Introduction to Mathematical Thinking

• Keith Devlin | 2012 | 117 páginas | pdf | 1,3 Mb

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In the twenty-first century, everyone can benefit from being able to think mathematically. This is not the same as “doing math.” The latter usually involves the application of formulas, procedures, and symbolic manipulations; mathematical thinking is a powerful way of thinking about things in the world -- logically, analytically, quantitatively, and with precision. It is not a natural way of thinking, but it can be learned. Mathematicians, scientists, and engineers need to “do math,” and it takes many years of college-level education to learn all that is required. Mathematical thinking is valuable to everyone, and can be mastered in about six weeks by anyone who has completed high school mathematics. Mathematical thinking does not have to be about mathematics at all, but parts of mathematics provide the ideal target domain to learn how to think that way, and that is the approach taken by this short but valuable book. The book is written primarily for first and second year students of science, technology, engineering, and mathematics (STEM) at colleges and universities, and for high school students intending to study a STEM subject at university. Many students encounter difficulty going from high school math to college-level mathematics. Even if they did well at math in school, most are knocked off course for a while by the shift in emphasis, from the K-12 focus on mastering procedures to the “mathematical thinking” characteristic of much university mathematics. Though the majority survive the transition, many do not. To help them make the shift, colleges and universities often have a “transition course.” This book could serve as a textbook or a supplementary source for such a course. Because of the widespread applicability of mathematical thinking, however, the book has been kept short and written in an engaging style, to make it accessible to anyone who seeks to extend and improve their analytic thinking skills. Going beyond a basic grasp of analytic thinking that everyone can benefit from, the STEM student who truly masters mathematical thinking will find that college-level mathematics goes from being confusing, frustrating, and at times seemingly impossible, to making sense and being hard but doable. Dr. Keith Devlin is a professional mathematician at Stanford University and the author of 31 previous books and over 80 research papers. His books have earned him many awards, including the Pythagoras Prize, the Carl Sagan Award, and the Joint Policy Board for Mathematics Communications Award. He is known to millions of NPR listeners as “the Math Guy” on Weekend Edition with Scott Simon. He writes a popular monthly blog “Devlin’s Angle” for the Mathematical Association of America, another blog under the name “profkeithdevlin”, and also blogs on various topics for the Huffington Post.

Contents
Preface
What this book is about
1 What is mathematics?
1.1 More than arithmetic
1.2 Mathematical notation
1.3 Modern college-level mathematics
1.4 Why do you have to learn this stuff?
2 Getting precise about language
2.1 Mathematical statements
2.2 The logical combinators and, or, and not
2.3 Implication
2.4 Quantifiers
3 Proofs
3.1 What is a proof?
3.2 Proof by contradiction
3.3 Proving conditionals
3.4 Proving quantified statements
3.5 Induction proofs
4 Proving results about numbers
4.1 The integers
4.2 The real numbers
4.3 Completeness
4.4 Sequences
APPENDIX: Set theory

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Concepts of Modern Mathematics

(Dover Books on Mathematics)

Ian Stewart

Dover Publications | 1995 | páginas | pdf | 7,5 Mb

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In this charming volume, a noted English mathematician uses humor and anecdote to illuminate the concepts underlying "new math": groups, sets, subsets, topology, Boolean algebra, and more. According to Professor Stewart, an understanding of these concepts offers the best route to grasping the true nature of mathematics, in particular the power, beauty, and utility of pure mathematics. No advanced mathematical background is needed (a smattering of algebra, geometry, and trigonometry is helpful) to follow the author's lucid and thought-provoking discussions of such topics as functions, symmetry, axiomatics, counting, topology, hyperspace, linear algebra, real analysis, probability, computers, applications of modern mathematics, and much more.
By the time readers have finished this book, they'll have a much clearer grasp of how modern mathematicians look at figures, functions, and formulas and how a firm grasp of the ideas underlying "new math" leads toward a genuine comprehension of the nature of mathematics itself.

Some years ago, "new math" took the country's classrooms by storm. Based on the abstract, general style of mathematical exposition favored by research mathematicians, its goal was to teach students not just to manipulate numbers and formulas, but to grasp the underlying mathematical concepts. The result, at least at first, was a great deal of confusion among teachers, students, and parents. Since then, the negative aspects of "new math" have been eliminated and its positive elements assimilated into classroom instruction.

Contents
Preface to the Dover Edition v
Preface to the First Edition vii
1. Mathematics in General 1
2. Motion without Movement 8
3. Sbort Cuts in the Higher Arithmetic 27
4. The Language of Sets 43
S. What is a Function? 63
6. The Beginnings of Abstract Algebra 76
7. Symmetty: The Group Cona:pt 95
8. Axiomatics 113
9. Counting: Fmite and Infinite 127
10. Topology 144
11. The Power of Indirect Thinking 1S9
12. Topological Invariants 174
13. Algebraic Topology 189
14. Into Hyperspace 200
I5. Linear Algebra 215
16. Real Analysis 229
17. The Theory of Probability 244
18. Computers and Their Uses 2SS
19. Applications of Modem Mathematics 269
20. Foundations 286
Appendix 299
Notes 322
Glossary of Symbols 335

Index 337


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The Best Writing on Mathematics 2010

Mircea Pitici e William P. Thurston


Princeton University Press | 2011 | 434 páginas | pdf | 6,5 Mb

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This anthology brings together the year's finest writing on mathematics from around the world. Featuring promising new voices alongside some of the foremost names in mathematics, The Best Writing on Mathematics makes available to a wide audience many articles not easily found anywhere else--and you don't need to be a mathematician to enjoy them. These writings offer surprising insights into the nature, meaning, and practice of mathematics today. They delve into the history, philosophy, teaching, and everyday occurrences of math, and take readers behind the scenes of today's hottest mathematical debates. Here readers will discover why Freeman Dyson thinks some mathematicians are birds while others are frogs; why Keith Devlin believes there's more to mathematics than proof; what Nick Paumgarten has to say about the timing patterns of New York City's traffic lights (and why jaywalking is the most mathematically efficient way to cross Sixty-sixth Street); what Samuel Arbesman can tell us about the epidemiology of the undead in zombie flicks; and much, much more.

In addition to presenting the year's most memorable writing on mathematics, this must-have anthology also includes a foreword by esteemed mathematician William Thurston and an informative introduction by Mircea Pitici. This book belongs on the shelf of anyone interested in where math has taken us--and where it's headed.

TABLE OF CONTENTS:
Foreword by William P. Thurston xiIntroduction by Mircea Pitici xv
Mathematics Alive

The Role of the Untrue in Mathematics by Chandler Davis 3

Desperately Seeking Mathematical Proof by Melvyn B. Nathanson 13

An Enduring Error by Branko Grunbaum 18

What Is Experimental Mathematics? By Keith Devlin 32

What Is Information-Based Complexity? By Henryk Woz´niakowski 37

What Is Financial Mathematics? By Tim Johnson 43

If Mathematics Is a Language, How Do You Swear in It? By David Wagner 47
Mathematicians and the Practice of Mathematics

Birds and Frogs by Freeman Dyson 57

Mathematics Is Not a Game But . . . by Robert Thomas 79

Massively Collaborative Mathematics by Timothy Gowers and Michael Nielsen 89

Bridging the Two Cultures: Paul Valery by Philip J. Davis 94

A Hidden Praise of Mathematics by Alicia Dickenstein 99
Mathematics and Its Applications

Mathematics and the Internet: A Source of Enormous Confusion and Great Potential by Walter Willinger, David L. Alderson, and John C. Doyle 109

The Higher Arithmetic: How to Count to a Zillion without Falling Off the End of the Number Line by Brian Hayes 134

Knowing When to Stop: How to Gamble If You Must--The Mathematics of Optimal Stopping by Theodore P. Hill 145

Homology: An Idea Whose Time Has Come by Barry A. Cipra 158
Mathematics Education

Adolescent Learning and Secondary Mathematics by Anne Watson 163

Accommodations of Learning Disabilities in Mathematics Courses by Kathleen Ambruso Acker, Mary W. Gray, and Behzad Jalali 175

Audience,Style and Criticism by David Pimm and Nathalie Sinclair 194

Aesthetics as a Liberating Force in Mathematics Education? By Nathalie Sinclair 206

Mathematics Textbooks and Their Potential Role in Supporting Misconceptions by Ann Kajander and Miroslav Lovric 236

Exploring Curvature with Paper Models by Howard T. Iseri 247

Intuitive vs Analytical Thinking: Four Perspectives by Uri Leron and Orit Hazzan 260
History and Philosophy of Mathematics

Why Did Lagrange "Prove" the Parallel Postulate? By Judith V. Grabiner 283

Kronecker's Algorithmic Mathematics by Harold M. Edwards 303

Indiscrete Variations on Gian-Carlo Rota's Themes by Carlo Cellucci 311

Circle Packing: A Personal Reminiscence by Philip L. Bowers 330Applying Inconsistent Mathematics by Mark Colyvan 346

Why Do We Believe Theorems? By Andrzej Pelc 358
Mathematics in the Media

Mathematicians Solve 45-Year-Old Kervaire Invariant Puzzle by Erica Klarreich 373

Darwin: The Reluctant Mathematician by Julie Rehmeyer 377

Loves Me, Loves Me Not (Do the Math) by Steven Strogatz 380

The Mysterious Equilibrium of Zombies and Other Things Mathematicians See at the Movies by Samuel Arbesman 383

Strength in Numbers: On Mathematics and Musical Rhythm by Vijay Iyer 387

Math-hattan by Nick Paumgarten 391
Contributors 395

Acknowledgments 403

Credits 405

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Basic Maths Practice Problems For Dummies

Colin Beveridge

For Dummies | 2012 | 386 páginas | RAR - PDF | 4,6 Mb

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password: matav

In his popular Basic Maths For Dummies, professional maths tutor Colin Beveridge proved that he could turn anyone – even the most maths-phobic person – into a natural-born number cruncher. In this book he supplies more of his unique brand of maths-made- easy coaching, plus 2,000 practice problems to help you master what you learn. Whether you're prepping for a numeracy test or an employability exam, thinking of returning to school, or you'd just like to be one of those know-it-alls who says, 'Oh, that's easy!' about any maths problem that comes your way, this book is for you.

• Master basic arithmetic, fast – in no time, solving addition, subtraction, multiplication and division problems will seem as easy as tying your shoes
• Face down fractions – you'll never again feel shy around fractions, decimals, percentages and ratios
• Juggle weights and measures like a pro – whether it's a question of how much it weighs, how long (or far) it is, or how much it costs, you'll never be at a loss for an answer
• Make shapes your playthings – circles, squares, triangles and rectangles – you'll measure them, draw them and manipulate them with ease

Open the book and find:

• 2,000 pencil-and-paper practice problems
• The keys to mastering addition, subtraction, multiplication and division
• The lowdown on fractions, decimals and percentages
• Basic geometry made easy
• How to handle weights, measures and money problems
• How to read charts, tables and graphs at a glance

Learn to:

• Master maths with more than 2,000 practice questions
• Add, subtract, multiply and divide with confidence
• Work with decimals, fractions and percentages
• Size up weights and measures

Managing Mathematical Projects - with Success!

P. P. G. Dyke 

Springer | 2003 | 285 páginas  | PDF | 3 Mb

lib.free-college.org
lib.free-college.org

The first student-centred guide on how to write projects and case studies in mathematics, with particular attention given to working in groups (something maths undergraduates have not traditionally done). With half of all universities in the UK including major project work of significant importance, this book will be essential reading for all students on the second or final year of a mathematics degree, or on courses with a high mathematical content, for example, physics and engineering.

The Heart of Mathematics: An Invitation to Effective Thinking

(Key Curriculum Press)

Edward B. Burger, Michael Starbird

3.ª edição 

Wiley | 2009 | 976 paginas | PDF | 40,2 Mb

link
rapidgator.net



The Heart of Mathematics: An invitation to effective thinking --now in its second edition--succeeds at reaching non-math, non-science-oriented readers and encourages them to discover the mathematics inherent in the world around them. Infused throughout with the authors' humor and enthusiasm, The Heart of Mathematics introduces readers to the most important and interesting ideas in mathematics while inspiring them to actively engage in mathematical thinking. 

CHAPTER ONE: Fun and Games: An introduction to rigorous thought

CHAPTER TWO: Number Contemplation
Section 2.1. Counting [Pigeonhole principle].
Section 2.2. Numerical Patterns in Nature: [Fibonacci numbers].
Section 2.3. Prime cuts of numbers [Prime numbers].
Section 2.4. Crazy clocks and checking out bars [Modular arithmetic].
Section 2.5. Secret Codes and How to Become a Spy [RSA public key cryptography].
Section 2.6. The irrational side of numbers [Irrational numbers].
Section 2.7. Get real [The real number line].

CHAPTER THREE: Infinity
Section 3.1. Beyond Numbers [An introduction to one-to-one correspondence].
Section 3.2. Comparing the Infinite [Examples of one-to-one correspondences].
Section 3.3. The Missing Member [Cantor's diagonalization proof that |N|<|R|].
Section 3.4. Travels Toward the Stratosphere of Infinities [Power set theorem].
Section 3.5. Straightening up the circle [Geometrical correspondences].

CHAPTER FOUR: Geometric Gems
Section 4.1. Pythagoras and his hypotenuse [Blaskara's elegant proof].
Section 4.2. A view of an art gallery [A view-obstruction question from computational geometry].
Section 4.3. The sexiest rectangle [The Golden Rectangle].
Section 4.4. Soothing symmetry and spinning pinwheels [Aperiodic tilings].
Section 4.5. The Platonic Solids Turn Amorous [Symmetry and duality in the Platonic Solids].
Section 4.6. The shape of reality? [Non-Euclidean geometries].
Section 4.7. The Fourth Dimension [Geometry through analogy].

CHAPTER FIVE: Contortions of Space
Section 5.1. Rubber sheet geometry [Topological equivalence by distortion].
Section 5.2. The Band That Wouldn't Stop Playing [Möbius Band and Klein Bottle].
Section 5.3. Circuit training. [The Euler circuit theorem].
Section 5.4. Feeling edgy? [The Euler characteristic].
Section 5.5. Knots and links [A little knot theory].
Section 5.6. Fixed Points, Hot Loops, and Rainy Days [The Brouwer Fixed Point Theorem].

CHAPTER SIX: Fractals and Chaos
Section 6.1. Images [A gallery of fractals].
Section 6.2. The infinitely detailed beauty of fractals [Creating fractals through repeated processes].
Section 6.3. Between dimensions [Fractal dimension].
Section 6.4. The mysterious art of imaginary fractals [Julia and Mandelbrot Sets].
Section 6.5. The Dynamics of Change [Repeated applications of simple processes].
Section 6.6. Predetermined chaos [Deterministic chaos].

CHAPTER SEVEN: Taming Uncertainty
Section 7.1. Chance surprises [Unexpected scenarios involving chance].
Section 7.2. Predicting the future in an uncertain world [Probability].
Section 7.3. Random thoughts [Coincidences].
Section 7.4. Down for the count [Systematic counting].
Section 7.5. Dizzling, Defending, and Doctoring [Probability of Precipitation, game theory, Bayesian probability]

CHAPTER EIGHT: Meaning from Data
Section 8.1. Stumbling Through a Minefield of Data [Pitfalls of statistics].
Section 8.2. Getting Your Data to Shape Up [Organizing, describing, and summarizing data]
Section 8.3. Looking at Super Models [Mathematically described distributions]
Section 8.4. Go Figure [Making inferences from data, hypothesis testing]
Section 8.5. War, Sports, and Tigers [Cause and effect and correlation, Simpson's Paradox, famous applications of inference]

CHAPTER NINE: Deciding Wisely
Section 9.1. Great Expectations [Expected value]
Section 9.2. Risk [Deciding personal and public safety]
Section 9.3. Money Matters [Compound interest]
Section 9.4. Peril at the polls [voting]
Section 9.5. Cutting cake for greedy people [fair division]

Biography

Edward B. Burger is professor of mathematics in the Department of Mathematics and Statistics at Williams College. He graduated from Connecticut College in 1985, where he earned B.A. Summa Cum Laude with Distinction in Mathematics, and received his Ph.D. in mathematics from the University of Texas at Austin. He did his postdoctoral work at the University of Waterloo in Canada. Dr. Burger has received numerous awards including:  the Award of Excellence, for "educational mathematics videos that break new ground from the Technology & Learning magazine, and theDistinguished Achievement Award, for Educational Video Technology from The Association of Educational Publishers.  He was honored as one of the "100 Best of America", Listed in Reader's Digest's Annual Special Issue as Best Math Teacher.  He also received the Residence Life Academic Teaching Award, University of Colorado at Boulder and the Robert W. Hamilton Book Award, for "The Heart of Mathematics".

Michael Starbird Michael Starbird is a University Distinguished Teaching Professor of Mathematics at The University of Texas at Austin. He has received more than a dozen teaching awards including the Mathematical Association of America’s 2007 national teaching award and several university-wide teaching awards based largely on his course in mathematics for liberal arts students. Starbird brings intriguing mathematics to general audiences through his classes, lectures, books, and video courses. In 1989, Starbird was UT’s Recreational Sports Super Racquets Champion.

Mathematical Omnibus: Thirty Lectures on Classic Mathematics

Dmitry Fuchs, Serge Tabachnikov

American Mathematical Society | 465 páginas | PDF | 7 Mb

links diretos:

math.psu.edu
f3.tiera.ru

rapidshare.com

filefactory.com

scribd.com

rapidgator.net

The book consists of thirty lectures on diverse topics, covering much of the mathematical landscape rather than focusing on one area. The reader will learn numerous results that often belong to neither the standard undergraduate nor graduate curriculum and will discover connections between classical and contemporary ideas in algebra, combinatorics, geometry, and topology. The reader's effort will be rewarded in seeing the harmony of each subject. The common thread in the selected subjects is their illustration of the unity and beauty of mathematics. Most lectures contain exercises, and solutions or answers are given to selected exercises. A special feature of the book is an abundance of drawings (more than four hundred), artwork by an award-winning artist, and about a hundred portraits of mathematicians. Almost every lecture contains surprises for even the seasoned researcher.

The Little Book of Maths Theorems, Theories & Things

Surendra Verma

Orient Publishing | 2008 | 168 páginas | PDF | 3,25 Mb

link (password : matav)

Somebody came up to Ralph P Boas, Jr. (1912-92), a distinguished American mathematician, after a talk he had given, and said,'You make mathematics seem like fun.' Boas was inspired to reply: 'If it isn't fun, why do it?'

Mathematics is indeed fun as this little book testifies. This book presents a unique collection of mathematical ideas, theories, theorems, conjectures, rules, facts, equations, formulas, paradoxes, fallacies and puzzles with short, simple and witty explanation that require no background in mathematics. It is peppered with anecdotes, quotes, limericks and poems showing the quirky and amusing side of mathematics and of people who have added, in the words of Roger Bacon (1214-92), 'things to this world which cannot be made known without knowledge of mathematics'.

Mathematical Handbook Elementary Mathematics

M. Vygodsky 

Mir Publishers | 1984 | 422 Páginas | PDF | 9 Mb

link
uploading.com

However, it is well to bear in mind that neither handbook, nor textbook alone suffices  to give the reader a knowledge of the subject: he must use pencil and paper and work through the examples and problems for himself.

This is how the the preface of the Mathematical Handbook – Elementary Mathematics by M. Vygodsky ends. The book has about 420 pages and was first published in 1979 by Mir Publishers.

The book has following sections

Tables, Arithmetic, Algebra, Geometry (Plane and Solid), Trigonometry, Functions and Graphs

Ingenuity in Mathematics

Ross Honsberger

The Mathematical Association of America | 1998 | 216 páginas | PDF | 5 Mb

link (pass: matav)

Djvu | 2,1 Mb

link direto
link
depositfiles.com


The nineteen essays here illustrate many different aspects of mathematical thinking. The author is very well-known for his best-selling books of problems; in this volume he seeks to share his appreciation of the elegant and ingenious approaches used in thinking about even elementary mathematics. Standard high school courses in algebra and geometry furnish a sufficient basis for understanding each essay. Topics include number theory, geometry, combinatorics, logic and probability, and the methods used often involve an interaction between these disciplines. Some of the essays are easy to read, others more challenging; some of the exercises are routine, others lead the reader deeper into the subject. 

A smoother pebble: Mathematical explorations

Donald C. Benson

Oxfоrd Univеrsity Prеss | 266 páginas | 2003 | PDF | 9,74 Mb

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This book takes a novel look at the topics of school mathematics--arithmetic, geometry, algebra, and calculus. In this stroll on the mathematical seashore we hope to find, quoting Newton, "...a smoother pebble or a prettier shell than ordinary..." This book assembles a collection of mathematical pebbles that are important as well as beautiful.

Introduction

I. BRIDGING THE GAP

1. The Ancient Fractions

2. Greek Gifts

3. The Music of the Ratios

II. THE SHAPE OF THINGS

4. Tubeland

5. The Calculating Eye

III. THE GREAT ART

6. Algebra Rules

7. The Root of the Problem

8. Symmetry Without Fear

9. The Magic Mirror

IV. A SMOOTHER PEBBLE

10. On the Shoulders of Giants

11. Six-Minute Calculus

12. Roller-Coaster Science

Glossary

References

Index

Foundation Mathematics

L. R. Mustoe, M. D. J. Barry

Wiley-Blackwell | 1998 | 668 Páginas | PDF | 14,78 Mb

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Mathematics is finding ever wider areas of application as we seek to understand more about the way in which the natural world and the man-made environment operate and interact. In addition to the traditional use of mathematical models as design tools and for the prediction of the behaviour of many phenomena, mathematicsis increasingly being used to model situations in many other disciplines including finance, management, politics and geography. Foundation Mathematics begins with a concise summary of arithmetic, basic algebra and a discussion of quadratics and cubics, strongly emphasising geometric ideas. Then follow the principles of Euclidean and Cartesian geometry and the concept of proof. Next are trigonometry, further algebra, functions and their inverses. Finally, the concepts of differential and integral calculus are introduced. Each chapter starts with a list of learning objectives and ends with a summary of key points and results. A generous supply of worked examplesincorporating motivating applications is designed to build knowledge and skill. Theexercises provided range in difficulty to aid consolidation and facilitate revision.Answers to the exercises, some including helpful hints, are placed at the end of each chapter. Foundation Mathematics together with its sequel Mathematics in Engineering and Science take the reader forward, in both content and style, from a level close to UK GCSE mathematics and its international equivalents to first year university-levelmathematics. The concise and focused approach will help the student build the necessary confidence to tackle the more advanced ideas of the authors related bookMathematics in Engineering and Science (Wiley, 1998). This no-nonsense textbookwill enable students to gain a basic grounding in the foundations of mathematics and will enable them to approach further study with confidence.
Covers the foundation of mathematics needed by students (without the dreaded "C" word---CALCULUS!). All major topics are included: arithmetic, basic algebra,geometry, trigonometry, and functions while only touching on the basics of calculus.

Mathematicians Delight

W.W. Sawyer

1943 | Harmondsworth, Middlesex, Eng., New York, Penguin Books

online: archive.org

Penguin Books | 1969 | 238 páginas | djvu

link

An introduction to mathematics which starts with simple arithmetic and algebra and proceeds through to graphs, logarithms, trigonometry to calculus and imaginary numbers. The author, who is internationally renowned for his innovative teaching methods, offers insights into the pleasures of mathematics that will appeal to readers of all backgrounds.

Breakthroughs in Mathematics

Peter Wolff

New American Library | 1970 | 276 páginas | djvu | 3,15 Mb

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PDF - 40Mb
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scribd.com

CONTENTS
INTRODUCTION
I GEOMETRY
CHAPTER ONE
Euclid—The Beginnings of Geometry 15
CHAPTER TWO
Lobachevski—Non-Euclidean Geometry 63
CHAPTER THREE
Descartes—Geometry and Algebra Joined 96
II ARITHMETIC
CHAPTER FOUR
Archimedes—Numbers and Counting 113
CHAPTER FIVE
Dedekind—Irrational Numbers 138
CHAPTER SIX
Russell—The Definition of Number 161

III ADVANCED TOPICS
CHAPTER SEVEN
Euler—A New Branch of  Mathematics: Topology 197
CHAPTER EIGHT
Laplace—The Theory of Probability 218
CHAPTER NINE
Boole—Algebra and Logic Joined 242
SUGGESTIONS FOR FURTHER READING 277
INDEX 279

Mathematics, its magic and mastery

Aaron Bakst

D. Van Nostrand Company | 1945 | 790 páginas | DJVU | 16,6 Mb

link direto
link

This book is a great way to learn some of the interesting aspects of numbers. Among some of the subjects covered include numbering systems, ciphers, angles, dimensions, and of course, algebra. This book is designed to make mathematics interesting. The science is not treated formally; abstract conceptions and uninteresting and abstruse procedures are completely avoided; yet at the same time the book is sufficiently complete so as to give a broad picture of mathematical fundamentals. Mathematics, in order to be appreciated by those who do not have a flare for the intangible must be seen in the light of its versatility in the various fields of human endeavor. This is very much the central theme of this book

Here are the chapter headings:

Numerals and Numeration (History and use of numerals)
Systems of Numerations (one of the best descriptions of other bases I have ever seen)
Some Remarkable Properties of Numbers
Number Giants (large nos, history including Archimedes contribution)
Number Pygmies
There's Secrecy in Numbers (cryptography, codes, etc.)
The Arithmetic of Measurement
Simple Calculating Devices (abacus, soroban, etc.)
Rapid Calculations ("human calculators", math tricks)
Problems and Puzzles (actual cases)
How the Number Magician Does It
Algebra & Its Numbers (the algebra chapters all contain many problems in real applications)
The Algebra of Number Giants and Pygmies
The Grammar of Algebra
Algebra, Boss of Arithmetic
Algebra Looks at Installment Buying
Chain Letter Algebra
Streamlining Everyday Computation (logs)
The Bankers Number - Jack of All Trades (net present value, future value, compounding in other ares such as growth and aging)
How to Have Fun with Lady Luck (probability)
The Thinking Machines
Postoffice Mathematics (relativity, probability, geometry)
New Worlds for Old (zero dimension, Flatland, geometry meets arithmetic and algebra)
Passport for Geometric Figures (geometry and movement)
Man's Servant - The Triangle
The Triangle - Man's Master
Circles, Angles & an Age Old Problem (pi)
The Mathematics of Seeing (optical illusion)
The Lost Horizon (curvature of the earth, natural illusions)
The Shape of Things (geometry beyond circles and triangles)
The Size of Things (three dimensions, area of geometric figures)
Escape from Flatland (solid geometry)
How Algebra Saves Geometry (applications of geometry & algebra)
Cork-Screw Geometry (distances between points in solid geometry)
Mathematics, Interpreter of the Universe (geometry in astronomy / geology, physics)
The Firing Squad & Mathematics (math and movement)
Of Math and Magic (more math & motion, time and distance)

An Introduction to Mathematics

Alfred North Whitehead

1911 | London : Williams & Northgate: [New York, H. Holt 

online:  archive.org

This distinguished little book is a brisk introduction to a series of mathematical concepts, a history of their development, and a concise summary of how today's reader may use them.

The Nature of Mathematics

Karl J. Smith

12.ª Edição

Brooks Cole | 2011 - 12ª edição | 1024 páginas | PDF | 84,2 Mb

link
link1

Experience mathematics--and develop problem-solving skills that will benefit you throughout your life--with THE NATURE OF MATHEMATICS. Karl Smith introduces you to proven problem-solving techniques and shows you how to use these techniques to solve unfamiliar problems that you encounter in your day-to-day world. You'll find coverage of interesting historical topics, and practical applications to real-world settings and situations, such as finance (amortization, installment buying, annuities) and voting. With Smith's guidance, you'll both understand mathematical concepts and master the techniques.

Contents
Prologue: Why Math? A Historical Overview
THE NATURE OF PROBLEM SOLVING 2
1.1 Problem Solving 3
1.2 Inductive and Deductive Reasoning 18
1.3 Scientific Notation and Estimation 28
1.4 Chapter 1 Summary 43
THE NATURE OF SETS 48
2.1 Sets, Subsets, and Venn Diagrams 49
2.2 Operations with Sets 59
2.3 Applications of Sets 64
2.4 Finite and Infinite Sets 72
2.5 Chapter 2 Summary 79
THE NATURE OF LOGIC 82
3.1 Deductive Reasoning 83
3.2 Truth Tables and the Conditional 91
3.3 Operators and Laws of Logic 100
3.4 The Nature of Proof 107
3.5 Problem Solving Using Logic 116
3.6 Logic Circuits 124
3.7 Chapter 3 Summary 129
THE NATURE OF NUMERATION SYSTEMS 134
4.1 Early Numeration Systems 135
4.2 Hindu-Arabic Numeration System 144
4.3 Different Numeration Systems 148
4.4 Binary Numeration System 154
4.5 History of Calculating Devices 159
4.6 Chapter 4 Summary 170
THE NATURE OF NUMBERS 174
5.1 Natural Numbers 175
5.2 Prime Numbers 183
5.3 Integers 197
5.4 Rational Numbers 205
5.5 Irrational Numbers 211
5.6 Groups, Fields, and Real Numbers 220
5.7 Discrete Mathematics 231
5.8 Cryptography 240
5.9 Chapter 5 Summary 245
THE NATURE OF ALGEBRA 250
6.1 Polynomials 251
6.2 Factoring 258
6.3 Evaluation, Applications, and Spreadsheets 264
6.4 Equations 274
GUEST ESSAY: “CHAOS”
6.5 Inequalities 283
6.6 Algebra in Problem Solving 288
6.7 Ratios, Proportions, and Problem Solving 300
6.8 Percents 308
6.9 Modeling Uncategorized Problems 317
6.10 Chapter 6 Summary 326
THE NATURE OF GEOMETRY 330
7.1 Geometry 331
7.2 Polygons and Angles 340
7.3 Triangles 349
7.4 Similar Triangles 356
7.5 Right-Triangle Trigonometry 363
7.6 Mathematics, Art, and Non-Euclidean Geometries 370
7.7 Chapter 7 Summary 384
THE NATURE OF NETWORKS AND GRAPH THEORY 388
8.1 Euler Circuits and Hamiltonian Cycles 389
8.2 Trees and Minimum Spanning Trees 402
8.3 Topology and Fractals 413
GUEST ESSAY: “WHAT GOOD ARE FRACTALS?”
8.4 Chapter 8 Summary 421
THE NATURE OF MEASUREMENT 426
9.1 Perimeter 427
9.2 Area 435
9.3 Surface Area, Volume, and Capacity 445
9.4 Miscellaneous Measurements 456
9.5 U.S.–Metric Conversions 467
9.6 Chapter 9 Summary 468
THE NATURE OF GROWTH 472
10.1 Exponential Equations 473
10.2 Logarithmic Equations 482
10.3 Applications of Growth and Decay 490
10.4 Chapter 10 Summary 500
THE NATURE OF FINANCIAL MANAGEMENT 502
11.1 Interest 503
11.2 Installment Buying 517
11.3 Sequences 526
11.4 Series 538
11.5 Annuities 548
11.6 Amortization 555
11.7 Summary of Financial Formulas 562
11.8 Chapter 11 Summary 567
THE NATURE OF COUNTING 572
12.1 Permutations 573
12.2 Combinations 582
12.3 Counting without Counting 590
12.4 Rubik’s Cube and Instant Insanity 598
12.5 Chapter 12 Summary 602

Foundations and Fundamental Concepts of Mathematics

Howard Eves

1997 | 368 Pages | PDF | 10 Mb

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Third edition of popular undergraduate-level text offers overview of historical roots and evolution of several areas of mathematics. Topics include mathematics before Euclid, Euclid's Elements, non-Euclidean geometry, algebraic structure, formal axiomatics, sets, and more. Emphasis on axiomatic procedures. Problems. Solution Suggestions for Selected Problems. Bibliography.

The Survival of a Mathematician: From Tenure to Emeritus

Steven G. Krantz

American Mathematical Society | 2008 | 310 páginas | PDF

versão draft - online:
math.wustl.edu
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A successful mathematical career involves doing good mathematics, to be sure, but also requires a wide range of skills that are not normally taught in graduate school. The purpose of this book is to provide guidance to the professional mathematician in how to develop and survive in the profession. There is information on how to begin a research program, how to apply for a grant, how to get tenure, how to teach, and how to get along with one's colleagues. After tenure, there is information on how to direct a Ph.D. student, how to serve on committees, and how to serve in various posts in the math department. There is extensive information on how to serve as Chairman. There is also material on trouble areas: sexual harassment, legal matters, disputes with colleagues, dealing with the dean, and so forth. One of the themes of the book is how to have a fulfilling professional life. In order to achieve this goal, Krantz discusses keeping a vigorous scholarly program going and finding new challenges, as well as dealing with the everyday tasks of research, teaching, and administration. In short, this is a survival manual for the professional mathematician--both in academics and in industry and government agencies. It is a sequel to the author's A Mathematician's Survival Guide.

Math for Grownups: Re-Learn the Arithmetic You Forgot From School So You Can

Laura Laing

Adams Media | 2011 | 256 páginas | PDF | 1,4 Mb

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Ever wish you'd paid more attention in math class? From third grade to senior year of high school, it went in one ear and out the other, didn't it?

But now you're staring at the new washer and dryer, trying to figure out the percentage of sales tax on the purchase price. You multiply something by something, right? Or you're scratching your head, wondering how to compute the odds that your football team will take next Sunday's game. You're pretty sure that involved ratios. The problem is, you can't quite remember.

Here you get an adult refresher and real-life context--with examples ranging from how to figure out how many shingles it takes to re-roof the garage to the formula for resizing Mom's tomato sauce recipe for your entire family.

Forget higher calculus--you just need an open mind. And with this practical guide, math can stop being scary and start being useful.