"Proof" has been and remains one of the concepts which characterises mathematics. Covering basic propositional and predicate logic as well as discussing axiom systems and formal proofs, the book seeks to explain what mathematicians understand by proofs and how they are communicated. The authors explore the principle techniques of direct and indirect proof including induction, existence and uniqueness proofs, proof by contradiction, constructive and non-constructive proofs, etc. Many examples from analysis and modern algebra are included. The exceptionally clear style and presentation ensures that the book will be useful and enjoyable to those studying and interested in the notion of mathematical "proof."
Author: Rowan Garnier,John Taylor
Much may be gathered, indirectly, from the arguments in these pages, as to the real nature of the Earth on which we live and of the heavenly bodies which were created for us. The reader is requested to be patient in this matter and not expect a whole flood of light to burst in upon him at once, through the dense clouds of opposition and prejudice which hang all around. Old ideas have to be gotten rid of, by some people, before they can entertain the new; and this will especially be the case in the matter of the Sun, about which we are taught, by Mr. Proctor, as follows: “The globe of the Sun is so much larger than that of the Earth that no less than 1,250,000 globes as large as the Earth would be wanted to make up together a globe as large as the Sun.” Whereas, we know that, as it is demonstrated that the Sun moves round over the Earth, its size is proportionately less. We can then easily understand that Day and Night, and the Seasons are brought about by his daily circuits round in a course concentric with the North, diminishing in their extent to the end of June, and increasing until the end of December, the equatorial region being the area covered by the Sun’s mean motion. If, then, these pages serve but to arouse the spirit of enquiry, the author will be satisfied.
Author: William Carpenter
Publisher: Ravenio Books
Category: Earth (Planet)
This text explores the many transformations that the mathematical proof has undergone from its inception to its versatile, present-day use, considering the advent of high-speed computing machines. Though there are many truths to be discovered in this book, by the end it is clear that there is no formalized approach or standard method of discovery to date. Most of the proofs are discussed in detail with figures and equations accompanying them, allowing both the professional mathematician and those less familiar with mathematics to derive the same joy from reading this book.
The Changing Nature of Mathematical Proof
Author: Steven G. Krantz
Publisher: Springer Science & Business Media
The amazing story of one of the greatest math problems of all time and the reclusive genius who solved it In the tradition of Fermat’s Enigma and Prime Obsession, George Szpiro brings to life the giants of mathematics who struggled to prove a theorem for a century and the mysterious man from St. Petersburg, Grigory Perelman, who fi nally accomplished the impossible. In 1904 Henri Poincaré developed the Poincaré Conjecture, an attempt to understand higher-dimensional space and possibly the shape of the universe. The problem was he couldn’t prove it. A century later it was named a Millennium Prize problem, one of the seven hardest problems we can imagine. Now this holy grail of mathematics has been found. Accessibly interweaving history and math, Szpiro captures the passion, frustration, and excitement of the hunt, and provides a fascinating portrait of a contemporary noble-genius.
The Hundred-Year Quest to Solve One of Math's Greatest Puzzles
Author: George G. Szpiro
Computer Security and Access Control
Author: Jerome Lobel
Publisher: McGraw-Hill Companies
Category: Computer security
In this must-have for anyone who wants to better understand their love life, a mathematician pulls back the curtain and reveals the hidden patterns—from dating sites to divorce, sex to marriage—behind the rituals of love. The roller coaster of romance is hard to quantify; defining how lovers might feel from a set of simple equations is impossible. But that doesn’t mean that mathematics isn’t a crucial tool for understanding love. Love, like most things in life, is full of patterns. And mathematics is ultimately the study of patterns—from predicting the weather to the fluctuations of the stock market, the movement of planets or the growth of cities. These patterns twist and turn and warp and evolve just as the rituals of love do. In The Mathematics of Love, Dr. Hannah Fry takes the reader on a fascinating journey through the patterns that define our love lives, applying mathematical formulas to the most common yet complex questions pertaining to love: What’s the chance of finding love? What’s the probability that it will last? How do online dating algorithms work, exactly? Can game theory help us decide who to approach in a bar? At what point in your dating life should you settle down? From evaluating the best strategies for online dating to defining the nebulous concept of beauty, Dr. Fry proves—with great insight, wit, and fun—that math is a surprisingly useful tool to negotiate the complicated, often baffling, sometimes infuriating, always interesting, mysteries of love.
Patterns, Proofs, and the Search for the Ultimate Equation
Author: Hannah Fry
Publisher: Simon and Schuster
Category: Family & Relationships
The heart of mathematics is its elegance; the way it all fits together. Unfortunately, its beauty often eludes the vast majority of people who are intimidated by fear of the difficulty of numbers. Mathematical Elegance remedies this. Using hundreds of examples, the author presents a view of the mathematical landscape that is both accessible and fascinating. At a time of concern that American youth are bored by math, there is renewed interest in improving math skills. Mathematical Elegance stimulates students, along with those already experienced in the discipline, to explore some of the unexpected pleasures of quantitative thinking. Invoking mathematical proofs famous for their simplicity and brainteasers that are fun and illuminating, the author leaves readers feeling exuberant—as well as convinced that their IQs have been raised by ten points. A host of anecdotes about well-known mathematicians humanize and provide new insights into their lofty subjects. Recalling such classic works as Lewis Carroll’s Introduction to Logic and A Mathematician Reads the Newspaper by John Allen Paulos, Mathematical Elegance will energize and delight a wide audience, ranging from intellectually curious students to the enthusiastic general reader.
An Approachable Guide to Understanding Basic Concepts
Author: Steven Goldberg
Publisher: Transaction Publishers
This book covers elementary discrete mathematics for computer science and engineering. It emphasizes mathematical definitions and proofs as well as applicable methods. Topics include formal logic notation, proof methods; induction, well-ordering; sets, relations; elementary graph theory; integer congruences; asymptotic notation and growth of functions; permutations and combinations, counting principles; discrete probability. Further selected topics may also be covered, such as recursive definition and structural induction; state machines and invariants; recurrences; generating functions.
Author: Eric Lehman,F. Thomson Leighton,Albert R. Meyer
First Published in 2007. Routledge is an imprint of Taylor & Francis, an Informa company.
A Statistical Guide
Author: Laura Langbein
Category: Business & Economics
Category: American literature
The world's leading authorities describe the state of the art in Serre's conjecture and rational points on algebraic varieties.
Author: Luis Dieulefait,Gerd Faltings,D. R. Heath-Brown,Yuri I. Manin,B. Z. Moroz,Yu. V. Manin,Jean-Pierre Wintenberger
Publisher: Cambridge University Press
In recent years, a number of works have appeared with important implications for the age-old question of the existence of a god. These writings, many of which are not by theologians, strengthen the rational case for the existence of a god, even as this god may not be exactly the Christian God of history. This book brings together for the first time such recent diverse contributions from fields such as physics, the philosophy of human consciousness, evolutionary biology, mathematics, the history of religion, and theology. Based on such new materials as well as older ones from the twentieth century, it develops five rational arguments that point strongly to the (very probable) existence of a god. They do not make use of the scientific method, which is inapplicable to the question of a god. Rather, they are in an older tradition of rational argument dating back at least to the ancient Greeks. For those who are already believers, the book will offer additional rational reasons that may strengthen their belief. Those who do not believe in the existence of a god at present will encounter new rational arguments that may cause them to reconsider their opinion.
Five Rational Ways to Think about the Question of a God
Author: Robert H. Nelson
Publisher: Wipf and Stock Publishers
This text is designed to teach students how to read and write proofs in mathematics and to acquaint them with how mathematicians investigate problems and formulate conjecture.
The Mathematician's Toolbox
Author: Robert S. Wolf
Publisher: St. Martin's Press
Author: Raymond F. Coughlin,David E. Zitarelli
Publisher: McGraw-Hill Companies
Author: Srinivasa Ramanujan Aiyangar,Bruce C. Berndt
This book, updated and improved, introduces the mathematics that supports advanced computer programming and the analysis of algorithms. The primary aim of its well-known authors is to provide a solid and relevant base of mathematical skills--the skills needed to solve complex problems, to evaluate horrendous-looking sums, to solve complex recurrence relations, and to discover subtle patterns in data. It is an indispensable text and reference, not only for computer scientists but for all technical professionals in virtually every discipline.
A Foundation for Computer Science
Author: Ronald L. Graham,Donald Ervin Knuth,Oren Patashnik
Publisher: Addison-Wesley Professional
G. H. Hardy was one of this century's finest mathematical thinkers, renowned among his contemporaries as a 'real mathematician ... the purest of the pure'. He was also, as C. P. Snow recounts in his Foreword, 'unorthodox, eccentric, radical, ready to talk about anything'. This 'apology', written in 1940, offers a brilliant and engaging account of mathematics as very much more than a science; when it was first published, Graham Greene hailed it alongside Henry James's notebooks as 'the best account of what it was like to be a creative artist'. C. P. Snow's Foreword gives sympathetic and witty insights into Hardy's life, with its rich store of anecdotes concerning his collaboration with the brilliant Indian mathematician Ramanujan, his idiosyncrasies and his passion for cricket. This is a unique account of the fascination of mathematics and of one of its most compelling exponents in modern times.
Author: G. H. Hardy
Publisher: Cambridge University Press
Emphasizes a Problem Solving Approach A first course in combinatorics Completely revised, How to Count: An Introduction to Combinatorics, Second Edition shows how to solve numerous classic and other interesting combinatorial problems. The authors take an easily accessible approach that introduces problems before leading into the theory involved. Although the authors present most of the topics through concrete problems, they also emphasize the importance of proofs in mathematics. New to the Second Edition This second edition incorporates 50 percent more material. It includes seven new chapters that cover occupancy problems, Stirling and Catalan numbers, graph theory, trees, Dirichlet’s pigeonhole principle, Ramsey theory, and rook polynomials. This edition also contains more than 450 exercises. Ideal for both classroom teaching and self-study, this text requires only a modest amount of mathematical background. In an engaging way, it covers many combinatorial tools, such as the inclusion-exclusion principle, generating functions, recurrence relations, and Pólya’s counting theorem.
An Introduction to Combinatorics, Second Edition
Author: R.B.J.T. Allenby,Alan Slomson
Publisher: CRC Press