What is Mathematics?

Introducing the requirement of mathematical philosophy in every day needs of the physical reality, the book delves into the concepts of the subject in a more logical approach. The chapters might be better appreciated if the initial note of their organization and preliminary requirement are followed as specified. The Chapter I The Natural Numbers deals with the laws of arithmetic and the principle of mathematical induction with quite a few applications, some being left for the reader to solve making the chapters more challenging yet no less gripping. Continuing in its Supplement to Chapter I The Theory of Numbers, the primes are taken up in a more abstract sense and humbling the original chapter both in volume and concept, some unsolved problems are referred as are discussed modulo arithmetic, the concept of congruence, theorem of relative primes as an extension to the theorem of prime by Fermat, the development of Pythagorean triple and the elegance of their primitive forms. Utilisation of Euclidean Algorithm for searching of common factors between integers, the concept was used more than once not only for finding the highest factor that divides set of integers, but also referring to that in detailing Euler function of relative primes and culminating in analysing Diophantine equations with pointers to their solvabilities. The Chapter II Number System of Mathematics dwells in denumerability of numbers, algebra, graphic representation of numbers by analytical geometry, complex plane and transcends to Liouville's theorem. This fascnitating chapter stresses the psychological need of mathematical evolution while acknowledging the hesitant steps that apparently is the barrier of quantifying abstract thinking. It is followed up with Supplement to Chapter II The Algebra of Sets that lays the foundation for set theory, which was touched up on the previous chapter. Mathematical tool for engineers seems to be the best way to describe Chapter III Geometrical Constructions: The Algebra of Number Fields, where the number fields and constructions are introduced to appreciate the fundamentals of geometrical constructions. The problem of Appolonius is stated with its proofs by different perspective, the unsolvability of various Greek problems within their domain of constraints are logically demonstrated, inversion with applications to various problems are elaborated. However, the most interesting part of the chapter seems to be Mascheroni Construction which required me to consult http://mathafou.free.fr/themes_en/compas.html to solve a proposed exercise. The prefaces by the authors with their subsequent revision by Richar Courant and Ian Stewart indicates that some chapters were later appended as logic was refined as it branched. The toughest is Chapter IV Projective Geometry. Axiomatic and Non-Eucildean Geometries that required visits to stackexchange.com, files.eric.ed.gov, nabla.hr, amsi.org.au to understand how to approach certain problems suggested. The chapter will give a brand new insight to geometry itself. The apparently less complicated Chapter V Topology is about a different geometrical aspect that deliberates on Euler characteristics of general surfaces including that with holes as well as special surfaces like Moebius strip with its unique features. Not only does the chapter contains geometrical revelation like that for the Jordan curve but also how the fundamental theorem of algebra has a topological perspective is given as the caveat. Chapter VI Functions and Limits are a revelation on the topics. The topics have been presented with the abstract elegance that several never think of while utilizing the concepts for solving mathematical intricacies. As the name Supplement to Chapter VI More Examples on Limits and Continuity obviously suggests, this helps in elaborating the preceding concepts with several examples and exercises for the readers that reveal their beauty and give them the touch of the classic. The next Chapter VII Maxima and Minima delves into the geometrical treatment of the extremum problems with illustrations of some natural phenomenon that corroborates the models with the physical laws. But its superbly beautiful aspect lies in the investigations of the existence of extremums that are quite sometimes taken for granted, which accurately portrays the Dirichlet conditions with quite a few interesting ideas. Academic twist awaits while going through Chapter VIII The Calculus that begins with integral calculus, generally reserved for the stage, following the understanding of differential calculus in conventional approaches to study this particular branch of mathematical creativity. Utilizing the basic summation method of computing definite integrals, the specific subject is elaborated with several revealing observations and inequalities. Then comes the derivative and how Fermat stimulated its necessity for obtaining extrema of functions. The derivative is formulated, analysed and symbolised. Lucid illustrations and challenges follow to compute derivatives of several algebraic expressions. Then comes the treatment for a few of the trigonometric observations. The requirement of continuity of functions for differentiability is elaborated. Applications start with derivatives applied in deriving acceleration and instantaneous velocity for moving bodies, the geometrical interpretation of 2^nd derivative, evaluation of minima and maxima among others. Then comes the techniques of differentiation and some of their very interesting applications. The notation that was used by Leibniz for differentiation represented a radical mathematical thinking, though the actual notion was only, at best, vaguely understood at the time by the thinkers in the field. With a logical note that justifies why great minds would proverbially arrive at similar results, the fundamental theorem of calculus is evaluated. The mechanical algorithm for the usual process of integration is justified and applications provided. The 1^st being Leibniz' formula for obtaining ℼ/4, followed by logarithm and exponential, with logarithm suprisingly yet logically preceding. The properties of the base of the natural logarithm are detailed and utilized followed by infinite series for logarithm. The concept of differential equations follows with applications in principles of physics, growth, vibrations and compound interest. The Supplement to Chapter VIII starts with basic principles of differentiability, integrals with its varied applications and continues with orders of magnitude, infinite series, the interesting advantage of the complex variable for their applications regarding seemingly unrelated domains, the harmonic series, the zeta function and the sine series by Euler, a beautiful discovery in the subject and the logical yet a bit approximate introduction to the prime number theorem. The prime numbers are the start to Chapter IX Recent Developments, which is a fascinating rambling on the more modern developments in this beautiful subject. Beginning with several elusive problems of prime viz., the Goldbach Conjecture and the Twin Prime Problem, the book stresses the need of novel methods for their decisive directions, which is followed up by the famous Last Theorem of Fermat that started as the unsolved mystery when the book was initially published but was demystified in 1994 and thus was inserted in fresh editions. Then, there is a discussion on Continuum Hypothesis, which may puzzle the intellectual greatly, the fashion of Set-theoretic notations through the ages, the tricky Four Colour Theorem, the proof of which literally challenged mathematical practices. The fractals with the apparently bizarre Hausdorf dimensions follow that will stretch the brain deeper towards abstract realms. The links and knots come next and then comes another abstract analysis of a mistake in a previous chapter, now logically rectified, concerning a tricky problem of mechanics having seriously simpler mathematical consequences. Then the algorithmic efficiency of computation is discussed with reference to the Problem of Steiner to search for the smallest length to bridge a network of points. The related intricacies brewed by Soap Films and Minimal Surfaces are briefed as we reach the concluding Nonstandard Analysis that immensely satisfies the wits of the readers. The book ultimately converges with Apppendix: Supplementary Remarks, Problems, And Exercises, a tutorial of varied problems to entertain and challenge with help from sites like math.stackexchange the wits of the readers ready to tackle problems on the field of arithmetic, topology, calculus, limits, geometry, algebra, functions, analytical techniques that completes this enthralling treatment.

Fermat's Last Theorem

Simon Singh's account of how the last theorem of Fermat was solved is truly a mathematical adventure. Starting with the famous yet often unspoken Pythagoras' proof, traversing the intricacies of Diophantine riddles and the Game Theory, the rigorousness of the dot conjecture, not to mention that of some apparently milder axioms, the cunningly composed paradoxes utilizing purest form of logic, the rationality of the irrationals, the powerful primes with their mysterious intervals, the variety of amazing abstractions that decorates the subject were  introduced suitably as the theorem was discussed whose apparent simple statement kept the mathematical community scratching for the proof for centuries. Complementing the beauty of mathematics, was the saga of the giants in the field, most prominent being the tragic genius Evariste Galois whose short life and notes documented during his even shorter mathematical career that lasted till the night before his death was one of the valuable threads for the proof. Then was the cornerstone conjecture of the mathematical pair of Taniyama and Shimura whose proof held the key for proving the elusive theorem, as stated by mathematician, Gerhard Frey, corroborated by Ken Ribet with a hint from Barry Mazur. The techniques pioneered by Kolyvagin, Flach, Iwasawa were utilized as Andrew Wiles improvised on the techniques to present the elegant proof for the most sought after theorem that baffled generations of mathematicians. The role of his former supervisor, John Coates, who introduced elliptical curves to him, that of Nick Katz, the only other mathematician who was taken into confidence by Wiles as he edged towards the proof during the secretive years and who actually failed to catch a gap in the logic during the time only to indicate the same, albeit with a bit of embarrassment, while refereeing the historic paper leading to yet another article by Wiles with his former student, Richard Taylor, another referee to the original and then a collaborator to its historic supplementary paper, the supporting ensemble to this classic drama also finds their deserving credits. Mention is made of the eccentric Paul Friedrich Wolfskehl and his chance encounter with the Fermat puzzle that not only averted his suicide but captivated him to such an extent as to announce a lumpsum reward for the person to solve it thus adding a material appeal with a solid timeline to the prestige that beckoned the chasers.

The book is draped with episodes of mathematical events but the underlying mathematical principles behind them are elaborated for the layman to admire the elegance of the various abstract forms. A theorem that sometimes threatened to be a wild goose chase, had to wait for centuries for a definitive proof. A 358 long years of perseverance by the mathematical community was prized with a voluminous proof that not only clarified the theorem but offered the probing of logical genes that constitute the subject. Andrew Wiles created history while solving the taunting theorem of Fermat but the chapters that he transcribed had described mathematics for its further refinement as can be assimilated by the limit of intelligence. The passionate dream of a child saw its culmination in the Annals of Mathematics as the masters of logic witnessed the composition of the master theorist. Simon Singh has beautifully pieced together the significant events in mathematical evolution, presenting it to the layman with a fervour which is both infectious and compelling.

Atomic Energy in Cosmic and Human Life Fifty Years of Radioactivity

Starting from the very basic, following the humour laced prefaces, this seemingly complicated subject that comprises science along with associated engineering, could never have been better elaborated for the layman but for this volume. Explaining the nature of the interaction of the nucleons based on the model of which he himself was the proponent, the revelation of the inner working of the atomic particles seemed no less wondrous than the style in which it was narrated. It will seem so simple as the apparently unfathomable mystery of the particles, unseen to the naked eyes of ordinary humans, is detailed, in principle, with analogies of physics that they would understand. The hidden energy inside the nucleus is compared with that liberated during chemical conversions. This highlights the significant difference in the order of energy conversion per unit mass that can be utilized in the betterment of mankind. Subtly touching on the destructive uses of atomic energy with numerical figures to horrify correctly, the scientific conjecture of building power sources and space vehicles will be appealing more to researchers yearning for development for a harmless future.

Euclid's Window

Starting from Euclid and going strong with Witten, mathematics and especially geometry had undergone a dimensional upgrade as crafted by the masters in each mathematical age. This is what Leonard Mlodinow has presented in this wittily wonderful biography of geometry. Infact, originating from the Pythogorean time, mathematics had evolved and so did abstraction that needed to be aligned for their coincident maturity. But, sometimes, it seemed the evolution was faster and the civilisation and requirement seemed a bit lacking in the course. Thus, the geometry required by Einstein for his revolutionary general and special theories of relativity, was already developed years back. This wonder of abstract conceptions with implications that could only be conceived in future seems to be the frustrating beauty of the subject which some may feel and care less for anything else but others in the pursuit in their fields may be held in despair. Thus, the wild geometry of strings, while being tamed by the ring masters in the arena is still being held in contempt by quite a few but apparently relent as the natural laws of basic physics seem to unravel in multidimensional world. The book is a refreshing find which will undoubtedly satisfy the mathematical spirit. Humour abound, the intellectual revelations could never be better documented. The only flaw seemed a lack of illustrations in this abstract base of knowledge. For this,  however, the margins provided by the publisher seemed enough to jot down miniatures of my understandings which might be referred to at their own risks.

The Great Physicists from Galileo to Einstein

Truly a biography of physics is this book, with the signature style of humble humour that characterizes Gamow in all his greatness. Starting from the birth of science in the ages before Galileo, he sketches the evolution past Einstein and thus maintains a balance in justifying the name of the book which focuses mainly on physics and the way it was shaped by the masters through the ages with a hint to what the future might hold on the science. With accompanying anecdotes to the serious science, the tough theories could never seem more fun to understand. Definitely, it will not be a layman's book if the term is defined strictly but if we remember the basics of the secondary level, the book will not just be adorable but enticing to take up physics to solve the fascinating mysteries that surrounds it in its various dimensions. With illustrations as delightful as the text itself, the book will remain a classic in the genre of popular physics along with biography category though the term is more reserved for the living world. But, after going through the essays, physics will surely be felt as throbbing and as lively as life itself. In this book, the teacher in Gamow has excelled while the storyteller in him has marvelled to give the readers a literary treasure blended with science. Gamow never got a Nobel in physics, but after going through several of his literary works, my only question to the nominators, had he ever been considered for the Prize in literature?

Mr Tompkins Explores the Atom

This time, it is for Mr Tompkins to explore the curious world of atoms. The peculiarity of the quantum world had been revealed to the naive amateur student of physics in the earlier volume and the adventure to the uncertain world continues, thanks to his physicist father-in-law's lectures. The lectures seems to give the audience nightmares, which is very much apparent as not only does Mr Tompkins fall a prey for it, but so does his wife who had accompanied him to one of them! But sometimes, the father-in-law takes pity on our protagonist and asks him not to attend some of the difficult ones but the physics lover will not be the sufferer as the book contains all the lectures including the one that Mr Tompkins skipped. Science couldn't have been explained in more simpler terms! It is humorous, the tone is humble, the content is just magnificent. With the illustration by the author himself, as the regular illustrator abandons his post, the appeal of the volume seems to increase more as the author seems to keep the drawings, some of them, adapted from originals, as perfect as possible.

Mr Tompkins in Wonderland

This is the 1st in the series centering Mr Tompkins, who takes peek into the world of science, during his off hours of serving as a bank clerk.
His understanding of science sometimes leads to dreams that takes him to lands where the scientific constants are trimmed so that he could realize what happens in the microscopic world of which he could only grossly comprehend at most. But this only helps the reader, who are also benefitted to understand the complicated yet unavoidable philosophy that led to the birth of relativity, replacing the classical notions of space and time and quantum mechanics, replacing the classical notions of certainty in measurement. Consisting of a series of dreams and lectures, the marvels of physics could never be better revealed. Readers with basic understanding of physics will undoubtedly find a great appeal to the intricacies of nature which is explained in easy terms laced with humour that overcomes the difficulty of understanding and inspires confidence to study the subject in detail.

Mr Tompkins Learns the Facts of Life

At a very early age, I was introduced to the character, Mr Tompkins by my father. Well, from then onwards, I became the fan of both the protagonist of the series and the creator. The illustrations that were sketched by Gamow added to the enticement. Science could never have been more thrillingly humorous as when one sees it through the eyes of Mr Tompkins. Ever since I had become enthralled by the intricacies of the Mother Nature and the science She offers.
So, when I took this up, the 3rd in the series, motivated by my recent endeavour with The Body by Bill Bryson (certainly this is in no way related specifically with his body but the medical mystery of the human body, in general), the adventures seem not to have aged much in appeal.
These brought back nostalgic scientific memories. Maybe, it was due to my upgraded view of science the insights seem clearer. As the clerk, Mr Tomkins had shifted his interest from physics after his 1st couple sets of adventures, the readers stood to benefit. Blood, gene and brain were opened to them for investigation as was some rudimentary logic of digital computing. Added to this is a chapter to summarise how energy is harnessed by plants deriving from the solar source and channelled through the multitude of sinks which make up the living and breathing earth. The inquisitive reader can locate several pointers to direct their interest for the future while being glued to the literary wealth of the series.

The Body

After reading this book you will know what you don't know. You, actually, will understand how little we understand about human body, one of the most complex manifestations of evolution that we have witnessed. Starting from the tiny albeit significant cell, it divides numerous times, but remains more or less united, to give you a body to house yourself. The various parts of this mysterious structure works untiringly for you well being but curiously we take it all for granted. Maybe, after going through this book, you will appreciate the various mechanisms of the human body that defends you from all sorts of foreign invasions and remains unappreciated for the better part of their lives. This book is not only how the bodies function but it also gives a brief history of the human understanding of the biology of their bodies. Laced with endearing humour throughout, the author once more gives a fascinating insight of the subject that he documents which will help the layman in understanding what happens the next time anything physiologically happens.

George's Secret Key To The Universe

One of the fascinating science based fictions I have ever read since the Tompkins series. Basically a book for astronomy enthusiasts, it can motivate the ordinary to get a taste of the basics.
An intergalactic adventure awaits George, whose scientific ambitions had been always snubbed down by his environment conscious parents. It was about to change when his pig breaks boundaries to poke into the next door neighbours. The mysteries of the universe awaits to be unfolded soon as did the mystery of their neighbours and their computer. As George stumbles on the marvels of the universe by the chance meeting, the crooked forces are also in the move, thwarting the aims of the noble minds while trying to use science for petty advantages.
Written lucidly by the brilliant Stephen Hawking and his scientist daughter Lucy Hawking with the charming illustrations of Garry Parsons, this book, once recommended by my father, had satisfied both his son and his grandson with its facts of science presented in the package of fiction with the touch of suspense to appeal the readers of varied ages.