Wonder Relation In Mathematics

Where do you reside?”

“In front of Post office.”

“Where is then post office?”

“It is in front of my house.” The answer has baffled all at one time or the other as the reference is related and is not fixed. In fact, both locations are the reference for one another. That makes the location not only unlocatable but puts one into never ending loop of hopping from one location to the other. Such loops are called closed and never ending.

But where is mathematics involved in it? Per Se there appears no mathematics in it but in mathematics, there are abundant such situations where there are such unending loops. I recollect vividly as a teacher of computer programming way back in eighties, I used to make programme involving loops based on mathematics questions, language at that time was simple ‘basic’ and commands were go to that line and from that back to original to puzzle the calculating machine. But machines are beyond nervousness and confusion. To my amusement, Computer when asked to display the status used to indicate, “Still calculating.” Hardly the machine knew, it had been put in never ending task.

Coming to my initial question of location of John’s house, if John says, his house is situated in front of Post Office in Peter Lane, then reference as Post Office is fixed but one when goes to Post Office, there one finds, Post Office is comprised of a huge building and there are a number of houses in front of that building and John’s house can not be found. Again John will be contacted and he informs, it is a grey coloured house in front of telegraph section of Post Office. But there are a number of grey houses in front of that section. On again inquiring, John replies, it is in front of cashier’s window. In situation highlighted here, there is a loop but it is localising on each cycle thus reaching nearby its destination. Such loops in mathematics are called convergent loops. Relationship of Post Office and house is called recursive relation.

Hope, I could clarify the concept of recursive relation. Such recursive relations have charismatic effects and can land mathematicians in new unexplored land of mathematics. Continued fractions, continued Ramanujan nested radicals, continued products are gifts given by such relations.

I take up some examples to bring home further the concept of recursive relation. Let us consider a right angled triangle ABC with base BC, perpendicular AC and right angle at C, ratio of perpendicular AC to hypotenuse BC is called sin B where B is angle at point B. This angle B can be halved by formula,

sin B = ½.sin B/2.cos B/2. ………………………………………………………….(1)

Examination of this formula reveals that sin of an angle appears in both left hand side LHS and right hand side RHS. This sine on right hand side RHS will reflect it back to the formula of sine written in left hand side LHS. This reflection of sine formula from LHS to RHS and then from RHS to LHS continues indefinitely. But in the process angle B goes on halving from B to B/2, from B/2 to B/4 and so on infinitely. This is analogous to reflection from Post Office to opposite building and opposite building to post office and continuous halving of angle is analogous to localising the house to smaller area.

Equation (1) is recursive relation and applying this formula successively, the angle B will be reduced to infinitesimal value B/2^n. How this angle will be reduced by application of recursive relation is explained in steps below.

sin B = ½.sin B/2.cos B/2.

Therefore, sin B/2 on RHS can be written as 2.sin B/4.cos B/4. That makes

sin B = 2.cos B/2. 2.sin B/4.cos B/4 = 2^2. cos B/2.cos B/4.sin B/4.

Again, sin B/4 on RHS can be written as 2.sin B/8.cos B/8. That makes

sin B = 2^3. cos B/2.cos B/4.2.sin B/8.cos B/8 = 2^3 cos B/2.cos B/4.cos B/8.sin B/8.

In this way, it can written,

sin B = 2^n cos B/2.cos B/4.cos B/8…………………………….sin B/2^n. …………………………….(2)

Angle B will go on reducing but for derivation of formula, it has been aborted at n terms.

Let us see what happens to this equation if a little mathematics is applied to it. Sin A = A is true when angle A is very very small tending to zero. In equation (2), B/2^n also tends to zero on successive halving particularly when n tends to infinitely large, therefore, sin B/2^n equals B/2^n. Applying it to equation (2), it transforms to

sin B = 2^n cos B/2.cos B/4.cos B/8…………………………….B/2^n.

On transposing B/2^n to RHS,

(sin B)/(B/2^n = 2^n cos B/2.cos B/4.cos B/8……………………………

Or (sin B)/(B/2^n = 2^n cos B/2.cos B/4.cos B/8……………………………

Or 2^n.(sin B)/(B) = 2^n cos B/2.cos B/4.cos B/8…………………………

On cancelling 2^n on LHS and RHS,

sin B/B = cos B/2.cos B/4.cos B/8…………………………………. (3)

This is a beautiful formula, first derived by great mathematician Euler and also indirectly used by another great mathematician François Viète. The formula derived here is the outcome of application of recursive relation and was widely used particularly for determining the value of π. Euler used it to factorise sin B in terms of angle whereas François Viète used it as it is.

Another example of recursive relation is angle halving formula for Cosine of an angle. For that I refer to right angled triangle as mentioned before wherein cos B is the ratio of base BC to hypotenuse AC and it can be written as

cos 2.B = 2.cos^2 B – 1 and it can be rearranged as

cos B = {(1+ cos 2.B)/2}^2 = ½.(2+ 2.cos 2.B)^2. …………………………………………………(4)

Examination of equation (2) reveals that it contains cosine of an angle in LHS and also in RHS, therefore, it can form a never ending loop. Cos 2.B in right hand side can be replaced by ½.(2+ 2.cos 4.B)^2 and again cos 4.B again replaced by ½.(2+ 2.cos 8.B)^2 and so on. Therefore, cos B can be written as

I take another example of angle tripling formula whereby sin 3.B is written as

sin 3.B = 3.sin B – 4.sin^3 B = sin B.(3- 4.sin^2 B) or

sin B = sin 3.B/(3- 4.sin^2 B) …………………………………………………………………………………..(6)

Inspection of equation (6) reveals that its RHS has sine of angles in numerator and denominator and LHS also has sine of an angle. In RHS, sin 3.B in numerator will be considered for recursive relation and will be replaced by sin 9.B/(3- 4.sin^2 3.B) and sin 9.B by sin 3^3.B/(3- 4.sin^2 3^2.B) and so on till sin 3^n.B = sin B. At that stage, sin B on LHS and RHS cancels and denominator which is product of terms (3- 4.sin^2 B).(3- 4.sin^2 3.B), …………………{3- 4.sin^2 3^(n-1) B} equals 1, giving rise to an identity.

For illustration, let us take the example of sin π/10.

Here angle B = π/10 which can be written as

sin π/10 = sin 3π/10/(3- 4.sin^2 π/10) = sin 9 π/10/{(3- 4.sin^2 π/10).(3- 4.sin^2 3π/10)}. That is

But sin 9.π/10 = sin π/10, therefore,

Here recursive relation stops when sin^n B equals sin B or – sin B. From this, it can be concluded


Π(3-4.sin^2 3^(j-1) = 1 …………………………………………………………………………..(7)


when sin (3^k B)= sin B and


Π(3-4.sin^2 3^(j-1) = – 1 …………………………………………………………………………(8)


when sin (3^k B)= – sin B.

In this way, a new trigonometric identity is established using recursive relation.

Coming to equation (5) which is an equation that contains roots of root, such continuous nested radicals were first used by François Viète when he derived formula for pi first time from entire Europe. Recently, these have been abundantly used by Indian mathematician Ramanujan. As such, use of recursive relation gave unexpected and astonishing results that mesmerised English mathematician Hardy.

I also experimented on recursive relations and results were encouraging. If you feel a bit interest in it, don’t shy away. Take a note book and jot down a formula. Find all possibilities for its recursive nature, try differently and look for different result. You will find one I am sure and that would be blissful. Do you feel interested in recursive relation, you may go through this article at http://www.ijma.info/index.php/ijma/article/download/5227/3074



1. A part of this article is excerpts from my paper written on “MORE TRIGONOMETRIC IDENTITIES ,” available at http://www.ijma.info/index.php/ijma/article/download/5227/3074

2. Title image courtesy brewbooks at https://www.flickr.com/photos/brewbooks/184343329/

About author

Writer 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.


Here Negative Energy Is Plentiful. Should We Remove Or Conserve It? 

After finishing morning chores, I took up this shining device in my hand and went through hurriedly messages sent by my friends on WhatsApp conversation. The message “HOW TO INCREASE POSITIVE ENERGY IN OUR HOUSE…..” was blinking on the tiny screen. After going through it, I found, it was more of house keeping than creating positive energy but author was claiming that these activities will remove negative energy and there will be plenty of positive energy. An abstract is given below.

“Fresh air and sunshine will remove negative energy from house. Keep rooms tidy and neat, remove old unwanted things, clutter are attracting magnets of negative energy. Walk barefoot in the house. Wash your feet before entering the house. Sweep and mop the floor with rock salt. Wash and soak your legs and arms with rock water solution. Keep your house well lit and illuminated.”

Coincidentally, there was another message on the same topic of negative energy, that indicated, “Ten signs you have negative energy.”
Courtesy Image Sorrowing old man (“At Eternity’s Gate”)  by Vincent Van Gogh at https://commons.m.wikimedia.org/wiki/File:Van_Gogh_-_Trauernder_alter_Mann.jpeg

I could not read this message probably apprehending the likelihood of at least one sign matching with that of my health condition. However, in the earlier message, some do’s and don’t’s were prescribed for removing negativity and bringing positivity. A bit interest in physics and repeatedly mention of word energy, drew my mind to that energy bundles h.f discovered by pioneer quantum physicist Planck. I could not accept the concept of negative energy that was highlighted in the messages. More I read, more my mind diverted to that negative energy which forms antimatter or exotic matter at the mouth of wormhole to keep it open and traversable.

I submit it has been the endeavour of Astrophysicists to discover negative matter or antimatter and theories have been put forth one after the other but all in vain as these theories were all hypotheses that needed experimental verification. And experiments could not prove existence of negative energy or the negative matter. Mathematics ignites the hopes of existence of negative matter but experiment extinguishes it. If negative matter or antimatter were discovered, there had been revolution in physics, present would have been bridged the gap between the present and past and  future could be dragged to present. Such looping of time are called Time like loops and are mathematically feasible. These were observed in analysis of Einstein Theory of General Relativity.

Surely strange would be the phenomenon when utilising such loops, one could hop to past and also to future. So strange were these loops that these once upset Einstein who himself felt uncomfortable by their queerness and junked these in initial stages when theory of general relativity was given recognition. However, his Colleagues did not relent and kept on working on charismatic effects of these loops. Such work is still continuing.

Before proceeding further, I would like to highlight what make these loops queer but charismatic. For that I will refer to time travel, that is going forward in time and also going back in time. Time travelling to past means, one while living in present on planet earth, enters the age of past, how much past one goes, depends upon the juncture or stage at which the loop is conjoined at its ends of present and past. Stage of past may be 100 years or 500 years or even more or less that depends upon the characteristics of timelike loop. Once one enters that age, one may find ones great grand father or still deep ancestor living at that time.

But problem arises, when one face to face, kills ones unmarried ancestor. What would happen to the existence of the great grand son in that circumstance, baffles the physicists. When ones ancestor is dead, cause of birth of one, is extinguished. Question arises, how does that person come into being when his ancestor had died unmarried. This is analogous to giving reply to the question that has not yet been asked. Result has preceded the cause. In physics, such situation is called “Grandfather paradox”.

Physicists have attempted explanation of this paradox as

1) Great grandson when enters the past would never be able to kill his great grand father. If he would attempt to shoot him, aim will be off the target or gun will fall or some event will take place that will preclude him from killing his great grand father.

2) Second explanation given is when great grand son enters the past, he though in front of his great grand father, yet he would be in the universe different from that of his great grand father. His action in the universe other than that of his of great grand father would not affect his great grand father.

What is the correct explanation of this paradox is unknown. Moot question that arises is why we could not time travel till now when progress in physics is manifold. According to me, its answer is attributed to the fact that we could not find negative energy that in term forms negative matter or antimatter.

For synonymity of energy and mass, I submit, of late, the physicists are not expressing mass in grams but it is written in electron volts/ c squared particularly in ‘Particle Physics.’ Since speed of light c is constant, mass is proportional to energy which is expressed in electron volts. Mass and energy are interchangeable and are not different.

Coming to negative energy that is claimed prevalent in destructive thoughts and is the hot topic on whatsApp conversation,  physicists ironically could not experimentally find it. Practically speaking, energy in the universe is all positive. Physicists crave to find negative energy as they know when negative energy were discovered, negative mass or antimatter would be discovered by default. Particles then could move with speed greater than that of light. Antimatter formed from negative energy will be repelled away from the earth instead of falling on earth as apple fell on the ground surprising Newton.

Strangely, whatsApp is full of messages on negative describing how to remove negative and create positive energy. It is also a burning topic of discussion between morning walkers or retired persons who find ways and company to express their views. Negative energy in electronic messages is synonymous with as arising from destructive thought processes, it is something connected with neuroscience or spirituality. But this negativity has now descended upon chemistry of epsom salt that is found in rocks. Or I should say negativity has climbed up rocks of epsom salt.

While sitting in my drawing room, I ponder if negative energy as discussed in whatsApp messages is so widespread and readily available in our houses or at filthy places and we are chasing it out, why we should not collect it and transport it to physicists who are ever willing to pay us heavily in exchange as Dengue afflicted patients are willing to pay exorbitant rate of Rs 2000 for a litre of mammal goat’s milk alleged to be raising platelets counts. Physicists probably do not know the extent of availability of negative energy which we are buzzing off by mopping our floor with ‘Kala Namak’ in this part of globe otherwise they would have placed a mammoth order of negative energy with the government earning it a tankful of foreign currency. This may also be other way round, messenger of such messages may not be in the know of the fact that negative energy is a sought after object that may earn them huge profits and besides it could transport them to the age of their great great grand father or could push them into unseen future. Should we then not keep a secret from such persons?


1.Title Image Of wormhole courtesy Wurmloch at https://commons.m.wikimedia.org/wiki/File:Wurmloch.jpg#mw-jump-to-license


About Author

Writer 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.

When Bribing Proves Noble 

Accepting or paying bribe is considered one of the most hatred acts and is condemned across the board by one and all including those who indulge in it. They also abhor the word but it is different they may like it if it is secret and unknown to others. Act of receiving or paying bribe oftenly termed as corruption is described as malignant tumour that can only be ridden by severing it of. Bribe and corruption are so detested that hardly will there be a person who can venture up to list it benefits. 

Does it have some salient features worth listing? This question per se appears bizarre and weird as if it implies promoting the aberration that most of the world suffer from. Recent readings have urged me to write about its other side which perhaps is unseen and untouched and this may tantamount to redefining bribe.

In early eighteen century, Guru Gobind Singh, tenth master of Sikhs was fighting mighty Mughal emperors against their repressive acts. His sons, Sahibzada Zorawar Singh Ji and Sahibzada Fateh Singh Ji along with Jagat Mata Gujri Ji (mother of Guru Gobind Singh Ji) were arrested by Kotwal (police officer) Jaani Khan of Morinda, India,They were brought to Sirhind before Faujdar Wazir Khan who imprisoned them in the Thanda Burj (cold Fort). Mata Gujri was not in favour of taking meals cooked in the Mughal kitchen in view of the atrocities of Mughal upon other religions and she preferred starvation.

In the area of Sirhind, there lived a man Baba Moti Ram Mehra who was employed in Hindu kitchen of Wazir Khan. Baba Moti Ram Mehra was a noble man who believed in helping the needy. He could not witness tender aged Sahibzadas and their aged grandmother starving while he took sumptuous meals. He expressed his desire before his wife that he wanted to serve young prisoners and their aged grandmother with milk and water. His wife cautioned him as Wazir Khan had already made the proclamation that those who attempted to help the prisoners, would be crushed in oil squeezer. But this warning did not deter Baba Moti Ram Mehra who had already resolved to help Sahibzadas and Jagat Mata Gujri.

“Aren’t you scared, my son?” asked his mother. Baba Mehra ji humbly replied, “Dear mother, our Guru is fighting against injustice of the Mughals. I will serve the great mother and the Sahibzadas. I don’t fear the punishment of the Faujdar. The history will not forgive us if we do not serve the great prisoners.”

Courage and conviction of Baba Mehra Ji transformed the heart of his wife and mother. A candle has lit other candles without diminishing its own light, His wife offered her jewellery and silver coins whatsoever she had in her possession, to give the guards as bribe for accomplishing this mission. She requested Baba Mehra Ji, “Please bribe the gate man of the Burj and request him to keep this act a secret.”

Baba Moti Ram served milk and water to the Sahibzadas and Mata Gujri Ji for three nights.

Finally tender Sahibzada Zorawar Singh Ji who was barely nine years old and Sahibzada Fateh Singh Ji who was barely seven years old when refused conversion to Islam, were bricked alive in a wall. It is touching to read Saka Sirhind (saga of martyrdom) particularly when elder brother Sahibzada Zorawar Singh Ji asks his younger brother, “You are lucky, being short in stature, to achieve martyrdom prior to I.” Tradition says that in the first attempt, wall crumbled down of its own when it reached chest height. Finally they were bricked and when news of death of Sahibzadas was broken to Mata Gujri, she collapsed and died. It was then also known that Baba Mehra Ji and his family had helped Sahibzadas, they were crushed in KOHLU, oil squeezer on the charge of helping the prisoners.

Here the gateman was bribed and he allowed Baba Mehra Ji to take milk and water for Sahibzadas and their grandmother yet this bribe paid for the service of needy was more sacred than worship. It was, in fact, an offering to Almighty to seek his blessing in the mission of helping the helpless.

Another related incident that I could not erase from my mind pertains to the period of Second World War. This remained hidden in the leaves of history books and was unearthed as late as 1999 when some students at rural Uniontown High School in Kansas USA began researching possible projects for the National History Day competition.

Their teacher Norman Conrad showed them title “Irena Sendler saved 2,500 children from the Warsaw Ghetto in 1942–43.” This article appeared in US News And World Report. Initially, the students thought the figure were 250 and there was typographical error. Saving 2500 children that makes at least five children a day seems impossible act to them. But as they prepared for their play, “Life In A Jar,” they researched more on the life of angelic lady Irena Sendler. As it was a case of more than seventy years back, they thought of going to her grave to pay respect to the noble lady but their joy knew no bounds when they came to know Irena Sendler was still alive. After meeting the lady personally, the students and their teachers had gathered sufficient information for the play. The play won the National History Day competition and also brought Irena Sendler to limelight.

Irena Sendler was a catholic christian and was highly impressed by altruistic nature of her father. Her father was a doctor who saved a number of patients suffering from typhus that was endemic at that time. But it also cost him his life as he also suffered from typhus which he acquired from his patients. Her father advised young Irena, “If a man is drowning, it is irrelevant what is his religion or nationality. One must help him.” Irena never forgot these words during her lifetime.

When young, she objected to segregating children on basis of religion in the school. During Second World War, she worked as nurse and joined Zegota which was the Council for Aid to Jews in Occupied Poland, a branch of the Polish underground. She was given the duty of children Department. Jews were confined to walled ghetto and initially, she brought out the children found on streets of ghetto to Orphanages with information to the parents. As the Nazis guards were alert, she adopted different ways, she would either put them in coffins or under corpses or in ambulances or in boxes and transported them to safe places.

Jewish Children In Ghetto

Two methods generally adopted by her were sneaking out children from ghetto through two buildings that squatted the gate. One building was church and other was courthouse. She used to teach young children basic Christianity in the spacious building of the church, removed their yellow star identity and put new clothes. These children were then pushed out of church front gate and even the vigilant eyes of Nazi guards could not detect their new identity.  

All these acts were done neatly and meticulously by Irena who would note down the particulars of their identities along with the place they were smuggled out so that they can be reunited with their parents at a later stage. Such lists were put in a jar and jars buried in ground in front of her friends house under an apple tree. Children too young to be taught Christianity for camouflaging their identities were taken in funeral vans or ambulances. If the infant being smuggled out cried, she would press the paw of the dog she had kept for this purpose. As the dog barked, other dogs outside also barked at Irena’s dog, the cocktailed sound suppressed the crying sound of the child and that made detection impossible.

  Tragically, Irena was arrested by the Germans on October 20, 1943–five months after the destruction of the Warsaw Ghetto. Her address had been revealed by an informer. She was tortured and beaten for several days; one leg and one foot were fractured. She refused to reveal the whereabouts of the children, or the names of anyone in Zegota. She was scheduled to be executed but members of Zegota bribed a guard to instead leave her in the woods, where they found and rescued her. Her name was printed on public lists of those who had been shot by German guards and she spent the rest of the war in hiding. After the war, she tried to reunite the children with their parents but most of the parents were executed.

Image of Irena Sendler with some people she saved as children: 

In 2003, Pope John Paul II sent Sendler a personal letter praising her wartime efforts. On 10 November 2003, she received the Order of the White Eagle, Poland’s highest civilian decoration and the Polish-American award, the Jan Karski Award “For Courage and Heart”, given by the American Centre of Polish Culture in Washington, D.C.

In 2006, Polish NGOs Centrum Edukacji Obywatelskiej and Stowarzyszenie Dzieci Holocaustu, the Polish Ministry of Foreign Affairs and the Life in a Jar Foundation established the “Irena Sendler’s Award: For Repairing the World” ( awarded to Polish and American teachers. The Life in a Jar Foundation is a foundation dedicated to promoting the attitude and message of Irena Sendler.

In 2007, and again in 2008, she was nominated for the Nobel Peace Prize from Poland, with support from numerous prominent personalities along with IFSW associations. On 14 March 2007, Sendler was honoured by the Polish Senate and a year later, on 30 July, by the American Congress. On 11 April 2007, she received the Order of the Smile, at that time she was the oldest recipient of the award. In 2007 she became an honorary citizen of the cities of Warsaw and Tarczyn. On the occasion of the Order of the Smile award, she mentioned that the award from children is among her favorite ones, along with the Righteous among the Nations award and the letter from the Pope. Irena passed away on May 12, 2008, and was interred in Warsaw’s Powazki cemetery–a place reserved for the elite among Poland’s artists, writers, scholars and war heroes.

In April 2009 she was posthumously granted the Humanitarian of the Year award from The Sister Rose Thering Endowment, and in May 2009, Sendler was posthumously granted the Audrey Hepburn Humanitarian Award. Around this time American filmmaker Mary Skinner filmed a documentary, Irena Sendler, In the Name of Their Mothers, featuring the last interviews Sendler gave before her death.   

Coming to bribing Nazi Guards, life of Irena Sendler was saved from the jaws of death when Nazis received the information that she had been saving the Jew children. It was the bribe paid to the guards that spared her life. Such bribe, in no terms can be considered a ‘BRIBE’ that had saved the life of a noble person. It was again an offering to Almighty rather than the hatred bribe. In common parlance, the bribe although appears offensive, but here it acted in furtherance of virtuous deeds.it can then be concluded, purpose that bribe will serve, matters and if purpose is pious, it is as revered as offering to God, if purpose is bad, it certainly is a crime. It can then be said that it is mens rea that decides an offence but once it goes missing from bribe, bribe turns a virtuous act. The word bribe then should not raise our eyebrows at least when we know intention behind it is pious and noble. If purpose of saving the life of innocents, helping the needy, taking care of helpless and dependent can be accomplished by bribe, the bribe would always be noble and virtuous act.


1 Title Image courtesy https://www.sikh24.com/2015/02/05/sgpc-allots-5-acre-of-land-for-building-baba-moti-ram-mehra-memorial/#.WdeZ0frhWf0

2, http://www.sikhiwiki.org/index.php/Sahibzada_Zorawar_Singh 

3. http://www.chabad.org/theJewishWoman/article_cdo/aid/939081/jewish/Irena-Sendler.htm

4, https://en.m.wikipedia.org/wiki/Irena_Sendler

5. Photographer Herrmann Ernst Image Jewish Children At Ghetto Cortney https://commons.m.wikimedia.org/wiki/File:Bundesarchiv_N_1576_Bild-003,_Warschau,_Bettelnde_Kinder.jpg#mw-jump-to-license 

6. Image of Irena Sendler with some people she saved as children:  Source own work, http://commons.wikimedia.org/wiki/User:Kmarius Author Mariusz Kubik, http://www.mariuszkubik.pl

About Author 

Writer 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.      


Mathematicians Think Differently

Yes, it is correct mathematicians think differently and extraordinarily. It is expected of them as traditional approach will lead to traditional results and halt the process of evolution of new results and formulae. It reminds me of Johann Carl Friedrich Gauss, sometimes referred to as Princeps mathematicorum. He was a German mathematician and also contributed to many fields including number theory, algebra, statistics, analysis, differential geometry, geodesy, geophysics, mechanics, electrostatics, magnetic fields, astronomy, matrix theory and optics.When Gauss was studying in school, to inflict punishment upon him for misbehaving, his school teacher gave him a cumbersome and time consuming problem of addition of all integers from 1 to 100. But to the astonishment of the teacher, he added the quantities with in few seconds and showed the result. The result was correct and it earned appreciation for young Gauss in the heart of the teacher.

What Gauss did even baffle now to most of us. He first wrote the sum as

S = 1 + 2 + 3 + ………………………………………up to 100 and again wrote it but in a different way

S = 100 + 99 + 98 +………………………………………………up to 1 and added both as

2S = 101 + 101 + 101 + ……………………up to 100 times and then

2S = 101 x 100 = 10100 and 

S = 5050.

That produced the result wonderfully in a neat and clean manner. Had there been some other student, he would have adopted the normal procedure of adding 1 with two then with 3 and then with four until he would have reached the integer 100. Even the teacher expected it so from Gauss but Gauss was not a simple student, he was extraordinary.

In the same way, another student at the university of Copenhagen was asked in examination how the height of a skyscraper can be measured with barometer. Naively, he replied by tying the barometer with a rope and then hanging it from top of the tower touching the ground and measuring the length of rope plus the height of barometer that will give the height of the skyscraper. Not impressed with the answer, the examiner awarded zero marks to this answer. 

On appeal by the student, it was held that answer is correct but there is no physics involved and the student was called for oral examination. When the student was asked to give answer to the same problem with in five minutes, he spent most of the time thinking for answer. Ultimately, he was warned the time was running out and he should put forth his answer immediately. However, the student replied, he has many answers to the problem and was thinking which answer was the best. When he was asked to give all the answers, he politely replied,” Height of skyscraper can be known by throwing the barometer from top and noting the time taken by it to touch the ground. Since it may cost barometer, the height can alternatively be known by measuring the length of barometer and also its shadow and then measuring the shadow of the skyscraper, ratio of barometer to its shadow will equal ratio of height of building to its shadow and simple arithmetics will give height of skyscraper. He continued, there is yet another way, difference of barometer reading at top and bottom of the sky scrapper will equal to atmospheric pressure equivalent to height of skyscrapers. He still added, If one is too scientific, one can note down the time period of a pendulum at the base of the skyscraper and and also at the top of skyscraper and difference in time period will be a measure of height.”
That student was genius mathematician and physicist Neil Bohr who propounded quantum theory later in his life.  

It is also said of Bohr that during a foot ball match where Bohr was a goalkeeper, the ball remained in the opponent half but when the attacker brought the ball to Bohr’s half, a spectator shouted, ” Be aware Bohr, the ball is in your half,” and at that time, Bohr was found solving mathematical problem on the ground.

Mathematicians think differently from an ordinary person. I am certain, I will be correct if I say, most of inventions have been made when unconventional route was adopted.

Let us take the case of determination of value of pi written as π which is a mathematical constant and is defined as a quantity when multiplied with diameter of a round object gives its circumstance. Normal thinking leads us to determine its value by measuring circumference of a round figure and then dividing it by diameter.

Method is correct but error in measurement of circumference and diameter will result in error in value of pi. And pi so calculated will culminate in large error in the area or circumference of a round object if that happens to be of large diameter. That led mathematicians to determine pi independent of measurement of its circumference or its area. This is, thinking differently.

Without going into the history of pi, I will attempt to explain how earlier mathematician and physicist thought of pi. Pi, according to mathematician and physicist Archimedes, can be calculated by averaging the area of a polygon circumscribed and inscribed in a circle. His thought points to the fact that circumference of a circle is equal to half the circumference of polygon just outside plus half the the circumference just inside the same circle. That was an apt thinking and gave great approximation of value of pi provided sides of regular polygon are quite large. Probably, such thought would generally not come to ordinary person.

Idea that occupied to most of the mathematicians in the subject of determination of pi was approximating regular polygon of extremely large number of sides with circle. Extremely large number here is that number which is greater than the greatest.

Let us attempt finding value of pi by above said approximation.

 I submit, a point forms an angle of 2. Pi around it and let us take polygon or Triangle of three equal sides AB, BC, CA. That will make, its three corners or points as equidistant from each other. Let there be a point O inside the triangle such that these points are also equidistant from point O ie OA = OB = OC = radius r. That means a circle with centre O , circumscribes triangle ABC. Let us find out perimeter of triangle ABC which is equal to sum of AB, BC and CA.

Side BC = twice length BD and BD = OB x sine of angle BOD.

Since ABC is a polygon of three sides, therefore angle BOC = 2.pi/3.

If regular polygon has n sides then angle BOC = 2.pi/n.

 For n =3, angle BOD = ½. 2.pi/3 = pi/3.

BC = 2.BD = 2r.sin pi/3. For n sided regular polygon, it would equal 2r.sin pi/n.

AB + BC + CA = 3. BC = 6.r. Sin pi/3. For n sided regular polygon, perimeter would be 2.n.r sin pi/n.

First approximation is made by equating perimeter of three sided regular polygon with circumference of circle (2.pi.r). That is

6.r. Sin pi/3= 2.pi.r.

Or pi = 3. under root 3/2 = 3. (.866)= 2.698.

Next approximation would be by taking n = 4,

 pi = 4.(sin pi/4)= 4./2^1/2 = 2.828.

Next approximation when n= 6,

 pi = 6. Sin pi/6 = 3.

It is clear from above as n increases, value of pi approximates better to actual value of pi.

For n sided regular polygon,

 pi = n.sin pi/n……………………………………………………………(1)

If we put x = pi/n, pi equals pi.sin x/x when x is very small.

It has been the endeavour of mathematician to find value of sin x/x as it contains the value of pi in it.

In Europe, first attempt to determine value of pi was made by François Viète*, a French mathematician. His work on new algebra was an important step towards modern algebra, due to its innovative use of letters as parameters in equations. He was a lawyer by trade, and served as a privy councillor to both Henry 3 and Henry 4 of France. He attempted equating polygon with circle by Geometry and finally gave a formula for pi in nested radicals. What he practically solved was determination of value sin x/x. Sin x is a trigonometric ratio of perpendicular to hypotenuse in a right angled triangle with angle x facing perpendicular.

He geometrically reached the formula for sin x as

 sin x = x.cos x/2.cos x/4. Cos x/8. Cos x/4…………………………………up to infinity with last term as 1.

When x equals pi/4, equation takes the form,

(½)^1/2 = pi. cos pi/8. cos pi/16. cos pi/32………………………

Where cos pi/8. cos pi/16. cos pi/32 are known from double angle formula,

 cos pi/8 = ½.(2+ 2^1/2)^1/2),

 cos pi/16 = ½{2+(2+ 2^1/2)^1/2}^1/2,

cos pi/32 = ½[2+ {2+(2+ 2^1/2)^1/2}^1/2]^1/2,


On putting these values,

1/pi = 2^1/2. ½.(2+ 2^1/2)^1/2). ½{2+(2+ 2^1/2)^1/2}^1/2. ½[2+ {2+(2+ 2^1/2)^1/2}^1/2]^1/2…….(2)

There came then most eminent mathematician of 18th century Leonhard Euler. He thought differently to determine the value of pi from sin x/x. He factorised sin x in infinite product.

 sin x/x = {1- (x/pi)^2}.{1- (x/2pi)^2}.{1- (x/3pi)^2}…………. up to infinity………………..(3)

Sin x can also be written in series expansion as x – x^3! + x^5!-…………. up to infinity.

Or sin x/x = 1- x^2/3! + x^3/5!-…………………………………………………….(4)

Comparing coefficient of x^2 of equations (3) and (4),

1/pi^2.[1/1 + ½^2 + 1/3^2+………]= 1/3!

Or pi^2= 6.[1/1 + ½^2 + 1/3^2+………]

 Another English mathematician John Wallis thought differently from Euler and Viete, he, in stead, straight way considered equation (3) of sin x/x and put x = pi/2. That gave the value of pi as

2/pi = (1-1/4).(1- 1/16).(1-1/36).(1-1/64)……………

Or pi = 2/1.4/3. 16/15. 36/35. 64/63……………..

Thus it can be summarised as equation of a circle in trigonometric ratios is pi = n.sin pi/n where n is number of sides of regular polygon. Also sin x/x where x = pi/n, when equals 1 then n tends to infinity. If it does not equal 1, it means, n has finite value and sin x/x has also value other than 1. Thus (sin x)/x is an equation that contains value of pi. Or in other words pi is determinable from (sin x)/x.

Different ways have been adopted by mathematician to calculate pi from (sin x)/x and that had given different formulae of pi. If an attempt is made to find (sin x)/x, new formula will originate. Is it not then worth trying to find some other way to express (sin x)/x?

I submit, mathematicians are not restricted to sin x divided by x to determine pi, statistical methods, probability theory and many more have already been introduced and pi freshly calculated. Still more methods will be put forth because mathematicians think differently.


1. Viète* For further reading, ‘When Law Met Mathematics’ at https://narinderkw.wordpress.com/2017/08/07/when-law-met-mathematics/

2. https://en.m.wikipedia.org/wiki/Carl_Friedrich_Gauss

3. http://felix.physics.sunysb.edu/~allen/Jokes/bohr.html

4. https://en.m.wikipedia.org/wiki/François_Viète

5. http://www2.mae.ufl.edu/~uhk/EULER-SINE.pdf

6. Figure courtesy https://www.quora.com/Find-the-perimeter-of-an-equilateral-triangle-of-side-l-cm-is-inscribed-in-a-circle-with-radius-r-cm

7 Cover photo courtesy artist Domenico Fetti, Italian Painter, current location Gemaldegalerie Alte Meister, Source/Photographer http://archimedes2.mpiwg-berlin.mpg.de/archimedes_templates/popup.htm 


About author:

Writer 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.      

It Is Surprising But True, Trigonometric Ratios Of Sine, Cosine Of An Angle Can Be Larger Than One

In basic plane trigonometry, trigonometric ratios of Sine or Cosine of an angle can have any value between -1 and + 1 and these can not be more than 1 or less than -1. To define sine of an angle, it is ratio of perpendicular to hypotenuse and cosine of an angle is ratio of base to hypotenuse in a right angled Triangle. From these definitions also and since hypotenuse is always larger than any of other side, these ratios are natural to be less than one. A number of problems have been solved by all assuming sin (n.x) and cos (n.x) to be equal to any value between -1 and + 1 when angle n.x tends to infinite with n approaching infinity. Assumption of (sin x)^n or (cos x)^n as approaching zero when these ratios are not equal to one with n tending to Infinity has always been made. All these assumptions are based on fact that magnitude of trigonometric ratios of sine and cosine are always less than or equal to one.

Are such assumptions then incorrect in view of the statement that these ratios can be larger than one? It is unmathematical to have answer in “yes,” and “no” to same question. This, therefore, needs analysis to reach logical result about the magnitude of these ratios. For that, it will be examined what deters to equate these trigonometric ratios to quantities larger than one.

Let us assume sin x equal to a quantity larger than one and let it be square root 5. Cos x can be calculated from identity (1- sin^2 x)^1/2 and that makes cos x equal to (-4)^1/2 or 2i where i is (-1)^1/2 and is called imaginary number. A question then arises whether imaginary quantity is practical and realisable. Simple answer to this query is “yes” if it is assumed that imagination is the result of real quantity. If a real quantity can give birth to imaginary then reverse should also be true and imagination should also give birth to reality.

Friedrich August Kekulé, a German chemist saw in his dream that snakes are entangled eating tails of each other and that led him to proclaim valency of carbon four and the structure of benzene. Apart from this, imaginary number raised to power imaginary number* ie i^i equals e^-pi/2 Here base is imaginary and power is also imaginary but the result is real and astonishing as it relates to pi which is the worlds most important mathematical constant.

Imaginary quantity can, therefore, not be dubbed as insignificant as it is analogous to operator 90 degree. That is it rotates the real quantity by 90 degree, for example 2i means, a quantity of magnitude 2 rotated by 90 degree, in other words, it is a quantity of magnitude 2 that lies on y axis. Similarly, i.i or -1 means rotation by 180 degree or – x axis,

i.i.i or -i means rotation by 270 degree or -y axis and

i.i.i.i or 1 means rotation by 360 degree or x axis.

This analogy of imaginary number is utilised in electrical engineering where current leads or lags according to capacitive or inductive nature of load and is represented by C.i or -C.i where C is the current flowing in the load. It can then be summed up, imaginary quantity is as important, significant and realisable as real quantity.

Coming to cos x = 2i calculated from Pythagorean equality, it is quite apparent that angle x must be complex so as to make cos x as imaginary quantity. A complex quantity is that which has real and imaginary part. If ‘a’ and ‘b’ are real then a + i.b is a complex quantity where a can be zero also. Let us assume angle 

x = a+ i.b, then

cos (a+ i.b) = 2.i

or cos a. cos i.b – sin a. sin i.b

= cos a.cosh b – i sin a.sinh b,

since cos ia = cosh a and sin ia = i.sinh b.


cos a.cosh b – i sin a.sinh b = 2.i.

Equating real and imaginary parts,

cos a.cosh b= 0,

i sin a.sinh b = 2i

or sin a.sinh b = 2.

That means a =(2k + 1). pi/2 or (2k- 1).pi/2.,Substituting this value,

1.sinh b = 2 or b = sinh inverse 2.

Sinh b can also be written as ½.(e^b – e^-b) = 2.

On putting e^b= y, we get

y – 1/y = 4 or y^2- 4y – 1= 0.

That gives

y = e^b = 2 + 5^1/2 or 2 – 5^1/2.

This can also be written as

y = 2Φ+ 1 or 3- 2Φ

where Φ is golden ratio equal to

½.(1+ 5^1/2).

From this, b can be written

= log (2 + 5^1/2) or log (2 – 5^1/2) or log (2Φ+ 1) or log (3- 2Φ)

Therefore complex angle whose cosine is 2i and sine is square root 5 is

= Pi/2 +i.log (2 + 5^1/2) or Pi/2 +i.log (2Φ+ 1).

On taking the principal value of a and positive value of b, we can also write

Sin [Pi/2 +i.log (2 + 5^1/2)] = 5^1/2,

Or sin [Pi/2 +i.log (2Φ+ 1)]= 5^1/2

Or cosh [log (2Φ+ 1)] = 2Φ-1,

cosh [log (2 + 5^1/2)] = 5^1/2,

Also sinh (log (2 + 5^1/2) = 2

or sinh [log (2Φ+ 1)] = 2.

From above analysis, we can say, cosine or sine of an angle can be larger than one provided the angle is complex containing imaginary part. Imaginary part is not an imagination but is realisable by way of cosh and sinh of imaginary angle as given below.

Cos ix = cosh x = ½.(e^x + e^-x),

sin ix = i.sinh x = ½.i.(e^x – e^-x)

and this leads to Pythagorean identities

cos^2 x + sin^2 x = cosh^2 x – sinh^2 x = 1.

Next question that arises is how to draw a complex or imaginary angle. In Plane Trigonometry, real angle can be drawn in x-y Plane and trigonometric ratios can be found out by drawing a right angled triangle in that plane. When an angle is drawn in a plane, it always has real value.That means, it can never be complex and resultantly cosine and sine of an angle as ratios of base to hypotenuse and perpendicular to hypotenuse can never be larger than one as hypotenuse in a Plane trigonometry is always larger than its sides. Therefore sine and cosine of an angle is always equal or less than one in plane trigonometry.

Coming to drawing of an imaginary angle, I submit, such a case must satisfy identity cosh^2 x – sinh^2 = 1 and in Plane trigonometry, imaginary angle would be such that the straight lines forming it, never join. When the lines do not join, it can never be defined as an angle. Therefore, in a Plane x-y where real angle can be drawn, imaginary angle can never be drawn.

That necessitates there should be another plane an if that Plane is considered perpendicular to the plane that contains real angle, then imaginary part of the angle can be drawn in that perpendicular plane. Let us see how. It is well known to those who have basic knowledge of plane trigonometry that as the angle increases the length rotates in a circle. Or we can say that if length starts from point (0,0) and ends at a point (a cos x, a sin x), then with increase in angle x, the line rotates in a circle of radius a in counter clockwise directin. The locus of the point in Plane trigonometry with change in angle is circle. Why locus is a circle can be proved from the identity

cos^2 x + sin^2 = 1

On multiplying both sides by a^2, we get

a^2.cos^2 x + a^2. Sin^2 x = a^2,

and that is an equation of circle. It is reiterated that when angle is real in plane trigonometry, a point can be defined as a point on a circle.

It is explicit from the explanation already given that when angle changes to imaginary angle say i.x then cosine of imaginary angle takes the form cos i x and that is equal to cosh x and cosh x is equal to ½(e^x + e^-x). When angle is imaginary, sine of imaginary angle takes the form sin ix and that is equal to i sinh x and i sinh x is equal to 1/2.i(e^x -e^-x). Therefore, it leads to identity,

cosh^2 x – sinh^2 x = 1.

On multiplying both sides by a^2, we get

a^2.cosh^2 x –a^2. sinh^2 x = a^2.

Let a^2.cosh^2 x be x and sinh^2 x = y, then locus of point x, y represents

x^2 – y^2 = 1

and that is an equation of hyperbola. That is why cosh x and sinh x are called hyperbolic functions.

It follows from this, change in imaginary angle drawn on perpendicular plane moves the point a.cosh x, a sinh x on hyperbola. We can, therefore, conclude that whenever there is trigonometric ratio and if that pertains to complex angle, real part of the angle will decide the length on the circumference of a circle in a plane and its imaginary part the length on the hyperbola in a plane perpendicular to the plane of real angle.

Based on this analysis let me answer the questions I raised at the start of this paper. Sine and cosine of a real angle that lies in a plane is always either equal to or less than one. It is true in plane trigonometry what is taught in schools. Length of the line rotates in a circle as the real angle is changed if we keep magnitude of length constant.

When the angle is complex (that is it has imaginary part also), sine and cosine of complex angle can be larger than one and in that case, imaginary part of the angle lies in a plane perpendicular to the in which real angle lies and the point moves upon hyperbola drawn in perpendicular plane with the change in imaginary angle. Finally, sine or cosine of an angle is Plane trigonometry is equal or less than one but if it pertains to complex angle containing imaginary part, it can be greater than one.




1. Title Image courtesy Kan8eDie at https://upload.wikimedia.org/wikipedia/commons/e/eb/Principle_branch_of_arg_on_Riemann_%28small%29.png

 2.Kekulé Dream by Michael Verderese

Professor Heinz D. Roth at https://web.chemdoodle.com/kekules-dream/

3.* Eternal Bonds Of Togetherness Between Natural Number, Pi, Tangent And Logarithm Of A Quantity at https://narinderkw.wordpress.com/2017/08/12/eternal-bonds-of-togetherness-between-natural-number-pi-tangent-and-logarithm-of-a-quantity/

4.Image of complex angle courtesy Richard Hammack at A Geometric View of Complex Trigonometric Functions http://www.people.vcu.edu/~rhammack/reprints/cmj210-217.pdf

About Author:

Writer 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.      



Eternal Bonds Of Togetherness Between Natural Number, Pi, Tangent And Logarithm Of A Quantity

Eternal bonds of love between the characters are not found only in plays of Shakespeare or literature but these are profound in mathematical functions. I am presenting this mathematical affinity amongst Natural number e, Mathematical constant pi, Tangent and logarithm of a quantity and a bit more. I will be using power series expansion, calculus and complex numbers. These mathematical functions also exhibit strong bonds for each other. I submit wherever there is a logarithm of a quantity, presence of natural number can not be denied. Wherever a natural number comes into being, mathematical constant pi appears from oblivion.Tangent of the quantity is also not far behind, it is hidden in the logarithm as the baby concealed in belly pouch of its mother Kangaroo. The efflorescence in which these quantities are born from each other urges me to state that there is some unseen eternal bonds between these. Cosmic cycles will revolve but there bonds not diminishing even by an iota.  


Before I proceed further let there be brief introduction. For tangent, I refer you to triangle ABC with right angle at corner B, base AB, perpendicular BC and hypotenuse CA that

makes an angle x radians at A. Ratio of perpendicular to base ie BC/AB is the tangent of angle x. 


                                        Graph Of Tan Inverse x and Cot inverse x.

If value of tangent of an angle is given, angle can be determined by taking inverse of tangent.
Natural number abbreviated as e* is defined as (1+ 1/n) multiplied with itself infinite times where n tends to infinity. 


Mathematically, it is written as  

e = (1 + 1/n)^n where n tends to infinity and sign ^ denotes raised to power. Numerically,
           e = 1+ 1 + ½! + 1/3! + ¼! + 1/5! +….up to Infinity.                                                                                      And this sums up as 2.71828.  

Image of circle showing Pi as ratio of circumference to diameter, Courtesy author Klonjee.

Pi**abbreviated as π is defined as ratio of circumference of a circle to its diameter. Numerically, it equals 3.14159.  

Logarithm*** of a quantity is that power of natural number which equals the quantity. If the quantity is e^x then natural logarithm is x. 

Graph of function of logarithm courtesy Krisnavedala

Theory And Proof

Having introduced these quantities, I take up function

F(x) = 1/(1+ x^2)

      = (1+ x^2)^-1.

Above function can be expanded as binomial expansion,

F(x) = (1+ x^2)^-1 = 1- x^2 + x^4 – x^6 + x^8- …………..

On integrating above function with respect to x, we get

tan inverse x =c + x – (x^3)/3 + (x^5)/5– (x^7)+ x^8- ………up to Infinity

where c is a constant of proportionality.

If x is 0, then tan inverse is either zero or is integral multiple of pi and is written as k.pi. On putting x equal to zero, right hand side of above equation becomes c.

That is c equals k.pi where k is any number 0,1,2, 3 ……. Substituting this value of c in above equation, we get

tan inverse x = k.pi + x – (x^3)/3 + (x^5)/5– (x^7)+ x^8- …………..

Taking principal value by putting k equal to zero, value of c also equals 0. The equation then can be rewritten as

tan inverse x = x – (x^3)/3 + (x^5)/5– (x^7)+ x^8- ….up to Infinity                                                      (1)

Now we take another function and will try to make it equal to tan inverse x expansion. Let this be

f(x) = 1/(1- x^2).

It is obvious from this function that when x would be substituted by i x this function will be indistinguishable from 1/(1+ x^2) which is derivative of tan inverse x. On integration with respect to x, this will result in tan inverse x and 1/(1- x^2) on integration will yield logarithmic function in x. These on substitution with x as i x will have close relations.

Based on this theory, let us decompose this function into partial fraction as

f(x) = 1/(1- x^2) = 1/(1+ x).(1-x) = A/(1+ x) + B/(1- x).

Or 1 = B.(1+ x) + A.(1- x).

On comparing constant term and coefficient of x of left hand side LHS with right hand side RHS, we get

A = ½ = B.

Then fx) = 1/(1- x^2) = (1/2)/(1+ x) + (1/2)/(1- x).

However 1/(1- x^2) can be expanded according to Binomial Theorem as

 1/(1- x^2) = 1+ x^2 + x^4 + x^6 + x^8- …………..up to infinity.


F(x) = 1/(1- x^2) = (1/2)/(1+ x) + (1/2)/(1- x) =1+ x^2 + x^4 +x^6 + x^8+ …………..up to Infinity.

On integrating with respect to x, we get

½. log (1+ x) – ½.log (1-x) = c1+ x + (x^3)/3 + (x^5)/5 +(x^7)/7 + (x^9)/9+ …………..up to infinity

where c1 is constant of proportionality.

At x = 0, 0 = c1.


½. log (1+ x) – ½.log (1-x) = x + (x^3)/3 + (x^5)/5 +(x^7)/7 + (x^9)/9+ …………..up to infinity.

Or ½.log (1+x)/(1-x) = x + (x^3)/3 + (x^5)/5 +(x^7)/7 + (x^9)/9+ ………….. up to infinity                    (2)

On putting x as i x in equation (1), it transforms to

tan inverse ix = ix + i(x^3)/3 + i(x^5)/5+ i(x^7)+ ix^8- ….up to Infinity.

Or (1/i) tan inverse ix = x + (x^3)/3 + (x^5)/5+ (x^7)+ x^8- ….up to Infinity……………………………………………………….                                                                                  (2/1)

This equation is same as equation (2), therefore equating these two we get,

1/i. tan inverse ix = ½.log (1+x)/(1-x) or

tan inverse ix = ½.i.log (1+x)/(1-x)                                                                                                            (3)

Let (1+x)/(1-x) = z, then x = (z-1)/(z+1) and

½.i log z = tan inverse i.(z- 1)/(1+z)                                                                                                            (4)

It is explicit from above, wherever the term logarithm of a quantity comes, tan inverse is a part and parcel of it and is given by above equations. Logarithm as it was defined earlier denotes the power of natural number e that makes it equal to given quantity and for tangent, that power points to an angle the tangent makes so as to equal the quantity. It is submitted, in other words, logarithm of a quantity is joint to the angle that the tangent makes, by an unbreakable bond. The bond is stated in relations as defined by equations (3) and (4).

If x is substituted by – i in equation (3), it takes the form

tan inverse 1 = ½. i.log (1-i)/(1+ i) or

k pi + pi/4 = ½. i.log {(1-i)(1+i)}/{1+ i)(1+i)}

                  = ½. i.log 2/2i = ½. i.log1/i = -1/2.i log i where i is (-1)^1/2 and k is any number 0, 1, 2, 3………….

Taking principal value by putting k = 0,

pi/4 = – ½.i log i or

– pi/2 = i. log i                                                                                                                                       (5)

That means logarithm of pure imaginary number multiplied by itself that is, pure imaginary number always equals minus half pi. Imaginary number i that equals (-1)^1/2 is impractical and unachievable but when its logarithm is multiplied with imaginary number, gives result as minus half pi. This again is un breakable bond between pi and logarithm of imaginary number.

If we combine equations (3) with (5), it transforms to

tan inverse ix – pi/4 = 1/2.i log (1+x)/(1-x) + ½ i log i

                                = ½.i log i.(1+x)/(1- x).                                                                                            (6)

Again this is an equation where logarithm, tangent, pi and natural number are all all connected with each other.

We revert to equation (5),

-pi/2 = log i^i

 e^-pi/2 = i^i                                                                                                                                                 (7).

This is another equation which joins natural number e with mathematical constant pi through imaginary number iota i.

Again coming to equation (3), tan inverse ix = ½.i.log (1+x)/(1-x), by rearranging I can write this as

Tan [½.i.log (1+x)/(1-x)] = i x.

From this, I can get

 sec [½.i.log (1+x)/(1-x)] = [1+ tan^2 {1/2i.log (1+x)/(1-x)}]^1/2

                                     = (1- x^2)^1/2.

By its reciprocal, value of cosine can be found as

Cos [½.i.log (1+x)/(1-x)] = (1- x^2)^-1/2.

This can also be written as

Cosh [½.log (1+x)/(1-x)] = (1- x^2)^-1/2.

Similarly value of sinh can also be found out as

Sinh ½.i.log (1+x)/(1-x)] = x/(1- x^2)^1/2

If Cosh [½.log (1+x)/(1-x)] = (1- x^2)^-1/2 is integrated with variable x from range 0 to 1, it equals sin inverse x with range 0 to 1 and that equals pi/4. Or

Integral Cosh [½.log (1+x)/(1-x)] range 0 to 1 = integral (1- x^2)^-1/2 range 0 to 1 = sin inverse x range 0 to 1 = pi/2. Logarithm of (1+x)/(1-x) is also connected by equation

Integral Cosh [½.log (1+x)/(1-x)] range 0 to 1 = pi/2


Natural Number, Pi, Tangent And Logarithm Of A Quantity are all connected with each other as has been proved from equations,

tan inverse ix = ½.i.log (1+x)/(1- x)                                                                                              (3)

½.i log z = tan inverse i.(z- 1)/(1+z)                                                                                               (4)

k pi + pi/4 = -1/2.i log i

where i is (-1)^1/2 and k is any number 0, 1, 2, 3………….

and for principal value on putting k = 0,

– pi/2 = i. log i                                                                                                                                        (5)

tan inverse ix – pi/4= ½.i log i.(1+x)/(1- x)                                                                                       (6).

e^-pi/2 = i^I,                                                                                                                                           (7)

sin [½.i.log (1+x)/(1-x)] = (1- x^2)^-1/2,

cos ½.i.log (1+x)/(1-x)] = x/(1- x^2)^1/2 and

cosh ½.log (1+x)/(1-x)] = x/(1- x^2)^1/2.

These bonds are not affected by physical matters as unfortunately human bonds are. Their relationships are intrinsic present and will last till eternity. Nothing in the universe can break these.

Keywords: Natural Number, Logarithm, Tangent, Pi, Eternal Bond, Binomial Expansion, Integration, Imaginary Number Iota, Tangent Inverse, Complex Number.


1) Graph of tan Inverse x courtesy Geek3 at https://en.m.wikipedia.org/wiki/Inverse_trigonometric_functions

2) Graph of natural number courtesy http://www.mathsisfun.com/numbers/e-eulers-number.html

3) Image of circle showing Pi as ratio of circumference to diameter by author Kjoonlee at https://en.m.wikipedia.org/wiki/Pi

4) Graph of function of logarithm courtesy Krisnavedala at https://en.m.wikipedia.org/wiki/Logarithm

5) * Natural number details can be read at https://narinderkw.wordpress.com/2017/07/05/natural-number-is-not-much-natural/

6) ** Pi details can be read at https://narinderkw.wordpress.com/2017/04/10/pi-is-sweeter-than-pie/

7) *** Logarithm details can be read at https://narinderkw.wordpress.com/2017/06/16/goodbye-to-log-table-and-calculator-determining-logarithm-is-easy-now/

8. Title Image English: Sunrise over the bay, Little Gasparilla Island, Florida, Source courtesy  Author Mmacbeth at https://commons.m.wikimedia.org/wiki/File:Little_Gasparilla_sunrise.jpg#mw-jump-to-license 


About Author

Writer 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.      





























































When Law Met Mathematics

He was a lawyer, a qualified lawyer and had studied law at Poitiers, graduating in 1559. He began his career as an attorney at a quite high level, with cases involving the widow of King Francis I of France and also Mary, Queen of Scots. He generally pleaded important cases of royal families and won also. He was always logical in his dealings and practice. He argued his cases on logics and never hid any facts irrespective these went against him.

But law was not his cup of tea. He was always curious solving mathematical problems. What makes a circle round, a rectangle cornered and a line straight? Can a polygon be a circle? He used to ponder over such problems sitting on his desk with elbow on the study table, taking food with other hand and constantly concentrating on the same problem even for as many as three days. He was François Viète of France, a Lawyer and Mathematician, a combination that is uncommon and he was uncommon.

Nonetheless, he was highly successful in law. By 1590 he was working for King Henry IV. The king admired his mathematical talents, and Viète soon confirmed his worth by cracking a Spanish cipher, thus allowing the French to read all the Spanish communications they were able to obtain.

He did not abandon his passion for mathematics and In 1591, François Viète came out with an important book, introducing what is called the new algebra: a symbolic method for dealing with polynomial equations.

How I came to know of this great Mathematician, is also worth telling. A few months back, it was Pie (written as π) day and there was a great hustle and bustle and magazines full of articles on Pi. This inspired me though late to contribute something to Pi which is considered the great mathematical constant.

For a person like me who had studied mathematics from the point of view of its applicability to Engineering, writing on Pi by me with contents noticeable among mathematics fraternity was analogous to passing post graduation by a school primer. With passing of each day, celebrations of Pi were dimming but my determination remained bright. Specially during morning walk when neurones were upbeat, different description of Pi focussed on my mental screen and finally I took this shining and glassy device in my hand and typed my notions about pie with title,”Pi Is Sweeter Than Pie.”

It was my whole hearted effort aimed at revealing something new which was not known earlier about Pi.I had already gone through various description evaluating Pi by means of infinite series and infinite products. Why I can’t give something alike of these, was the voice of my inner self.

Writing sine of an angle, I started sub dividing it infinitely into sines and cosines of an angle. Equating sin x as x where x diminished to very small quantity, Lo, there appeared an identity not seen by me. And I substituted x as Pi/2 and there appeared the value of Pi in nested form.

Immediately, I checked the identity on google search engine, yes it was correct, no one had done it the way I did. But still there was a lurking doubt in my mind, if I could do, others would have also done it. It were Eureka moments till my further search led me to Viete Formula for Pi. I looked at it intently, final formula of Viete had more or less the resemblance with my formula but his method was altogether different. There was a geometrical drawing, certain perpendiculars, triangles, everything was different but the result had the resemblance. Derivation done by me was based upon the trigonometry and calculus whereas great Mathematician Viete has derived it from geometry. This assuaged my feelings a bit, after all my approach was independent unguided by any help. An idea had cropped up in my mind and that led me to this derivation. “There are different ways to reach a destination and journey by each route had its own pleasures. Those who discover new ways, enjoys the beauty of virgin paths,” said I to myself.

Lifting my morales, I completed, “Pie is sweeter than pie,” which have many formulae to ponder and there one can find a bit of mathematician in oneself. Later on I found, it indeed proved sweeter than pie to many readers to whom I am indebted for showing interest in my works. Now whenever, I attempt any mathematical derivation, similarity of the result with Viete formula keeps me on tenter hooks. I am writing a paper on golden ratio and there too appears a nested function (similar to Viete Pi formula) to determine value of  φ^1/k where k = 2^n, φ = (1+ 5^1/2)/2 and n is 0, 1, 2, 3, ……. any number. But great Mathematician Viete has not done, according to me, work on golden ratio  φ and I feel secure.

Besides this, I find, attempting a mathematical derivation without having any knowledge of previous works in that field, gives one great satisfaction on knowing later on that ones approach was unique and undefiled. I am benefitted in this regard, there is an ocean of mathematics to attempt and derive independently. Do you endorse, books bias ones thinking? Whatsoever your answer may be, I am certain, it is one of the most difficult question to answer.

All said and done, Francoise Viete will remain great Mathematician in the annals of history. He was the first one from Europe to give formula in nested radicals of Pi to the world.

(Viete Beautiful Formula For Pi, see only integer 2 is used.)

If you happen to see this formula, it uses only one integer 2 to calculate Pi. One wonders if 2 was sufficient to evaluate Pi, what are then other integers meant for? Such great men will always be remembered like Benjamin Jonson said in poem, “In Short Measure Life May Perfect Be.” It is virtue that never dies, it is like a wood that is seasoned to last for ever.


1. Image Pi courtesy Pi visualized. Saw it over at /r/dataisbeautiful and had to make it. – View on Imgur: https://m.imgur.com/r/wallpapers/ZCUW7js

2. Pi and Golden ratio by John Baez at https://johncarlosbaez.wordpress.com/2017/03/07/pi-and-the-golden-ratio/amp/

3. “Pi Is Sweeter Than Pie,” at https://narinderkw.wordpress.com/2017/04/10/pi-is-sweeter-than-pie/

4. Viete’s Formula for Pi at World Of Pi http://www.pi314.net/eng/viete.php


About Author

Writer 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.