Friday, December 30, 2016


I fear.
I fear the fear.
Fear - a spanning feeling of this year.
I fear to remember this year.
Fear of power.
I fear my power
For the fear of losing it.
I fear the powerless
For the fear of them gaining it.
From powerlessness I powered to power.
I used this. I used that.
I stamped over this. I tampered with that.
I fear all those whom and which I used, stamped over and tampered with.
I fear the existence of all those.
I am  in a spree to stop all those gaps.
The gaps I used to rise to the power.
The gaps I used to wrest the power.
Gaps are gaps.
Gaps are void.
They are black.
I am on a mission to fill the gaps,
Hoping to turn the black to the white.
I need to stop the gaps to drive my fear away.
But, is the opposite of black, white?
Is the opposite of white, black?
Woe to me!
The trap of the binary, I fell into.
The power of fear will grapple me.
The fear of power will topple me.
O man, count the moments
As the lies and lines will fade away.
"Put up thy sword into its place,
For all they that use the fear shall perish with the fear."
Thus spake, the lord of fearlessness.

The Certain Uncertainties of Life

Is life binary?
Between the known and unknown,
Do we restrict ourselves?
Is the known fully known?
Are all the unknowns completely unknowns?
Is life struggling between Yes and No?
The passage from the unknown to the known,
Is a passage from the known to the unknown.
The more we know, the more we know not.
Yes is a No and No is an Yes.
Is life n-ary?
I doubt, doubt and doubt the doubt.
Life is a sequence of the knowns and the unknowns,
And the unknowns and the knowns,
And the knowns and the knowns,
Partial and partially partial,
Separated by commas, exclamations, question marks and hyphens.
Of course, within the unavoidable ruthless margins, it blossoms.
Am I missing something?
Am I avoiding something?
Yes, I must accept the stops,
They give meaning to the commas.
Right, the margins and full-stops,
The frames and periods,
Protect us from the known unknowns and the unknown unknowns.
It is life.
Life is continuous, yet discrete.
It is discretely continuous.
It is continuously discrete.
Hang on…
No place for any conundrums.
Bang on, it is sagaciously!

Thursday, December 29, 2016

Love of Light

I love light.
To be in the light
Not in the limelight
Is my might.
I love fight
To be in the light.
For that I hate
To be in the limelight.
Light is bright,
Transparent and fast.
That's why I hate
What is not light.

Wednesday, December 28, 2016

Feast of Denha

Denha is the second great Feast after Easter that was being celebrated from early centuries in the Universal Church. Denha which means Sunrise or Dawn is celebrated to commemorate the Baptism of Our Lord and through that His public manifestation to the world by God the Father as His Beloved Son (Mt.3:17: Mk. 1:11; Lk. 3:22). The Holy Spirit descended upon Jesus in the form of a dove (Mt.3:16: Mk.1:10: Lk.3:22; Jn. 1:32). Thus the three Persons of the Holy Trinity also were manifested on that occasion. Therefore in the early Church the Holy Trinity also was commemorated on this day. 

In the Nazrani Church, this feast is celebrated in two distinct forms viz., Rakkuli Perunnal and Pindi Perunnal. To commemorate the Baptism of Our Lord in the River Jordan, the Faithful at Palai and Pulinkunnu used to take a holy dip by night (Rakkuli) in the streams flowing near the churches in these localities after the solemn celebration of Ramsha, the Evening Liturgy. Faithful in and around Thrisur light small oil lamps and pin them around plantain trunks (Pindi) erected in churchyards or in front of the houses and go around them chanting in Syriac El payya. These Syriac words mean God is radiant. The thought underlying the lighting of lamps and the chanting is that God sheds His light over the human race through Mishiha (Lk.1:76-79). We may also allude here to the tradition in the Syriac Church which says that when Jesus was baptized in River Jordan, a great light shone in that River. This also might be linked with the introduction of lamps into this festive ceremony. In recent years this Pindi Perunnal ceremony is celebrated in the southern regions also.

Friday, December 16, 2016

Peace to the World

You train gun,
We drain fun.
You train fun,
We drain gun.
You drain gun,
We train fun.
You drain fun,
We rain fun.
Gun's no fun
Have gun of fun
Let's go to the man-ger
Peace be with us

Tuesday, December 06, 2016

The Prodigy that was Shakuntala Devi
Shakuntala Devi was the best known face of popular Mathematics in India. She was renowned for her mindboggling calculations.  Indian Mathematics was synonymous with two names late Srinivasa Ramanujan and the other being Shakuntala Devi herself.  While Ramanujan is associated with the pre-independence period, Shakuntala Devi flowered in the post-independence India.
Ramanujan was famous for his longstanding contribution to the modern Theory of Numbers, including the methodology to estimate the accurate value of Pi. On the other hand, Shakuntala Devi, who lacked formal education, attempted to eradicate ‘Arithmophobhia’ or the fear of numbers among the masses.  Her public performances sought to convey the message that Mathematics is fun and not necessarily esoteric.

When Shakuntala Devi established herself through the 1960s and 1970s the television era did not exist   in the country. The only medium available to her was public performances at schools, colleges or clubs.      Shakuntala Devi was inherently gifted with the power of calculation that started as a child through association with her father who performed card tricks. She gradually mastered the art of Mind Dynamics to astonish audiences worldwide with her almost instantaneous calculations of complex Mathematical operations. The lady was not a qualified classical Mathematician in the true sense of the term, considering she did not focus on development of Mathematical theories like CR Rao or SR Varadhan. Instead she concentrated solely on her amazing ability to calculate with super speed.

On 18 June 1980 at the Imperial College of London, Shakuntala Devi outdid the then fastest computer to multiply two thirteen digit numbers with each other. This earned her a coveted place in the Guinness Book of World Records.    It would be relevant to mention that the techniques that Shakuntala Devi used to wow the world with her superhuman skills were gleaned from the Atharva Veda.  These included:  ‘modular ‘arithmetic to figure out the day of any given date to find the nth root of any integer power; besides, movements of celestial bodies aimed to generate astrological predictions. Shakuntala Devi through her public performances was able to restore some of the glory of Vedic Mathematics.
The fact that Shakuntala Devi did not go to school never deterred her ability to grasp the nuances of numbers along with their permutations and combinations. Her father who was born in to a family of traditional priests had the courage of his conviction to pursue his passion as a vocation and became a circus artiste. This resulted in financial instability which hampered Shakuntala Devi’s formal schooling. Despite this disadvantage, she authored a dozen odd books on Arithmetic skills and Mathematical puzzles which are well received across all age groups and even beyond Indian shores.  
Clearly her life suggests that learning is as important as teaching.  Otherwise the lady applied her mathematical capabilities to Astrology in an era when there were no computers. This helped her become a celebrity astrologer and earn some wealth.

Clearly Shakuntala Devi possessed a God- given gift that coupled with her circumstances nurtured her capabilities to calculate to flourish over time. To that extent, she was born with mathematical abilities and lends credence to the thinking that Mathematicians are born and not made. Interestingly Shakuntala Devi’s birth and death, from a mathematical viewpoint have mystical significance. She was born in November 1929 and died in April 2013 aged 83 years. To a Mathematician these numbers 19, 29, 2, 13 and 83 are all prime numbers --- the building blocks of Counting Numbers.      
Shakuntala Devi was a unique individual and India’s gift to the world of Mathematics. Her life serves as an inspiration to one and all. It sends the message of the importance to identify one’s hidden talents and thereafter nurture them irrespective of one’s circumstances or grooming.

Sunday, November 20, 2016

The Pain of Trekking

Rock on the rock on the rock is a hill
Uphill is to climb on the rock on the hill
On the top of the hill, at our feet are all
Hilarious, jubilant, exulting, we're above all

Tip of the top of the hill is not all
Lonely, the mind isn't willing to be that tall
Descend, descend and descend, we're with all
Hilarious, jubilant, exulting, all at our fall

On the plains, back from the hill, we're in pain
First on the feet, up on the knee, up is the pain
On the hip, on the rib, in the chest, we're in pain
All joints, all nerves, all cells, it's all pain

No problem, it's temporary, we'll be alright
What's the problem? Is climbing not right?
It's not a habit. That's the problem.
Hardwork as a habit, is never a problem.

Be true trekkers, mountaineers in real life.

Monday, November 14, 2016

Mathematics of the Blind

How does a blind know the other person is blind or not? One of the toughest explorations of humankind is the exploration for the truth. Human search for food ended with finding food. In some way, it is parallel to what other living beings also do. Hunger for food brings human beings, animal and plants on the same platform. They work towards getting good, sometimes good food. As a rational animal, human beings went further ahead. They searched for the truth behind it. They found that there is much more in the truth.

Among the three types of living beings, human beings and animals are superior to plants because of their capacity to change their positions. They find better positions, in the face of threats and sometimes for comfort. Hence we may say that plants live in the first dimension.  Human beings are superior to animals because of their capacity to change their positions in thinking. Though animals think, their thinking is a very static thinking. Hence, we may say that animals live in the second dimension. They live according their instinct. Human beings live in the third dimension. They have life. They can move. They can remember the past and dream about the future. They can imagine.

Among the many capacities human beings have, the most important one is the capacity to reason. Reason is the reason for human beings to search for truth. Human beings, in general, are in an incessant search for truth. Some search for the truth of life. Some others search for the truth of the nature. There are some others who search for the truth of the truth itself. This search for truth is the basis of recording and documentation. Documentation is the backbone of all the developments in the world. As human beings change their styles and modes of living, a continuous progress is visible in the world. It is by encountering and polishing the truths of the nature, the truths of the truths, human beings progress towards better life.  In this pursuit of progress, the most important and precise recording is in the form of Mathematics.

Many say, “Mathematics is the study of anything that has reason.” Mathematics is about recognizing patterns in the world and contributing suggestions for its repetition and improving.  In establishing the claims of what is right and what is good, mathematicians use a technique called “Proof.” In fact, everyone with rationality does proving, in many different ways. Hence, in general everyone is a mathematician. However, academic mathematicians do more documented proving. There are various methods of Proof.

One without the capacity to see things using one’s own eyes is termed as a blind. Of course, blindness is not a hindrance to reasoning. Hence, a blind person also does proving. But, can a blind person use a visual proof? To be precise, we come back to the initial question, “how does a blind know the other person is blind or not?” One can always bluff the blind by pretending to be blind or not. Here comes a beautiful mathematical proof method, named as “Zero Knowledge Proof.” It is illustrated in the case a blind person (B) who tests the blindness of another (A).

B has two identical marbles of different colours.
B knows that the marbles are of different colours. However, B cannot distinguish them by their colours. B holds the marbles in either of the palms. B asks A to identify the colours of the marbles. There is a possibility that A might come with the two colours correctly. But, B is still not sure if A is truthful. Hence, B keeps swapping the marbles quickly for some time between the palms with the perfect knowledge of the movement of the marbles. If A is blind, A has no clue whether the marble is exchanged or not. B then asks A to identify the location of the different colours. This procedure is repeated a couple of times more. If A is not blind, then A can never deviate from the initial identification of colours of the marbles.

In modern cryptography, Zero Knowledge Proof is efficiently utilized. It was initially conceived and published by the trio Shafi Goldwasser, Silvio Micali and Charles Rackoff.* The power of Zero Knowledge Proof is that the prover never passes the proof but a verifier can always test the prover of the knowledge of the proof and be satisfied.

*S. Goldwasser et al. “The knowledge complexity of interactive proof systems,” SIAM J. Comput. vol. 18, no. 1, pp. 186-208, Feb. 1989.

Thursday, November 10, 2016

A priceless surprise

O my beloved ones!
You pinned me
To a night-less day 
To a sleepless night
With such a thoughtful gift
Which is the second best gift
In the last eighteen months.
No wonder, the first,
O my beloved ones,
Who else?
What else?
Other than you,

On the day after, 
When paper money became just paper
When a dump triumphed over the wise
You pinned me 
To priceless papers of wisdom
Which no one can ever demonetise
To countless numbers on papers
Which no dump can ever triumph over...

With the fullest sense of its humblest
Exhibit ever possible,
Permit me to use the word, THANKS.

Love you all...

Wednesday, November 09, 2016

Thrown Thorny Throne

A forty thorny fence
An effort of defence
By the thorny thorns
Of the thrown thrones
On my deadly head
By the dread wicked
Bid them good bye
Brought me good sigh

Friday, October 21, 2016

Writing or Cooking?

May ye cook verse
Hey, ye look emerse
Verses cook ye to skies
Pulses hook ye to pulses

'Fort'y too

How do I thank thee...
Always I bank on thee...
Even when I'm prank n absentee
Ye rank me Lo, foree

Sunday, September 11, 2016

Mathematics for Everyone

Mathematics is everywhere. Hence, it  is for everyone. Mathematical concepts, simple to complex,  are explained by great mathematicians and writers of the recent centuries in popular formats. Many of these monographs are enjoyable by even those who do not have any previous knowledge in Mathematics or science. They are addictive! You start reading, you stop reading! This collection is dedicated to all my students who helped me improve my interest in Mathematics. Read and Enjoy!

  1. A Beautiful Mind: The Life of Mathematical Genius and Nobel Laureate John Nash by Sylvia Nasar
  2. A Beautiful Question: Finding Nature's Deep Design by Frank Wilczek
  3. A Certain Ambiguity - A Mathematical Novel by Gaurav Suri and Hartosh Singh Bal
  4. A Curious History of Mathematics by Joel Levy
  5. A History of π by Petr Beckmann
  6. A Mathematician Plays the Market by John Allen Paulos
  7. A Mathematician's Apology by C. P. Snow and G. H. Hardy
  8. A Mind for Numbers: How to Excel at Math and Science by Barbara Oakley
  9. A New Kind of Science by Stephen Wolfram
  10. A Strange Wilderness: The Lives of the Great Mathematicians by Amir D. Aczel
  11. Adventures of a Mathematician by Stanislaw M. Ulam
  12. Alan Turing: The Enigma by Andrew Hodges
  13. Alex Through the Looking-Glass: How Life Reflects Numbers and Numbers Reflect Life by Alex Bellos
  14. Alex’s Adventures in Numberland by Alex Bellos
  15. Algorithms to Live By: The Computer Science of Human Decisions by Brian Christian
  16. An Imaginary Tale: The Story of the Square Root of Minus One by Paul J. Nahin
  17. Another Fine Math You've Got Me into by  Ian Stewart
  18. Birth of Theorem by Cedric Villaini
  19. Book of Proof by Richard Heath Hammack
  20. Borges and Mathematics by Guillermo Martinez
  21. Brilliant Blunders: From Darwin to Einstein - Colossal Mistakes by Great Scientists That Changed Our Mario Livio
  22. Descartes' Secret Notebook: A True Tale of Mathematics, Mysticism, and the Quest to Understand the Universe by Amir D. Aczel
  23. Dr Euler's Fabulous Formula: Cures Many Mathematical Ills by Paul J. Nahin
  24. Duel at Dawn: Heroes, Martyrs, and the Rise of Modern Mathematics by Amir R. Alexander
  25. e: the Story of a Number by Eli Maor
  26. Elements of Mathematics: From Euclid to Gödel by John Stillwell
  27. Euclid's Window: The Story of Geometry from Parallel Lines to Hyperspace by Leonard Mlodinow
  28. Euler: The Master of Us All by William Dunham
  29. Fermat's Enigma: The Epic Quest to Solve the World's Greatest Mathematical Problem by Simon Singh
  30. Fermat's last theorem by Amir D. Aczel
  31. Fermat's Last Theorem by Simon Singh
  32. Finding Moonshine: A Mathematician's Journey Through Symmetry by Marcus Du Sautoy
  33. Finding Zero: A Mathematician's Odyssey to Uncover the Origins of Numbers by Amir D. Aczel
  34. Flatland by Edwin A. Abbott
  35. Flatterland: Like Flatland Only More So by Ian Stewart
  36. Fractals, Googols and Other Mathematical Tales by Theoni Pappas
  37. From One to Zero by Georges Ifrah
  38. From 0 to Infinity in 26 Centuries: The Extraordinary Story of Maths by Chris Waring
  39. Further Adventures of Penrose the Mathematical Cat by Theoni Pappas
  40. Gamma: Exploring Euler's Constant by Julian Havil
  41. Genius at Play: The Curious Mind of John Horton Conway by Siobhan Roberts
  42. God Created the Integers: The Mathematical Breakthroughs That Changed History by Stephen Hawking
  43. Godel, Escher, Bach: An Eternal Golden Braid by Douglas R. Hofstadter
  44. God's Equation by Amir D. Aczel
  45. Gösta Mittag-Leffler: A Man of Conviction by Arild Stubhaug
  46. Harmonograph: A Visual Guide to the Mathematics of Music by Anthony Ashton
  47. Henri Poincaré: A Scientific Biography by Jeremy Gray
  48. Here's Looking at Euclid by Alex Bellos
  49. Hidden In Plain Sight 6: Why Three Dimensions? by Andrew Thomas
  50. How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics by William Byers
  51. How Not to be Wrong by Jordan Ellenberg
  52. How to Bake Pi Eugenia Cheng
  53. How to Prove It: A Structured Approach by Daniel J. Velleman
  54. How to Solve it: A New Aspect of Mathematical Method by George Polya
  55. How to Think Like a Mathematician: A Companion to Undergraduate Mathematics by Kevin Houston 
  56. I Want to Be a Mathematician: An Automathography by Paul R. Halmos
  57. Idea Makers: Personal Perspectives on the Lives & Ideas of Some Notable People by Stephen Wolfram
  58. Imagining Numbers by Barry Mazur
  59. In Pursuit of the Unknown: 17 Equations That Changed the World by Ian Stewart
  60. Infinity: Beyond the Beyond the Beyond by Hugh Gray Lieber, Lillian R. Lieber and Barry Mazur
  61. Introduction to Mathematical Thinking by Keith Devlin
  62. Is God a Mathematician? and other musings by Narendra Murty
  63. Is God a Mathematician? by Benjamin Franklin
  64. Is God a Mathematician? by Mario Livio
  65. Journey through Genius: The Great Theorems of Mathematics by William Dunham
  66. Kerala Mathematics history and its possible transmission to Europe by George Gheverghese Joseph
  67. Language of Mathematics by Keith Devlin
  68. Life's Other Secret: The New Mathematics of the Living World by Ian Stewart
  69. Locks, Mahabharata and Mathematics: An Exploration of Unexpected Parallels by V Raghunathan
  70. Love and Math: The Heart of Hidden Reality by Edward Frenkel
  71. Loving and Hating Mathematics: Challenging the Myths of Mathematical Life by Reuben Hersh and Vera John-Steiner
  72. Make Your Own Neural Network by Tariq Rashid
  73. Making Number Talks Matter: Developing Mathematical Practices and Deepening Understanding by Cathy Humphreys and Ruth Parker
  74. Math Concepts Everyone Should Know (And Can Learn) by Metin Bektas
  75. Math Stuff by Theoni Pappas
  76. Math Talk: Selected Essays by Theoni Pappas
  77. Math, Better Explained: Learn to Unlock Your Math Intuition by Kalid Azad
  78. Math-A-Day by Theoni Pappas
  79. Mathematical Circles by Fomin Et Al
  80. Mathematical Footprints: Discovering Mathematics Everywhere by Theoni Pappas
  81. Mathematical Mindsets: Unleashing Students′ Potential through Creative Math, Inspiring Messages and Innovative Teaching by Jo Boaler
  82. Mathematical Mysteries: The Beauty and Magic of Numbers by Calvin C. Clawson
  83. Mathematical Snippets: Exploring Mathematical Ideas in Small Bites by Theoni Pappas
  84. Mathematical Sorcery by Calvin C. Clawson
  85. Mathematicians are People Too: Vol 1: Stories from the Lives of Great Mathematicians by Luetta Reimer
  86. Mathematicians are People Too: Vol 2: Stories from the Lives of Great Mathematicians by Luetta Reimer
  87. Mathematics and its history by John Stillwell
  88. Mathematics and the Imagination by Edward Kasner, James Roy Newman
  89. Mathematics for the Nonmathematician by Morris Kline
  90. Mathematics, Magic and Mystery by Martin Gardner
  91. Mathematics, Poetry and Beauty by Ron Aharoni
  92. Mathematics: An Illustrated History of Numbers by Tom Jackson
  93. Mathematics: From the Birth of Numbers by Jan Gullberg
  94. Mathematics: Its Content, Methods and Meaning by A. D. Aleksandrov, A. N. Kolmogorov  and M. A. Lavrent'ev
  95. Mathletics by Wayne L. Winston
  96. Men of Mathematics by E T Bell
  97. My Brain Is Open: The Mathematical Journeys of Paul Erdos by Bruce Schechter
  98. My Search for Ramanujan: How I Learned to Count by Ken Ono and Amir D. Aczel
  99. Nagel & Newman: Godel's Proof Special by Ernest Nagel and James R. Newman
  100. Niels Henrik Abel and his Times: Called Too Soon by Flames Afar by Arild Stubhaug
  101. Nine Algorithms That Changed the Future: The Ingenious Ideas That Drive Today's Computers by John MacCormick
  102. Number Mysteries by Marcus Du Sautoy
  103. Number and Numbers by Alain Badz
  104. Numbers: The Universal Language by Denis Guedj and Lory Frankel
  105. Of Men & Numbers: The Story of the Great Mathematicians by Jane Muir
  106. On Numbers and Games by John H. Conway
  107. Our Mathematical Universe by Max Tegmark
  108. Prime Numbers and the Riemann Hypothesis by Barry Mazur and William Stein
  109. Prime Obsession: Berhhard Riemann and the Greatest Unsolved Problem in Mathematics by John Derryshire
  110. Professor Stewart's Cabinet of Mathematical Curiosities by Ian Stewart
  111. Professor Stewart's Incredible Numbers by Ian Stewart
  112. Proofiness: How You're Being Fooled by the Numbers by Charles Seife
  113. Proofs from the Book by Martin Aligner and Gunter M Ziegler
  114. Puzzles from Penrose the Mathematical Cat by Theoni Pappas
  115. Pythagoras' Revenge: A Mathematical Mystery by Arturo Sangalli
  116. Q.E.D.: Beauty in Mathematical Proof by Burkard Polster
  117. Quite Right: The Story of Mathematics, Measurement, and Money by Norman Biggs
  118. Remarkable Mathematicians: From Euler to Von Neumann by I. M. James, Ioan James and James Ioan
  119. Roads to Infinity: The Mathematics of Truth and Proof by John Stillwell
  120. Short Stories About Numbers by Rajnish Kumar
  121. Struck by Genius: How a Brain Injury Made Me a Mathematical Marvel by Jason Padgett and Maureen Ann Seaberg
  122. Summing It Up: From One Plus One to Modern Number Theory by Avner Ash and Robert Gross
  123. Symmetry and the Monster: One of the Greatest Quests of Mathematics by Mark Ronan
  124. Symmetry by Hermann Weyl
  125. Symmetry: A Very Short Introduction by Ian Stewart
  126. Sync: The Emerging Science of Spontaneous Order by Steven Strogatz
  127. Taming the Infinite: The Story of Mathematics by Ian Stewart
  128. The Adventures of Penrose the Mathematical Cat by Theoni Pappas
  129. The Art of Mathematics: Coffee Time in Memphis by Béla Bollobás
  130. The Artist and the Mathematician: The Story of Nicolas Bourbaki, the Genius Mathematician Who Never Existed by Amir D. Aczel
  131. The Book of Numbers by John H. Conway, Richard K. Guy
  132. The Book of Numbers by Richard K Guy and John Horton Conway
  133. The Calculus of Friendship - What a Teacher and a Student Learned about Life while Corresponding about Math by Steven Strogatz
  134. The Code Book: The Secret History of Codes and Code-breaking by Simon Singh
  135. The Colossal Book of Mathematics - Classic Puzzles, Paradoxes & Problems by Martin Gardner
  136. The Connolly Book of Numbers, Vol 1: The Fundamentals by Eileen Connolly
  137. The Connolly Book of Numbers, Vol II: The Conslutant's Manual by Eileen Connolly
  138. The Constants Of Nature by John D. Barrow
  139. The Crest of the Peacock by George Gheverghese Joseph
  140. The Drunkard's Walk: How Randomness Rules Our Lives by Leonard Mlodinow
  141. The Education of T.C. Mits: What Modern Mathematics Means to You by Barry Mazur, Hugh Gray Lieber and Lillian R. Lieber
  142. The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry by Mario Livio
  143. The Fascinating World of Graph Theory by Arthur Benjamin
  144. The Genius of Euler: Reflections on his Life and Work by William Dunham
  145. The Geometry of Art and Life by Matila Ghyka
  146. The Golden Ratio: The Story of PHI, the World's Most Astonishing Number by Mario Livio
  147. The Golden Section: Nature’s Greatest Secret by Scott Olsen
  148. The Golden Ticket: P, NP, and the Search for the Impossible by Lance Fortnow
  149. The Great Mathematical Problems by Ian Stewart
  150. The History of Counting by Denise Schmandt-Besserat and Michael Hays
  151. The History of Zero: Exploring Our Place-Value Number System by Tika Downey
  152. The Honors Class by Ben H Yandell
  153. The Irrationals by Julian Havil
  154. The Joy of Pi by David Blatner
  155. The Joy of x: A Guided Tour of Math, from One to Infinity by Steven Strogatz
  156. The Little Green Math Book: 30 Powerful Principles for Building Math and Numeracy Skills by Brandon Royal
  157. The Magic of Math: Solving for x and Figuring Out Why by Arthur Benjamin
  158. The Man of Numbers: Fibonacci's Arithmetic Revolution by Keith J. Devlin
  159. The Man Who Counted - A Collection of Mathematical Adventures by Malba Tahan, Leslie Clark, Alastair Reid and Patricia Reid Baquero
  160. The Man Who Knew Infinity: A Life of the Genius Ramanujan by Robert Kanigel
  161. The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth by Paul Hoffman
  162. The Man Who Loved Only Numbers: The Story of Paul Erdős and the Search for Mathematical Truth by Paul Hoffman
  163. The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics by Clifford A. Pickover
  164. The Mathematical Brain by Brian Butterworth
  165. The Mathematical Experience by Elena Anne Marchisotto, Reuben Hersh and Philip J Davis
  166. The Mathematician Sophus Lie by Arild Stubhaug
  167. The Mathematics Devotional by Clifford A. Pickover
  168. The Mathematics of Egypt, Mesopotamia, China, India, and Islam by Victor J Katz
  169. The Mind of the Mathematician by Michael Fitzgerald and Ioan James
  170. The Music of Reason: Experience the Beauty of Mathematics Through Quotations by Theoni Pappas
  171. The Music of the Primes by Marcus Du Sautoy
  172. The Mystery of the Aleph by Amir D. Aczel
  173. The Nothing that is: A Natural History of Zero by Robert Kaplan
  174. The Nothing That Is: A Natural History of Zero by Robert M. Kaplan
  175. The Outer Limits of Reason - What Science, Mathematics, and Logic Cannot Tell Us by Noson S. Yanofsky
  176. The Pea and the Sun: A Mathematical Paradox by Leonard M. Wapner
  177. The Pleasures of Counting by T.W. Körner
  178. The Poincaré Conjecture: In Search of the Shape of the Universe by Donal O'Shea
  179. The Prince of Mathematics: Carl Friedrich Gauss by M. B. W. Tent
  180. The Proof and the Pudding by Jim Henle
  181. The Riddle of the Compass: The Invention That Changed the World by Amir D. Aczel
  182. The Simpsons and Their Mathematical Secrets by Simon Singh
  183. The Symmetries of Things by John H. Conway, Heidi Burgiel and Chaim Goodman-Strauss
  184. The Unimaginable Mathematics of Borges' Library of Babel by W. L. Bloch
  185. The Universal History of Numbers: From Prehistory to the Invention of the Computer by Georges Ifrah
  186. The Universal History of Numbers: The Computer and the Information Revolution by Georges Ifrah
  187. The Universal History of Numbers: The World's First Number-Systems by Georges Ifrah
  188. Things to Make and Do in the Fourth Dimension: A Mathematician's Journey Through Narcissistic Numbers, Optimal Dating Algorithms, at Least Two Kinds of Infinity, and More by Matt Parker
  189. To Infinity and Beyond: A Cultural History of the Infinite by Eli Maor
  190. Trolling Euclid: An Irreverent Guide to Nine of Mathematics' Most Important Problems by Tom Wright
  191. Unknown Quantity X: A Real and Imaginary History of Algebra by John Derbyshire
  192. Weapons of Math Destruction: How Big Data Increases Inequality and Threatens Democracy by Cathy O'Neil
  193. What Is Mathematics, Really? by Reuben Hersh
  194. What Is Mathematics? by Richard Courant and Herbert Robbins
  195. What We Cannot Know by Marcus Du Sautoy
  196. What's Math Got to Do with It?: How Teachers and Parents Can Transform Mathematics Learning and Inspire Success by Jo Boaler
  197. Yearning for the impossible by John Stillwell
  198. Zero History by William Gibson
  199. Zero: The Biography of a Dangerous Idea by Charles Seife
  200. 50 Greatest Mathematicians of the World by Rajesh Thakur
  201. 50 Maths Ideas You Really Need to Know by Tony Crilly

Friday, September 09, 2016

Fall over Donuts...

At the Arrivals
Anxiously on my lanky legs...
Doors at the gate open in several folds
When no shadows fall, they're shut like lazy lids.
Thrust open my wide eyes seeing
Grilled Hams
'Harmless' Steaks
French Fries and Fried Eggs
Hardly make eleven quids
Hauled my lusty eyes and there I saw this seductive Ad
"Jane Parker Donuts"
"Over 3000000 enjoyed daily"
In no time, they had to change the Ad
"Over 3000001 enjoyed daily"

Wednesday, September 07, 2016

The Kite Runner

Cutting across the envious winds
Flying high above the clouds
Surging through the sunny rays
Winning is a habit for the kite always.

"O dude, you break records.
Not one among us birds
Can beat you in the heights."
Uttered the bird its thoughts.

"O bro, you’ve golden words
Among your fellow birds.
Need to reach new heights,
If ever the kite runner permits."

"That’s cruel, be free like birds.
My beak’s sharper than swords
Shall I cut the line?"
“At once bro, cut the line.”

In no time, the kite line is cut.
But no line to support,
Into a marshy pit, the kite
Fell quickly, ending the flight.

A teacher is a kite runner.
Once cut away from this trainer,
O student, can you fly high?
No. Hence be patient to fly high.

Sunday, September 04, 2016

What type of a teacher am I?

Am I gentle enough to treat my students gently?

Am I guiding my students through the adventurous path of learning,
Because learning is a war between ignorance and wisdom,
Because learning is a tragedy for the ego,
Because true learning is the sorrowful departure of the ignorant self?

Am I helping my students to acquire faith, love and courage
Because life is to be lived successfully like that of conquering continents?
Am I training them to face the flip side of the harsh realities of the world?
Does my training help my students to be optimistic
To see that there are also friends, heroes and dedicated leaders in the world?

Am I helping my students to earn from their sweat?

Am I transferring the wisdom to accept
That failures are honorable,
That losses are graceful and
That tears are glittering
Instead of cheats and undeserving wins?

When they are really sad,
did I make my students laugh?
To scoff at cynics
did I train my students?

Did I ever introduce my students
The wondrous world of books?
To enjoy the mastery of birds and mystery of buds
did I train my students?

When did I last promote the genuine ideas of my students?

To keep away from crowds and listen to themselves
did I instruct my students?

Would I tell my students
to choose right friends?

Heart and soul, two of our invaluable assets,
Are my students cherishing?

How many of my students have
The courage to be impatient
And the bravery to be patient?

Do they have faith in themselves
That leads them to faith in mankind and in God?

Yes, I am ready for this challenge.
Dear Lincoln, send your child to me.

Based on Lincoln's letter to his son's teacher

Friday, August 12, 2016

Identity Crisis

Forgetfulness is not uncommon when you deal with more than just what you can deal with. But it is not an excuse for forgetting. There are people who forget almost everything. From debit card to pan card, from ATM pin to email password, from current date to spouse’s date of birth, from own name to own child, forgetfulness is without boundaries. Remember the Adam Sandler and Drew Barrymore romantic comedy of 2004, “50 First Dates” where Lucy played by Barrymore has short-term memory loss. We had Kollywood block-buster Gajini by Surya which also dealt with memory loss.

All of us have short or long memory loss in some way. Memory has three stages: encoding, storing and retrieval. Retrieval is the most difficult stage. Retrieving the right information in the right time characterises how brilliant a person is.

For me every travel is a nightmare. First of all, it is a forceful push to go against the momentum as proposed by Isaac Newton. Then comes the problem of adjustment such as seat, food, sleep, strangers etc. The best of travel I  enjoy is the anonymity. But the worst of a travel also comes with the opposite of anonymity, proving one’s own identity. We encounter people who ask us who we are. How can one explain to others who one is?  Are our names matter or our profession or our family or address? For a casual enquiry, anything is fine. But when it comes to the real issue of proving our identity with the help of an identity card, it is rather simple and at the same time complicated. Simple, if we have one. Complicated, else.

Years ago, I was travelling by train with my sister from our work place to another town for a function. It was an overnight journey and it was a tatkal ticket. About an hour after the journey began, the Ticket Examiner in glittering black blazers and necktie with violet and white stripes appeared with a writing pad with the list of the reserved passengers. I was cool as I had my ticket with me. When my turn came, I produced my ticket. After having ticked our names in the list, he asked for my identity card. After giving him a quizzical stare, I searched for my identity card in my wallet in vain. I continued my fruitless search in my carry bag. Observing my difficulty, the Ticket Examiner asked me to check with my sister whether she had her identity card. As I was the sole authority of the travel arrangements, her negative reply was an expected one.

The Ticket Examiner now turned a preacher. He, in a relatively louder voice, preached about seriousness in life, possessing identity card while travelling, obeying rules, respecting the Indian Railways, failure of the present generation and a host of other things. Humiliated, I asked him what he wanted for winding up the preaching. He then became irritated and declared that we were illegal passengers. But, I claimed that we had a valid ticket. He replied by reading the rule of the Railways printed on the e-ticket. That clearly mentioned that we were illegal passengers. Helpless, I surrendered to him and asked what I should do for becoming a legal passenger. He replied to me that he would settle the matter after all the checkings were over. To this, I told him that I could pay the penalty. He became very gentle and told me that he would settle the issue after some time. But I did not agree to this. I told him to impose fine on me and to make us legal passengers. He was still in the procrastinating mood. Then my sister joined me asking the Ticket Examiner to impose fine or face complaint to the higher authorities. We ended up paying full ticket charge and the penalty of Rs. 500 each with a valid receipt. The story did not end here. After the function was over, we had to travel back in the next evening. As we did not have the identity cards, we went straight to the Ticket Examiner, paid and obtained the receipt for the full tickets and the penalty without giving a chance to him to find fault with us.

Years passed. I travelled several times in the trains. I love trains for many reasons. The parallel rails, the connected buses, the ever adjusting meanderings of the chain of the buses etc. always amused me. Long or short, I always preferred trains. As I used to arrive only at the last minute, there are countless incidents that are associated with my travels. But this one which I experienced the other day was something very unique.

It was yet another overnight journey. I had to deliver an invited talk. The tickets were booked by the organisers and the details were messaged. As usual, I left my room about forty minutes before the actual departure of the train. Fortunately, I could catch the third autorikshaw after stepping onto the road. The driver was very cordial. The meter was not for a short distance race. I felt relaxed. After sending a couple of important messages and making some calls, my attention turned towards the road. The road was relatively quiet. Visibly, there was no huge rush even though the office hours were still not over. I expected to get some fifteen minutes at the railway station. Once when I passed the Hudson circle, which was about three kilometres away from the station, I got a shooting pain up my spine reaching my heart. I started searching my bag. No, it was not there. I could not find my identity card in it. This was a small bag which I rarely used for travel.  I chose this because the travel was just for a day. As a precaution, I had kept the original-look-like copies of my identity card in my other bags. As I never used this one, I did not care to insert one in it. In the hurry of getting out of my room, the thought of identity card never flashed too.

The pain slowly spread to the other parts of the body. I was fatigued. I only had Rs. 550 in my pocket which I borrowed from my friend just before the coming out my room. I had to deliver the lecture the next day as the organisers chose that day after checking my convenience. I had no money to pay fine. A bus ticket to the destination, if at all it was available, would cost more than the amount I had. I had a moment of prayer. Instantly, I was back to normalcy. Within no time, the autorikshaw dropped me in front of the railway station. The meter just showed Rs. 98 which was much lower than the average rate in my city. (A couple of months early, I had travelled in an autorikshaw which displayed the Rs. 320 for the same distance). As I felt the driver was just, I paid him double and walked towards towards the train.  There were some 18 minutes left for the scheduled departure of the train.

All of a sudden, as if guided, I turned towards the ticket counter. The ticket counters were almost empty. I got a second class ticket for Rs. 150 quickly. I headed slowly towards the train. As I was about to reach the general compartment, I saw the man in glittering black blazers and necktie with violet and white stripes with a writing pad with the list of reserved passengers. Just suddenly, without much preparation, I greeted him. He reciprocated. I then told him that I did not carry an identity card but had a confirmed ticket. He replied to me that one must carry a valid identity card to travel with an online ticket. Though this was not a new information to me, I just passively listened to him. He then asked what I was. I replied. Having learnt that I was a teacher, he told me to take his seat in the three-tier compartment. I was in utter shock. I could not believe my ears.  I told him that I did not have enough money to pay for that. He replied me that it was not needed. He then took me to the three-tier compartment and made me seated  there in his seat. Without much wait, when it was time, the train began its routine race. I sat there not fully recovering from the shock and total disbelief. After an hour, the Ticket Examiner came to me and allotted me a vacant seat.

My surprises were not over. He started conversing with me. After having learnt that I neither carried supper nor ordered for it, he ordered some parcelled supper with someone who was at a station enroute. When we reached that station, the supper was brought in plus a banana and bottle of water.  After the supper, he asked how I planned to return. As I had to return the very next day in the same train when it would return, he sensed my difficulty. He then gave me his phone number and asked me to ring him when I would reach the station. Thanking him and thanking God, I went to sleep. The next day, I reached the station some half an hour before the departure. After finding him sitting in one of the concrete benches in the platform, I greeted him. Recognising me he told that it was a rush day and there were some RAC (Reservation Against Cancellation) ticket holders. I then told him that I could go to the general compartment. However, he turned me away from that attempt and guided me to his seat.  After forcing me seated there, he went for his checking round. Half an hour after the departure of the train he came back to the seat and shared with me the dinner I got packed by one of my students. When dinner was over, he went again for his duty. A couple of minutes later, he emerged again and took me to a vacant two-tier seat.  Surprise after surprise.   I slept comfortably well. In the next morning, I went to him and thanked him profusely. We then walked out together towards the exit. Before parting ways, I thanked that extremely gentle human being.  

Who was he? A good Samaritan? No. I was not wounded and left uncared on the rocky and thorny road. A good man.  A great man.  A noble man. Yes, all of them put together. Someone whom we rarely see in the society. Did he break rules? Probably, yes. But, he knew why rules were for. With his helping hand, he took the pages of the rule book and interpreted in a humane way. He helped me from my grief and from my brief identity crisis. He helped me to reassure the faith in the goodness of human beings.

Sunday, July 31, 2016

God of Imperfections

Reflections on John 9:3: Neither this man nor his parents sinned," said Jesus, "but this happened so that the works of God might be displayed in him.

Quite often, one is shell-shocked at facing adversities. Terrorist attacks, calamities, accidents etc. are mostly unanswerable. Why did God send all these?

Can God be a better one?
Human beings are not perfect.
Our helplessness in many situations are clear indications that all of us are also not perfect.
Blindness is one type of imperfection.
Physical illness, disabilities, abilities, poverty, richness, appetite, hunger, you say anything, we can find imperfections in abundance.
Our communication using this medium itself is another display of our imperfection.
If the world or human beings are full of imperfections, what is perfection?
If you believe in God, you know that God is perfection.
Sometimes people are gifted with faith in God.
Sometimes people crave for perfection and reach God.
Sometimes people pretend to be faithful. But at an opportunity, one may switch side. It's once again, another testimony of imperfection.

God is not in need of us.  That is why God is perfect.
However, if you need to attain perfection you need to go in the path of God.
One of the methods is prayer.
Through prayers, one can attain spiritual and physical perfection.
Asking God for cure was a prayer. The case of the blind mentioned in John 9.
Why then God created imperfections?
Once if you are aware of yourself, you will come to know that we are not endowed with the mind of God. Then, how do we know that why God created me like me?

How God created the other like the other?
The only answer is that it is to live in this world and meet me, my friend, my foe etc.. and live with them.
Let's live. Let's ask for blessings, if we want. If we have extra, give to others who are in need of that...

For those who are not believing in God; God is still perfect. But those people also believe in their ideologies, their partners, their children, their

security guards, their vehicles etc.
These are minor forms of faith. Gradually they will reach God, once when they realize the invitation of God to be perfect.
Have the humility. Then, one can gradually realize how better God can be experienced.
Those miracles may not be recorded in any scriptures. But, it will be recorded in oneself.

Many of of us are imperfect. So am I. But I know that my God is perfect. I work towards achieving that perfection. That is the moment, I loose my blindness completely.
Then, it's no more Joseph. It's God only.
Jesus showed this through His life.
That Jesus of the Bible is the best example for undue and innocent suffering; answer to all imperfections in the world.
Do not worry, leave the matter to itself if we do not understand something.
In the right time we will understand everything perfectly. Keep asking, we will be answered.

Wednesday, July 27, 2016

Prayer of a Teacher

God give me students who are challenging,  demanding and engaging...
Challenging all that can challenge them...
Demanding always freedom from the bondage of learning...
Engaging in stuffs that can disengage them from reaching heights...
Let my students have functions of blaming all that can fame them...
Let them oscillate between unwanted things or at least diverge but never converge...
Let them continue to be discontinuous essentially....
Let them derive the joy of being constant...

Monday, July 18, 2016

Coconut can carry Keralam to heights...

In the budget of the government of Keralam, there was a proposal to have five per cent levy on coconut oil. It is estimated to fetch Rs. 150 crore for the exchequer. Although, this proposal is sharply criticized by many, it is a clever move to encash the growing demand for coconut oil in Keralam. It also boosts the price of the coconut indirectly, thereby helping the coconut farmers. Keralam has great potency in converting its rare natural gift, the coconut tree, to which it owes its name (Keralam – Land of Kera (coconut) Tree). It is very strange to note that on the streets of Chennai, one has to pay Rs. 40 for a non-tasty tender coconut. A concerted effort from the various government agencies can create huge coconut money in Keralam. Here are some suggestions to have new directions in the coconut-promotion.

  1. Promotion of coconut as the cause of the name of the state of Keralam.
  2. Expand the activities of the Coconut Development Corporation to every ward of every local body.
  3. Identify and honour the oldest coconut in each ward.
  4. Insurance cover for each coconut.
  5. Promotion of Coconut Tree Adoption.
  6. Soaps including tax cut for Coconut Farmers.
  7. Freely distribute coconut saplings to the students.
  8. Studies of Coconut should be inserted in the curricula.
  9. Manual Coconut Plucking should be declared as a highly skilled and risky job.
  10. Tax-cuts for hotels and restaurants that use coconut oil.
  11. Introduction of Coconut-Week Celebrations.
  12. Promotion of Coconut preservation programmes.
  13. Introduce tax for other imported oils.
  14. For lamps used in government functions, coconut oil only should be used.
  15. Preference for coir products in the government offices.
  16. Attractive and early pension schemes for the Coconut Plucking labourers.
  17. Regular broadcasting of advertisements, songs etc. promoting Coconut.
  18. Regular promotional activities on the medicinal effect of coconut oil.
  19. Endowment for Coconut Research.
  20. Establishment of Coconut Research institutes in every Assembly constituency.
  21. Promoting handicrafts from coconut shell.
  22. Thin coir string is to used for all office needs.
  23. Research Centers and Chairs in all Universities in Keralam.
  24. Avoid all drinks except water, tender coconut, the processed non-alcoholic drinks from coconut in all government offices and functions.
  25. Promotion of Coconut “apple.” 
  26. Copra-based sweets, biscuits, fast-food, Tiffin etc. may be promoted with tax cuts.
  27. Awards for coconut farmers.
  28. Short-films and pamphlets in the promotion of coconut.
  29. Special welfare schemes for workers in all types coconut mills.
  30. Promotion of start-ups in the area of Coconut-related industry.
  31. Climbing on coconut tree should be included as a competitive sports event.
  32. Promote developing more dwarf-coconut varieties.
  33. Promote machines for coconut plucking and processing.

Wednesday, June 01, 2016

The world needs you

One who achieves a grade is a graduate. Today is your graduation day. You’re formally entrusted with the grades you have achieved during the course of your training programme here at Christ University. You are now graduates. This is a tremendous achievement from many perspectives.

It is an achievement by you. We believe that you have acquired the required experience, sufficient knowledge, and the capacity to make good judgement. This was what we aimed at while giving training to you here.

It is an achievement by your parents. They have brought you to the best educational institution. They applied their muscles and toiled for you while you were muscling with the books. They prayed for you while you were buried with your assignments and tests. They spent their precious time worrying about you while you systematically invested your time and energy to reach this glory.

It is an achievement for your nation. You are from different parts of the world. You become another fortunate son or daughter of your nation by adding yourself to the skilled force of your nation.

As graduates when you step out of your alma mater, we entrust you with a prophetic mission that was well presented in our core values. You were trained to be human beings whose ultimate goal is the fullness in God, knowing the truth and leading a virtuous life. We do want you to be real guiding lights. It should be not like, as the most famous European humanist of the 16th century Erasmus puts it, in regione caecorum rex est luscus. Become true beacons of light. We want you to work as positive catalysts in the world.
For some of you, study was cakewalk. Some of you never worked any way near to your potential. For some it was real struggle. It was fight against all odds. The sweetness of your achievement depends on the way in which you reached this day.
Let me take your attention to two persons and a sports team which became viral in the social media recently.

Srinivasa Ramanujan
An immensely talented Mathematician of our Land was Srinivasa Ramanujan. The recent movie, The Man Who Know Infinity, is based on his life. He independently compiled nearly 3,900 mathematical results. Humble to the core, he attributed all his works to the divine assistance. He died at a young age of 32. He was not a graduate in Indian terms. But he was one of the youngest Fellows of Royal Society. Although he was highly gifted, a good trainer like GH Hardy was needed to make him an effective contributor. If you are ready to be trained, your effectiveness is unimaginable.

Anant Ambani
An immensely rich young man is Anant Ambani, son of Mukesh Ambani. He was born with a silver spoon in his mouth. As a teenager he was often presented as the side effect of wealth as he was hugely obese. Reaching 21, his inspirational mother transformed him to a good looking handsome young man. He walked 21 km a day, practiced yoga, followed a strict low-carb, zero-sugar diet, i.e., avoided any kind of processed foods, such as, breads, pastas, cereals, sweets, soft drinks etc., concentrating instead on proteins and healthy fats. Finally, he shed 108 kgs in 18 months. It’s like a fairytale. But it’s not a myth. It happened just recently. Some of you have worked like this to achieve your graduation. Hardwork counts.

Leicester City
Until the second half of 2015, very few in the world had heard about Leicester City football team. It was placed last in the English Premier League. But on 7 May, 2016 they became the Champions with three games left in the league. In one of the toughest leagues in the world with 38 games, at the time of winning the title they lost only 3 games. The story of Leicester City as a team is a case study material. There are many players in English Premier League who has salary more than the total salary of all members of the Leicester City Team. The Team is owned by an Asian. The Coach never won a major title. Then, how this has happened? Educationalist John Dunford, lists five reasons for the great success of the team as follows:
1. Hardwork, 2.Teamwork, 3. Right Decisions, 4. Support of the Leadership and 5. Evidence-based policies.
If you have achieved the real goal you set years ago, do not be complacent. The world needs you.
If you have not achieved the real goal, do not lose hope. Still you have not lost time. The world needs you.

If you have underperformed while achieving your goal, do not regret unncesserily. Start today. Perform well. The world needs you.

Sunday, May 08, 2016

Mother is God incarnated.....

Just in case, if you couldn’t meet God anywhere
Why don’t you go to your kitchen
Search for that woman who has dirt everywhere
Shy to come out and always forgotten
Unselfish to love and selfish to be loved
Shed you her body and blood
Fed you for your good
Fled from her home not by her choice
Led by philosophies that are not wise
She needs not your mercy but your prayer
Believe then that God is everywhere
Confirm that God is not a piece of a stone
Affirm that God is flesh that can be seen

Friday, February 12, 2016

Love and Hatred

I love you when you integrate me
But for your partitioning of me finer and finer.
I hate you for differentiating me
Though you wait for my limits.

Sum of the Squares of the Degrees

One of the most fundamental results in graphs is that the sum of the degrees of all the vertices of a graph is twice the number of edges in it.

This is is the famous Handshaking Theorem.
It is called Handshaking Theorem due to the analogy of the involvement of two hands in every handshake with that of the two vertices in every relation (edge).

Hence, the total number of relations of the persons (degrees of  the vertices) in a graph is double the number of relations (edges) in it.

What happens if we add the squares of the degrees of all the vertices in a simple graph?

It is found out that the sum of the squares of all vertices in a graph is the the sum of the degrees of the adjacent vertices of all the vertices the graph it self.

Why is it so? It is because, the square of the degree of a vertex is same as adding the degree of a vertex, the degree number of times.
                                                         n2=n+n+n+...+n (n times)
But one can easily verify that every vertex has exactly its degree number of adjacent vertices. Hence, when we sum the degrees of  the adjacent vertices of all the vertices of a graph, every degree is added that degree number of times. This proves the result.

We can also observe that if we add all the degrees and degree squares of all the vertices of a graph, then it is same as summing up all the degrees of all vertices in the closed neighbourhood of all the vertices of that graph.

Monday, February 08, 2016

The First Twenty Five Primes

Every prime number is a positive integer that is divisible by 1 and itself only. There are infinitely many prime numbers. Prime numbers are peculiar numbers which do not have a common predictable property other than that of the indivisibility. Because they are indivisible, the fundamental theorem of arithmetic is that they are the building blocks of numbers. That is to say that any positive integer greater than 1 can be written uniquely as the product of prime numbers.

There are 25 primes below 100. This is a method to remember all the first 25 primes.
We begin with 25 itself.
The first and third primes are 2 and 5 respectively. They are prime numbers but no other prime number would have 2 and 5 as its end digits
Now, 5+2 and 5-2 give the other two single digits primes. i.e., 7 and 3.
Hence, we have all the single digit primes as 2, 3, 5 and 7.

All primes with at least two digits would end only in 1, 3, 7 or 9.
We list all the two digit primes that end in 1.
Wen use 2 and 5 once again for that.
Splitting 2, we get the first two digit prime, 11.
Then, 1+2=3, 1+3=4, 1+5=6 and 1+6=7 are the first digits of the remaining primes.
Hence, the two digit primes are 11, 31, 41, 61 and 71. Thus we get all primes that end in 1.
A total of 9 primes, we have found out so far.

Now, we need to list all the primes that end in 3.
Adding the digits of the primes that end in 1, we get 1+1=2, 3+1=4, 4+1=5, 6+1=7 and 7+1=8.
Appending 3 to the right of the above numbers, we get the two digit primes that end in 3.
They are 23, 43, 53, 73 and 83. Hence, excluding 3, we got 13 primes. Of course, 13 is also a prime number.

What left are the primes that end in 7 or 9. But, this is very easy.
List all the digits in numbers from 1 to 10.
They are 1, 2, 3, 4, 5, 6, 7, 8, 9, 1 and 3.
Appending 7 or 9 as the case may be(by avoiding the trivial composite numbers we get,
17, 29, 37, 47, 59, 67, 79, 89, 97, 19 and 07.
These are the 11 prime numbers that end in 7 or 9.
Hence all the prime numbers below 100 are:
  1. 2 and 5
  2. 11, 31, 41, 61 and 71
  3. 3, 13, 23, 43, 53, 73 and 83.
  4. 7, 17, 37, 47, 67 and 97
  5. 19, 29, 59, 79 and 89.