could not come to reading 12 but jumped it. In seventh round, on completion of 59th step, the hand will come to reading 12 and at that point, recurrence starts. On the same lines, hand of clock C that moves in steps of 6 seconds, will come to reading 12 on completion of 9th step and recurrence starts. Applying this analogy, recurrence for cos {pi/(2m+1)} takes place depending upon the value of n whichisequal to m+1. Since a number has distinct value of n, therefore, every number has distinct recursive relation.
Next question is whether it is always possible that recurrence will take place. For recurrence to take place, following condition for cos (pi/n) must be satisfied. Next question that crops up is how cos (pi/n) can represent the entire spectrum of rational, irrational and transcendental quantities. Admittedly, value of cos (pi/n) lies between -1 and +1 and that includes 0, -1 and +1. If we take reciprocal of cos (pi/n), its domain increases to minus infinity to zero and zero to plus infinity. This covers all mathematical quantities.
Beauty of number 2 is not limited to representation of mathematical quantities. It is generally said that 2 makes a company whereas single is all alone. 2 recreate whereas a single incarcerates in aloofness. 2 support one another whereas a single look for support. Let us pay attention for a minute to what Ms Neha Pradeep Saini has to say on Number 2 [5]. She describes Number 2 as those born on dates 02, 11, 20, 29 of any month. Also, people born under the sun sign of cancer fall into this category. She goes on to say, “These people are highly emotional, naturally affectionate, sensitive, caring and loving and by default, they take charge of helping, caring, guiding others in their organisations, schools, colleges they are in.” It is an another feather in the cap of number 2.
Left to me, I do not believe in Numerology, it is not a perfect science, therefore, readers may exercise their discretion in believing or disbelieving her.” I believe in perfect science and its beauty. Mathematics which is language of science, is universal and so is number 2 in representation of the entire spectrum of numbers. I do not subscribe to “It takes two to make a row” but certainly applaud English television and film director, Nigel Lythgoe. “It takes two to make a relationship but only one to screw it up.”
End
References:
1. Duality Principle at https://en.m.wikipedia.org/wiki/Duality_principle
2. Jim Robidoux Adam and Eve, forbidden fruit and ‘The Fall’ at https://manchesterinklink.com/adam-and-eve-forbidden-fruit-and-the-fall/
3. K. Rick, π to Thousands of Digits from Vieta’s Formula. Mathematics Magazine 81(3), 201–207, (2008).
4. Narinder Kumar Wadhawan and Priyanka Wadhawan, Infinite Nested Radicals – A Way To Express All Quantities, Rational, Irrational, Transcendental by A Single Integer Two, Mathematical Journal of Interdisciplinary Sciences Oct, 2020 at https://mjis.chitkara.edu.in/index.php/mjis/article/view/229/144.
5. Neha Pradeep Saini, Numerology of Number 2, The Times of India, Astrology at
http://timesofindia.indiatimes.com/articleshow/68205549.cms?utm_source=contentofinterest&utm_medium=text&utm_campaign=cppst
6. Sandy Irani, Number Representation, ICS 6D at https://www.ics.uci.edu/~irani/w17-6D/BoardNotes/12_NumberRepresentationPost.pdf
Mary Mallon is depicted breaking skulls into a skillet
In the early twentieth century, a thirty seven year old Irish lady, Mary Mallon worked as a cook with the wealthy banker, Charles Henry Warren in New York. Six of the eleven people of this banker’s family fell ill to typhoid in spite of the best cleanliness and hygiene being maintained in the house. Outbreak of typhoid in his family upset Warren who wanted the matter investigated in order to know the cause of disease.
George Soper was a civil engineer by qualification but was trained in the matter of public health. There could not be a better expert than Soper to investigate the cause of disease. Soper inspected the residence of Warren and found the cleanliness in order and the dishes properly cooked at high temperature ruling out any possibility of bacteria affecting the residents. However, he also noticed that ice cream generally made of mashed peaches and milk was not pasteurised and that could be the only source of bacteria Salmonella typhi causing typhoid.
When Mary Mallon the cook confronted with questions, she was not cooperative. However Soper did not lose hope and traced her previous five years history of employment. His discovery was astounding. Every family where Mary had worked in earlier five years, had its members suffering from typhoid. That confirmed there was some problem with Mary and Sopore wanted to confirm that. He requested Mary to give samples of urine and feces for testing but she not only refused but came attacking Soper with carving fork. He even tried to convince Mary that he was in fact writing a book and would share royalties with her if she had cooperated. In stead, she charged him of harassing her as she was hale and hearty having no symptoms of typhoid or fever of any kind.
When all attempts failed, Soper took the help of police and on examination of samples, she was found carrier of bacteria Salmonella typhi but without any symptoms. She was quarantined in a small island near New York with Terry a dog as her only companion. Not satisfied living in isolation, she moved the court and her application was dismissed holding she was a threat to the society.
Mary Mallon when in isolation, again made a petition to Health Commissioner who allowed her to lead a normal life in society on the undertaking that she would not work as a cook. Being uneducated and not convinced with her condition of asymptomatic bacteria carrier, she did not leave her profession of cooking and continued working as cook by the assumed name of Mrs Brown but in different county. As was obvious, there also she spread the disease and again the task of investigation was given to Soper who this time did not find any difficulty in reaching the conclusion that Mrs Brown aka Mary Mallon, the Irish lady who had migrated to USA in her adolescence, was the cause of affliction. This time Mary was slapped with severe punishment of living in isolation where she finally died. By that time, it was widely published in the papers that Mary Mallon aka Typhoid Mary was a superspreader of the disease Typhoid.
From the life story of Typhoid Mary, it is immensely clear that she never admitted herself suffering from typhoid and even refused to get removed her gall bladder that was considered bag of bacteria Salmonella typhi. Interestingly, the same perception exists today when coronavirus has encompassed the globe. An asymptomatic infected person never gets himself tested for the novel virus and will always insist, he is healthy. One can recall such instances that recently occurred in Punjab, UP and elsewhere. Superspreaders thus infect many and will continue spreading irrespective of doctors advice.
References:
Author: Narinder Kumar Wadhawan is an Electronics and Electrical Communication Engineering graduate and was earlier Scientist, then Instrument Maintenance Engineer, then Civil Servant in Indian Administrative Service (IAS). After retirement, he writes on subjects, Astronomy, Mathematics, Yoga, Humanity etc.
Happiness is widely discussed on social media and is advocated as a solution to all ills. May it be successful life, longevity or prosperity., it is a recommended wonder pill against all maladies of life. Happiness is a goose that lays golden eggs, also a golden key that opens all doors to ones well being. One who differs with its charismatic effects is dubbed the odd man who queerly glorifies melancholy.
But to my mind in simple words, happiness is that which does not have its existence at its own, it coexists with melancholy, it is inseparable from sadness, it can only be felt when one has passed through gloominess. Happiness is akin to relief one gets with a rest of few hours after engagement in bone tiring hard work. Happiness is akin to joy one feels on finding lost valuables although net gain is nothing, Is it then not the sadness that defines happiness as situated diagonally opposite but inseparable from it? Is it not the loss that realises how happy one was without suffering it?
It is melancholy that creates emotions of happiness. That is why wires in our brain get crossed over when one cries feeling extreme happiness. Hungarian song ‘Gloomy Sunday,’ was liked most. https://youtu.be/55FP1LfkkVQ
Those who liked most and enjoyed it to the hilt committed suicides. Is extreme happiness not suicidal? Why are Bollywood movies, ‘Anand,’ ‘Do Ankhe, Barah Hath,’ ‘Devdas’ to name a few, liked by the audience? Why are plays of Shakespeare liked most? These all have one thing in common, it is their tragic end.
Credit To ‘Balaji Creations’ For This Video Clip Of Bollywood Melancholic Songs
Psychology says a listener involuntarily finds oneself in the situation being described. Narrator’s miseries get reflected in the mind of a listener who feels as if he were passing through the same situation being narrated and that gives rise to empathy, its repetitive occurrence in ones mind puts it in permanent memory of the brain. That is why one rarely forgets happenings that have tragic endings but easily forgets those which culminates in to happy reunions.
Father of our nation, Bapu Gandhi described in his autobiography indelible impression of pain that play Sharvana made in his mind. Scene of bind handicap old parents mourning over the death of their dutiful son who had gone to fetch water for them from a nearby pond, was killed by the arrow of King Dashratha mistaking him for an animal, was unforgettable for Gandhiji throughout his life. It evidences human empathy over the pain of others is everlasting and can never be waned or dampened. Sound of cheers, laughters are transient, momentary and short lived whereas moans of one in pain are last for ever in ones mind.
Of the literary works of C S Lewis, the most acclaimed was ‘A Grief Observed,’ written pseudonymously as N W Clerk to hide his identity. He had explored in these collections the fundamental questions of faith and theodicy. Lewis was staunch Christian, grieved and shaken to irreparable limit by the death of his wife. As resentment over the death of his wife, he cursed God saying, “We worship You, always keep You in high esteem, revere You but when we need You, Yours doors are closed and shut for us for ever.” This book in fact was his recollection of his experiences after the death of his beloved wife. It was later claimed by his step son that the book was not only the result of grief over the death of his mother but it was recollection of the the miseries suffered by the great writer throughout his life. Later on the book was republished with his actual name CS Lewis. The book moved every heart who read it and consequently a movie by the name Shadowlands was televised and later on a film of the same name. It was again the grief observed by C S Lewis that took the centre stage leaving everlasting effect upon the audience.
Melancholy, gloom or sadness are effaceable and sensible minds that keep on pondering are prone to be effected. Persons with such sensible minds think and rethink to find solution to the miseries. Overthinking by the brain defines one’s genius. More over-thinkers are more innovative. As a corollary, genius, researcher, innovator tend to be gloomy. Should then being a genius and over-thinker at the cost of gloominess be despised?
Image Courtesy TeePublic
Nay, that is what my mind says. Gloominess discovers humanity in us, it guides us to alleviate the pains of others, it evokes in us empathy for others. Melancholy unites us. Happiness gives us joy momentarily, it sets us in endless race of competition to prove one better than the other, richer than the other, stronger than the other. It puts us in competitive mode. Happiness closes the door for empathy and makes one cold shouldered and immune to human sufferings. Empathy for human sufferings is not felt at clubs, bars, extravagant theatres or hotels, it is felt in nursing homes, healthcare centres and hospitals which are synonymous with Temples, Churches, Mosques or any other place of Worship.
In the annals of literature, one finds innumerable stories, poems, essays where melancholy, suffering, gloom and sadness have given birth to talents. No art can be better reflected than by mournful cries of ones soul. No poem can be better written than expressing ones soulful agony. Maharishi Valmiki composed first Sanskrit Shaloka or verse when filled with deep agony over the death of mating crane by the arrow of the hunter and his lips cried out,
मा निषाद प्रतिष्ठां त्वमगमः शाश्वतीः समाः।
यत्क्रौञ्चमिथुनादेकमवधीः काममोहितम्॥’
mā niṣāda pratiṣṭhā tvamagamaḥ śāśvatīḥ samāḥ
yat krauñcamithunādekam avadhīḥ kāmamohitam.
You will find no rest for the long years of Eternity
For you killed a bird in love and unsuspecting.
Thus came first verse as a result of deep grief observed by Maharishi. Conclusively, melancholy, grief, gloom, sadness are universal truth, happiness is relative. Without grief, happiness does not have its legs to support upon.
End
Note: In this article, relativity of happiness to melancholy is explained, it is not the purpose of the author to propagate melancholy as a way of life.
Author Narinder Kumar Wadhawan is an Electronics and Electrical Communication Engineering graduate and was earlier Scientist, then Instrument Maintenance Engineer, then Civil Servant in Indian Administrative Service (IAS). After retirement, he writes on subjects, Astronomy, Mathematics, Yoga, Humanity etc.
The Green At Chevy Chase Luxury Apartments, Buffalo Grove
Buffalo Grove is a village in Lake and Cook counties in the state of Illinois and is situated in the northwest suburbs of Chicago. At Buffalo Grove, there existed “The Green, Luxury Apartments,” where I stayed for few months last year. Beside these apartments, expanse grassy land was available to the delight of golfers occasionally seen driving carriages propelled by electric machines otherwise the place presented a deserted look. One could find very few persons moving on the foot passage that went along the periphery of this “Chevy Chase Business Park.” Out of those few, I was the one who could be seen in the wee hours leaving for “Wheeling” suburb for morning walk but taking due precautions to be on the right of the path. That habitat was mostly populated with persons of Indian origin and in a set of four apartments, one could find three occupied by Indians. Apart from this oriental population of “The Green,” its remarkable feature was its club which was fully equipped for workouts, be it for toning of muscles or enhancing endurance and stamina. I had the fancy for the treadmills and used to hop on these particularly when the sky was overcast and rain imminent. That weather was sought after for me as it broke monotonous long walks which could not be undertaken in wet weather. Club At The Green At Chevy Chase Luxury Apartments, Buffalo Grove
Upon treadmills were fitted buttons, knobs and controls of different hues mesmerising one as if one were the pilot in the cabin about to fly the winged machine. On such days while on treadmill I used to start with a pace of 3.5 miles per hour and after a little warming to brisk walk and then finally to speed of 5 mph. Butafter 4.5mph, the brisk walk turned to jog. One fine morning after spending a full forty minutes on the machine and surging my heart rate to 125 beats per minute at pace of 4.5 mph, I was feeling exhausted. A careful look at the indicator suggested me, maximum permissible heart rate in my case was 155. That is 220 minus my age. It was assuring I could continue but exhaustion and perspiration said something else. Sweating at an ambience of 9 degree C was overwhelming for the onlookers. Author In Reading Room Of Club At Chevy Chase Luxury Apartments, Buffalo Grove
Then it was time for me to photoshoot with a complimentary cup of coffee. Workout on that day had drained the energy out of my blood, however intake of the hot brew refreshed me as collecting residual energy at one place. Life in Buffalo Grove at that time appeared as if I was deprived of what I otherwise used to have. I often used to think when that august August will come and I will fly to my native land. I have been in India for the last one year, I have returned from foreign land. But my heart corrects it to state I have returned from familiar land. Now when I am in India, for reasons not known to me, my heart aches and longs for Chevy Chase Club as if treadmills there were calling me and were asking the visitors in their humming voice of my whereabouts little knowing I was away at an unfathomable distance of seven seas.
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About author
Author Narinder Kumar Wadhawan is an Electronics and Electrical Communication Engineering graduate and was earlier Scientist, then Instrument Maintenance Engineer, then Civil Servant in Indian Administrative Service (IAS). After retirement, he writes on subjects, Astronomy, Mathematics, Yoga, Humanity etc.
While attempting to find a general solution to cubic equations, I reached a road block where I was required to solve Binomial coefficient involving division of one factorials by the other . Search with google engine led me to a formula given by mathematician Stirling and to my disappointment, use of that formula again led me to another cubic and thus it was again of no use. However, curiosity arose in me to simplify the formula for its applicability to cubic equations. Wikipedia, it will not be exaggeration if I say, is the treasure of all knowledge, was consulted on the topic. Before I proceed further I submit, factorial is nothing but multiplication of consecutive integers starting from one and ending with the number to be factorialised. As it was expected, topic in Wikipedia had huge information from history to intricate questions on factorial and I admit that was all Greek to me, beyond my comprehension although I boast of engineering qualification. It had ‘O’ function, upper and lower bounds and asymptotic methods to solve it.
“Am I required to know all such special functions to derive a formula as simple as multiplication of 4, 3, 2 for 4 factorial? Can mathematics be so tough in its approach as to confine derivation of a formula to knowledgeable few ? How then can it be popularised amongst masses? Can I make it simple and comprehensive to all?” were some of the questions that cropped up in my mind at that time.
Not to be trapped by the web of special functions and mathematical lexicography, I was determined to find a formula that would be simple and easy to understand. That needed, my approach must be simple and independent of earlier works an approach which should be different from traditional mathematics. At the same time, my mind should be free from from bias or prejudices that one acquired from traditional learning. I submit what colleges and Universities do to our mind, they bias us towards a particular thinking in conformity with that of peers. We lose our independent thinking, kill our creativity and move on the dotted lines to reach the set results. They ask us in examinations to derive relations of theory of relativity but never ask us whether it can be interpreted in some different form.
Coming to the derivation of formula for factorial, I had unaffected and unbiased mind, I had not learnt derivation of Stirling formula in college. It was a great plus point in my favour what others may construe my handicap. How to proceed to derive the formula was initial stumbling block. I always subscribe to axiom that sophistication manifests in simplicity. More one will be simple in ones approach, more probable is he would reach his goal. I knew, simple formulae can disentangle the web of mathematical puzzles of factorial.
It required of my understanding of number theory in simplest sense. For academic use I submit I knew numbers progress as 1, 2, 3, so on. My mind worked on what made these progress in this way. Is it not simple addition of unity to consecutive number. Can these progress by multiplication? I had found the answer. Yes, these can. I illustrate their progression by multiplication.
1 = 1,
2 = 1.(1 + 1),
3 = 1.(1 + 1).(1 + ½),
4 = 1.(1 + 1).(1 + ½).(1 + 1/3),
………………………………,
By mathematical induction, n can be written as
where sign of dot . means multiplication. On multiplying all these equations and by definition of factorial n written as n!, factorial n can be written as
Next hurdle was how to compute its value. For that I used simple logarithm and integration formulae
and this sum approximates as integration of ln (x) with x when x varies from x1 to x2
Noteworthy point here is, this integration gives approximation and it would be natural that value of n factorial would not be free from error as approximation does not equal to actual value. Next step was elimination of error. It is submitted plot of (n – 1).ln {n/(n – 1)} (on Y axis) with n (on X axis) is a smooth curve but n changes in steps from 1 to 2, 2 to 3 and so on. Therefore, smooth curve must conform to ladder shaped figure formed by step changes in n with smooth curve and and conformity of the two can be achieved by algebraic addition of area of triangles above the smooth curve. In this way, error although highly removed had still residual error on account of curvature of the curve. After final removal of this error, approximation of n factorial was 10 to 100 times more accurate than that given by Stirling formula. I further submit, mathematics not only expects accurate results but also appreciates neat and clean formulae for the beauty of its intrinsic value. In the scales of mathematicians, this result is clumsy and needs ornamentation. A careful look at the above said formula reveals that (1-1/n) has exponent that is n squared and to avoid that I used ‘e’ given by great Swiss mathematician Leonhard Euler. This number is given by relation, n tends to infinity, (1 + 1/n) raised to power n, tends to e.
For that I assumed n to be very large and replaced (1 + 1/n) raised to power n with e wherever it occurred. That gave the above said formula, luckily or unluckily, look resembling Stirling formula. All said and done, formula derived, next step was its publication. I would not be wrong if I state publication was the most difficult task, not that the paper was lacking in quality but there were certain other reasons that I had never anticipated.
Innovation is not difficult, it may be meritorious, it may be easily comprehensible but if it is not referred to earlier works, it is likely to be trashed and junked. Academicians want, a researcher must walk on the dotted lines imprinted by peers to reach his findings and any deviation from that path leads one to nowhere. A number of references means researcher has put in lot of hard work and more meritorious is the submission. Can one imagine a research paper being rejected on account of the fact author’s contribution in theory or its application was significant? Yes, it is a valid ground for rejection. Papers are rejected as the authors have contributed significantly? What is the inference? Don’t contribute significantly. But then who will contribute to make research a quality research? Natural corollary is to take the support of peers and limit your contribution to minimum. Question that arises is whether such untenable rejections will not retard the progress of Scientific inventions for betterment of mankind? Should independent and creative thinking not be encouraged? Should the rejections for the sake of proving the journal is the best and strict acceptance is adhered to, not be discouraged? But who will bell the cat?
Coming to the publication of this paper, I passed through all the hurdles, journals wanted me to describe the upper bound in deriving the formula, others wanted me to draw the graph with computer and as graph drawn manually was not acceptable. I explained in details my paper is not based on set procedure whereby upper and lower bounds were considered and it is not required of me to explain matters which did not form parts of my paper. Further I had derived the formula considering as if Stirling formula were never given. I lament I could not satisfy them and suffered rejections. Initially, rejections were upsetting but when I again checked my paper, I was reassured there was no fault with paper, probably actual fault was their age old procedure of assessment.
Finally I came across a journal that did not raise any objection and accepted my paper which was then published. Repeated rejections in fact made me stronger and more trustful of my work. I could do a work using elementary functions what others did standing with the supports of ‘O’ functions, upper and lower bounds. I could not resist my urge to refer to Friedman great conjecture here.
“Every theorem published in the Annals of Mathematics whose statement involves only finitary mathematical objects (i.e., what logicians call an arithmetical statement) can be proved in EFA. EFA is the weak fragment of Peano Arithmetic based on the usual quantifier-free axioms for 0, 1, +, ×, exp, together with the scheme of induction for all formulas in the language all of whose quantifiers are bounded.” All theorems are provable by simple elementary functions. There is no limitation, you, he and I can prove any theorem by use of elementary functions provided we resort to proper reasoning, concentration, patience and determination. Let us think differently, all are endowed with some special qualities which differ from one to other. Let us explore and utilise that special quality and not surrender before fighting. It also reminds me of Steve Jobs who was not even graduate in qualification but what he did astonishes us all even the peer in the fields.
“They teach you how other people think, during your most productive years,” he said. “It kills creativity. Makes people into bozos.” Lisa Brennan-Jobs daughter of Steve jobs recalls her father in her book and its abstract was published in Times Of India. That was my unforgettable experience with factorial and it taught me to continue my journey following untouched and virgin paths of mathematics.
References
1. Wadhawan Narinder, Wadhawan Priyanka, A New Approach To Approximation Of Factorial Using Elementary Functions Only, International Journal Of Engineering And Future Technology 2020 http://www.ceser.in/ceserp/index.php/IJEFT/article/view/6203
Writer Narinder Kumar Wadhawan is an Electronics and Electrical Communication Engineering graduate and was earlier Scientist, then Instrument Maintenance Engineer, then Civil Servant in Indian Administrative Service (IAS). After retirement, he writes on subjects, Astronomy, Mathematics, Yoga, Humanity etc.
“They teach you how other people think, during your most productive years,” he said. “It kills creativity. Makes people into bozos.” Lisa Brennan-Jobs daughter of Steve jobs recalls her father in her book and its abstract was published in Times Of India.
How true it is about our education system! This system makes us think on the lines others have already thought. There is no scope of independent thinking, young minds are biased and made to think in a particular direction and memorise certain theorems to fetch good marks.
Where is creativity? Will education system ever change and create a platform for independent and creative thinking? Or will the generations continue to toe the lines already drawn and dance like puppets with the strings in the hands of those who had already done work on it?
Georg Ferdinand Ludwig Philipp Cantor
It reminds me of German mathematician Georg Ferdinand Ludwig Philipp Cantor whose original revolutionary idea of existence of infinite infinities of different sizes, was detested by mathematicians particularly his colleague Kronecker who jeered at calling him Mr Many Infinities. He was so devastated and depressed that he offered to quit teaching his parent subject mathematics and preferred philosophy. History reveals new creative ideas are always mocked at. Ironically the same ideas after the death of the inventors, are respected as new breakthroughs and followed in toto.
Will there be education system where creativity not only appreciated but encouraged also? Would that day arrive when success in examinations will depend upon one’s innovation and creations instead of proving Bernoulli’s theorem on dotted lines.
This text is written on being inspired by thoughts of Steve Jobs that colleges and universities bias the minds of students in the direction of particular thinking killing true creativity.
Writer Narinder Kumar Wadhawan is an Electronics and Electrical Communication Engineering graduate and was earlier Scientist, then Instrument Maintenance Engineer, then Civil Servant in Indian Administrative Service (IAS). After retirement, he writes on subjects, Astronomy, Mathematics, Yoga, Humanity etc.
(Universe: The Hubble Ultra-Deep Field image shows some of the most remote galaxies visible with present technology, each consisting of billions of stars)
It was dark, pitch dark and presumably nothing existed and the darkness indicated absence of everything. Then, there appeared a ray of light 13.8 billion years back. Source was an unknown point. A point that has no volume but was dense enough to assimilate infinite energy. What sourced that infinite energy is not known. The question is unanswered and still baffles those who curiously look into to it. What existed prior to 13.8 billion years, is also not known. But 13.8 billion years back, first ray of light appeared with unprecedented Big Bang and in a fraction of second, it spread infinite heat and that formed basic particles of existence. Those particles, then fused to form atoms which gushed out in abundance in all directions, spreading and ever spreading to oblivion. First atom that was born, was hydrogen, the lightest atom of our material world. Creation started with the winking of an eye. Hindu’s scripture, “Manu Samiriti’s” description is no different. There is no discernible change in explanation of evolution of universe except eulogisation of the primordial ray of light with Lord Brahma. After Big Bang, atoms continued to evolve, from hydrogen to helium, helium to carbon, carbon to nitrogen, nitrogen to oxygen so on by fusion of their nuclei under intense heat. From lighter to heavier, different atoms continued evolving. Now the infinitely large heat that was compressed in dot of zero volume, was spreading after transforming into mass but that amalgamation still had major share of energy. Atoms of different elements were still evolving.
With the birth of mass from energy, gravity evolved. This attractive force lumped the atoms forming dense spots in the clouds of energy and atoms. Lumps being heavier gravitated more strongly and garnered more atoms in them and the process increased the size of lumps. This regenerative action went on enlarging the lumps till the stage the lumps got so massive that they started compressing their own self by the process of self gravitation. Resultant compression heated up the lumps to the temperature so high that it started radiating heat. Thus first star was born.
Self gravitation was compressing the atoms thereby raising the temperature and rise in temperature was causing their expansion. A tug of war in this way, started between gravitation and thermal energy. And that adjusted the size of the star when an equilibrium reached where thermal energy equalled the gravitational energy. That formed a stable star. But some stars were not lucky and could not muster sufficient thermal energy. There, gravitation won. Gravitation shrunk the size of the star and with each shrinkage, gravitation got stronger. Now the gravity was so huge that the star started mauling whatsoever was with in its vicinity and with each capture, mass of the star was increasing and finally it did not remain a star, it was a monster star, a celestial demonic monster which was devouring even the light rays if that happened to pass by its event horizon. This monster ‘Blackhole’ ruled over the celestial empire and took central place in the clusters of stars. The stars started revolving around it like servants moving around their masters to fulfil his wishes.
(NGC 4414, a typical spiral galaxy in the constellation Coma Berenices, is about 55,000 light-years in diameter and approximately 60 million light-years away from Earth.)
Tug of war between the gravity and thermal energy did not end with formation of star or Blackhole. The lump of masses which failed to gather sufficient mass to fructify into a star, gravity lost the war. The lump reduced to the status of a planet. Jupiter is a fine example of a failed star.
Apart from evolution of stars, blackholes or planets, these continued to be members of the cluster subservient to Blackhole at its centre. This clusters formation continued and is still continuing. Such clusters were called galaxies and our star sun and its family is also a member of the galaxy called ‘Milky Way.’ The spreading of energy that started with Big Bang is still continuing. Galaxies are spreading out, better to say, are receding from each other and the infinitely high temperature at the start of Big Bang is lowering. It is approximately 2.7 degree Kelvin as is measured from ‘Cosmic Microwave Background Radiations CMBR.’ CMBR is average residual radiations of space that exists now. These radiations are omnipresent and are tell tale signs that once the universe had been hot. Corresponding temperature of CMBR is likely to further go down as the galaxies will recede. CMBR is a yardstick to calculate the age of the universe.
Can you guess how many galaxies are there in the universe? These are 100 billions which is written as one followed by eleven zeros 100000000000. With better instrumentation resulting in better detection, the figure is likely to increase to 200 billions. Wow, a mammoth figure! Isn’t it?
Can you have an idea of number of stars present in a galaxy? On an average, it can be taken as if a galaxy has 10 billion stars that is 10000000000. Again it is a mammoth figure. About number of planets, it can be assumed that a star has, on an average, 10 planets.
Multiplication of these figures gives us the estimate of 10000000000000000000000 planets in the universe leaving aside gaseous diffused matter. It is extremely high a figure and is not comparable with materialistic possessions. Considering surface area of a planet as 5490000000000000 square feet, total surface area of all the planets calculate as 5490000000000000000000000000000000000 square feet. Assuming surface area of a man as 5 square feet and on comparing it with surface area of all planets of the universe, ratio is 1 to 100000000000000000000000000000000000. For all intent and purposes, it is tending to zero, a big cipher, a man is big cipher and an insignificant figure.
Strangely, inspite of its size approaching to zero, he is self centred, greedy and runs a never ending rat race. He shows his oneupmanship over others, enslaves weak and deceives others for petty gains. He feels pride in stating, “This is mine, that is mine, he will further acquisition that.” Did he ever realise his existence in the grandeur design of the universe? Will his acquisition change the status of his existence. Nay, No. Zero added to zero will be no more than zero. His pursuit of acquisition is mirage, it is never ending, appearance of water at a distance is illusion. His existence as compared to the universe is null, a void and a big nothing. This is his existence. Will he ever realise it?
Writer Narinder Kumar Wadhawan is an Electronics and Electrical Communication Engineering graduate and was earlier Scientist, then Instrument Maintenance Engineer, then Civil Servant in Indian Administrative Service (IAS). After retirement, he writes on subjects, Astronomy, Mathematics, Yoga, Humanity etc.
Happiness is widely discussed on social media and is advocated as a solution to all ills. May it be successful life, longevity or prosperity., it is a recommended wonder pill against all maladies of life. Happiness is a goose that lays golden eggs, also a golden key that opens all doors to ones well being. One who differs with its charismatic effects is dubbed the odd man who queerly glorifies melancholy.
where sign of dot . means multiplication. On multiplying all these equations and by definition of factorial n written as n!, factorial n can be written as
and this sum approximates as integration of ln (x) with x when x varies from x1 to x2
Narinder 
