Pi - Unleashed

Jörg Arndt, Christoph Haenel, C. Lischka and D. Lischka

Spinger | 2001 | PDF | 288 páginas | 10,7 MB

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In the 4,000-year history of research into Pi, results have never been as prolific as present. This book describes, in easy-to-understand language, the latest and most fascinating findings of mathematicians and computer scientists in the field of Pi. Attention is focused on new methods of high-speed computation.


Why the flood of books on pi (do a search, you'll see)? And why calculate its decimal expansion to enormous numbers of places? Is number mysticism having a revival?

Certainly there are many fascinating theorems involving pi, which is one of the two most important transcendental numbers (the other being e) and which shows up unexpectedly in many different branches of mathematics. These books are well worth reading to learn those theorems, those lovely, unexpected formulas, and the interesting history.

If you are a trained mathematician, the best of these books by far is the recent one by Eymard and Lafon, but it is very difficult.

My complaint about all these books is that not one of them proves that pi exists! I mean pi is defined as the ratio of the circumference to the diameter of any circle; in order for that definition to make sense, one must prove that ratio to be constant. But that ratio is only constant in Euclidean geometry, not hyperbolic or elliptic geometries, so the proof depends on the Euclidean parallel postulate and is not at all obvious.

There is a proof in the book by Moise "Elementary Geometry from an Advanced Viewpoint."

This book is a good one, its main competition being the good one by Posamentier and Lehmann.

Intentional Mathematics

 (Studies in Logic and the Foundations of Mathematics)

Stewart Shapiro

Elsevier Science Ltd | 1985 | djvu | 1,38 Mb

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Platonism and intuitionism are rival philosophies of Mathematics, the former holding that the subject matter of mathematics consists of abstract objects whose existence is independent of the mathematician, the latter that the subject matter consists of mental construction...both views are implicitly opposed to materialistic accounts of mathematics which take the subject matter of mathematics to consist (in a direct way) of material objects...'' FROM THE INTRODUCTION Among the aims of this book are: - The discussion of some important philosophical issues using the precision of mathematics. - The development of formal systems that contain both classical and constructive components. This allows the study of constructivity in otherwise classical contexts and represents the formalization of important intensional aspects of mathematical practice. - The direct formalization of intensional concepts (such as computability) in a mixed constructive/classical context.

Philosophy of Mathematics: Structure and Ontology

Stewart Shapiro

Oxford University Press | 1997 | 296 páginas | PDF | 907 kb

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Do numbers, sets, and so forth, exist? What do mathematical statements mean? Are they literally true or false, or do they lack truth values altogether? Addressing questions that have attracted lively debate in recent years, Stewart Shapiro contends that standard realist and antirealist accounts of mathematics are both problematic.

As Benacerraf first noted, we are confronted with the following powerful dilemma. The desired continuity between mathematical and, say, scientific language suggests realism, but realism in this context suggests seemingly intractable epistemic problems. As a way out of this dilemma, Shapiro articulates a structuralist approach. On this view, the subject matter of arithmetic, for example, is not a fixed domain of numbers independent of each other, but rather is the natural number structure, the pattern common to any system of objects that has an initial object and successor relation satisfying the induction principle. Using this framework, realism in mathematics can be preserved without troublesome epistemic consequences.

Shapiro concludes by showing how a structuralist approach can be applied to wider philosophical questions such as the nature of an "object" and the Quinean nature of ontological commitment. Clear, compelling, and tautly argued, Shapiro's work, noteworthy both in its attempt to develop a full-length structuralist approach to mathematics and to trace its emergence in the history of mathematics, will be of deep interest to both philosophers and mathematicians.

The Oxford Handbook of Philosophy of Mathematics and Logic

Stewart Shapiro

O U P | 2005 | 856 Páginas | PDF | 3 MB
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Mathematics and logic have been central topics of concern since the dawn of philosophy. Since logic is the study of correct reasoning, it is a fundamental branch of epistemology and a priority in any philosophical system. Philosophers have focused on mathematics as a case study for general philosophical issues and for its role in overall knowledge- gathering. Today, philosophy of mathematics and logic remain central disciplines in contemporary philosophy, as evidenced by the regular appearance of articles on these topics in the best mainstream philosophical journals; in fact, the last decade has seen an explosion of scholarly work in these areas.

This volume covers these disciplines in a comprehensive and accessible manner, giving the reader an overview of the major problems, positions, and battle lines. The 26 contributed chapters are by established experts in the field, and their articles contain both exposition and criticism as well as substantial development of their own positions. The essays, which are substantially self-contained, serve both to introduce the reader to the subject and to engage in it at its frontiers. Certain major positions are represented by two chapters--one supportive and one critical.

The Oxford Handbook of Philosophy of Math and Logic is a ground-breaking reference like no other in its field. It is a central resource to those wishing to learn about the philosophy of mathematics and the philosophy of logic, or some aspect thereof, and to those who actively engage in the discipline, from advanced undergraduates to professional philosophers, mathematicians, and historians. 

Intermediate Algebra: Graphs and Models

4.ª Edição 

Marvin L. Bittinger, David J. Ellenbogen, Barbara L. Johnson 

Adison Weley | 2012 | 960 páginas | PDF | 73,8 MB

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The Bittinger Graphs and Models Series helps readers learn algebra by making connections between mathematical concepts and their real-world applications. Abundant applications, many of which use real data, offer students a context for learning the math. The authors use a variety of tools and techniques—including graphing calculators, multiple approaches to problem solving, and interactive features—to engage and motivate all types of learners.

Table of Contents

Preface

1. Basics of Algebra and Graphing
1.1 Some Basics of Algebra
1.2 Operations with Real Numbers
1.3 Equivalent Algebraic Expressions
1.4 Exponential Notation and Scientific Notation
            Mid-Chapter Review
1.5 Graphs
1.6 Solving Equations and Formulas
1.7 Introduction to Problem Solving and Models
            Summary and Review
            Test

2. Functions, Linear Equations, and Models
2.1 Functions
2.2 Linear Functions: Slope, Graphs, and Models
2.3 Another Look at Linear Graphs
2.4 Introduction to Curve Fitting: Point-Slope Form
            Mid-Chapter Review
2.5 The Algebra of Functions
            Summary and Review
            Test

3. Systems of Linear Equations and Problem Solving
3.1 Systems of Equations in Two Variables
3.2 Solving by Substitution or Elimination
3.3 Solving Applications: Systems of Two Equations
            Mid-Chapter Review
3.4 Systems of Equations in Three Variables
3.5 Solving Applications: Systems of Three Equations
3.6 Elimination Using Matrices
3.7 Determinants and Cramer's Rule
3.8 Business and Economics Applications
            Summary and Review
            Test

Cumulative Review: Chapters 1—3

4. Inequalities
4.1 Inequalities and Applications
4.2 Solving Equations and Inequalities by Graphing
4.3 Intersections, Unions, and Compound Inequalities
4.4 Absolute-Value Equations and Inequalities
            Mid-Chapter Review
4.5 Inequalities in Two Variables
            Summary and Review
            Test

5. Polynomials and Polynomial Functions
5.1 Introduction to Polynomials and Polynomial Functions
5.2 Multiplication of Polynomials
5.3 Polynomial Equations and Factoring
5.4 Trinomials of the Type x² + bx + c
5.5 Trinomials of the Type ax² + bx + c
5.6 Perfect-Square Trinomials and Differences of Squares
5.7 Sums or Differences of Cubes
            Mid-Chapter Review
5.8 Applications of Polynomial Equations
            Summary and Review
            Test

6. Rational Expressions, Equations, and Functions
6.1 Rational Expressions and Functions: Multiplying and Dividing
6.2 Rational Expressions and Functions: Adding and Subtracting
6.3 Complex Rational Expressions
            Mid-Chapter Review
6.4 Rational Equations
6.5 Applications Using Rational Equations
6.6 Division of Polynomials
6.7 Synthetic Division
6.8 Formulas, Applications, and Variation
            Summary and Review
            Test

Cumulative Review: Chapters 1—6

7. Exponents and Radical Functions
7.1 Radical Expressions, Functions, and Models
7.2 Rational Numbers as Exponents
7.3 Multiplying Radical Expressions
7.4 Dividing Radical Expressions
7.5 Expressions Containing Several Radical Terms
            Mid-Chapter Review
7.6 Solving Radical Equations
7.7 The Distance Formula, the Midpoint Formula, and Other Applications
7.8 The Complex Numbers
            Summary and Review
            Test

8. Quadratic Functions and Equations
8.1 Quadratic Equations
8.2 The Quadratic Formula
8.3 Studying Solutions of Quadratic Equations
8.4 Studying Solutions of Quadratic Equations
8.5 Equations Reducible to Quadratic
            Mid-Chapter Review
8.6 Quadratic Functions and Their Graphs
8.7 More About Graphing Quadratic Functions
8.8 Problem Solving and Quadratic Functions
8.9 Polynomial Inequalities and Rational Inequalities
            Summary and Review
            Test

9. Exponential Functions and Logarithmic Functions
9.1 Composite Functions and Inverse Functions
9.2 Exponential Functions
9.3 Logarithmic Functions
9.4 Properties of Logarithmic Functions
            Mid-Chapter Review
9.5 Natural Logarithms and Changing Bases
9.6 Solving Exponential and Logarithmic Equations
9.7 Applications of Exponential and Logarithmic Functions
            Summary and Review
            Test

Cumulative Review: Chapters 1—9

10. Conic Sections
10.1 Conic Sections: Parabolas and Circles
10.2 Conic Sections: Ellipses
10.3 Conic Sections: Hyperbolas
            Mid-Chapter Review
10.4 Nonlinear Systems of Equations
            Summary and Review
            Test

11. Sequences, Series, and the Binomial Theorem
11.1 Sequences and Series
11.2 Arithmetic Sequences and Series
11.3 Geometric Sequences and Series
            Mid-Chapter Review
11.4 The Binomial Theorem
            Summary and Review
            Test

Cumulative Review: Chapters 1-11

Answers
Glossary
Photo Credits
Index
Index of Applications

Precalculus: Functions and Graphs

12.ª edição

Earl Swokowski, Jeffery Cole
Brooks Cole  | 2011 | 796 pages | PDF | 13 MB
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The latest edition of Swokowski and Cole's PRECALCULUS: FUNCTIONS AND GRAPHS retains the elements that have made it so popular with instructors and students alike: clear exposition, an appealing and uncluttered layout, and applications-rich exercise sets. The excellent, time-tested problems have been widely praised for their consistency and their appropriate level of difficulty for precalculus students. The book also provides calculator examples, including specific keystrokes that show students how to use various graphing calculators to solve problems more quickly. The Twelfth Edition features updated topical references and data, and continues to be supported by outstanding technology resources. Mathematically sound, this book effectively prepares students for further courses in mathematics.

Each chapter ends with Review Exercises, Discussion Exercises, and a Chapter Test.
1. TOPICS FROM ALGEBRA.
Real Numbers. Exponents and Radicals. Algebraic Expressions. Equations. Complex Numbers. Inequalities.
2. FUNCTIONS AND GRAPHS.
Rectangular Coordinate Systems. Graphs of Equations. Lines. Definition of Function. Graphs of Functions. Quadratic Functions. Operations on Functions.
3. POLYNOMIAL AND RATIONAL FUNCTIONS.
Polynomial Functions of Degree Greater Than 2. Properties of Division. Zeros of Polynomials. Complex and Rational Zeros of Polynomials. Rational Functions. Variation.
4. INVERSE, EXPONENTIAL, AND LOGARITHMIC FUNCTIONS.
Inverse Functions. Exponential Functions. The Natural Exponential Function. Logarithmic Functions. Properties of Logarithms. Exponential and Logarithmic Equations.
5. TRIGONOMETRIC FUNCTIONS.
Angles. Trigonometric Functions of Angles. Trigonometric Functions of Real Numbers. Values of the Trigonometric Functions. Trigonometric Graphs. Additional Trigonometric Graphs. Applied Problems.
6. ANALYTIC TRIGONOMETRY.
Verifying Trigonometric Identities. Trigonometric Equations. The Additions and Subtraction of Formulas. Multiple-Angle Formulas. Product-To-Sum and Sum-To-Product Formulas. The Inverse Trigonometric Functions.
7. APPLICATIONS OF TRIGONOMETRY.
The Law of Sines. The Law of Cosines. Vectors. The Dot Product. Trigonometric Form for Complex Numbers. De Moivre's Theorem and nth Roots of Complex Numbers.
8. SYSTEMS OF EQUATIONS AND INEQUALITIES.
Systems of Equations. Systems of Linear Equations in Two Variables. Systems of Inequalities. Linear Programming. Systems of Linear Equations in More Than Two Variables. The Algebra of Matrices. The Inverse of a Matrix. Determinants. Properties of Determinants. Partial Fractions.
9. SEQUENCES, SERIES, AND PROBABILITY.
Infinite Sequences and Summation Notation. Arithmetic Sequences. Geometric Sequences. Mathematical Induction. The Binomial Theorem. Permutations. Distinguishable Permutations and Combinations. Probability.
10. TOPICS FROM ANALYTICAL GEOMETRY.
Parabolas. Ellipses. Hyperbolas. Plane Curves and Parametric Equations. Polar Coordinates. Polar Equations of Conics.
Appendix I: Common Graphs and Their Equations.
Appendix II: A Summary of Graph Transformations.
Appendix III: Graphs of the Trigonometric Functions and Their Inverses.
Appendix IV: Values of the Trigonometric Functions of Special Angles on a Unit Circle.

Prealgebra

4.ª Edição

Tom Carson

Pearson | 2012 | 720 páginas | PDF | 15 Mb

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Tom Carson's Prealgebra, Fourth Edition addresses individual learning styles with a complete study system to meet the needs of today’s students. Carson’s Study System, presented in the “To the Student” section at the front of the text, adapts to the way each student learns, and targeted learning strategies are presented throughout the book to guide students to success. Tom speaks to students in everyday language and walks them through the concepts, explaining not only how to do the math, but also where the concepts come from and why they work.

Table of Contents

1. Whole Numbers
1.1 Introduction to Numbers, Notation, and Rounding
1.2 Adding and Subtracting Whole Numbers; Solving Equations
1.3 Multiplying Whole Numbers; Exponents
1.4 Dividing Whole Numbers; Solving Equations
1.5 Order of Operations; Mean, Median, and Mode
1.6 More with Formulas
Summary and Review
Practice Test

2. Integers
2.1 Introduction to Integers
2.2 Adding Integers
2.3 Subtracting Integers and Solving Equations
2.4 Multiplying and Dividing Integers; Exponents; Square Roots; Solving Equations
2.5 Order of Operations
2.6 Additional Applications and Problem Solving
Summary and Review
Practice Test
Chapters 1–2 Cumulative Review Exercises

3. Expressions and Polynomials
3.1 Translating and Evaluating Expressions
3.2 Introduction to Polynomials; Combining Like Terms
3.3 Adding and Subtracting Polynomials
3.4 Exponent Rules; Multiplying Polynomials
3.5 Prime Numbers and GCF
3.6 Exponent Rules; Introduction to Factoring
3.7 Additional Applications and Problem Solving
Summary and Review
Practice Test
Chapters 1–3 Cumulative Review Exercises

4. Equations
4.1 Equations and Their Solutions
4.2 The Addition Principle of Equality
4.3 The Multiplication Principle of Equality
4.4 Translating Word Sentences to Equations
4.5 Applications and Problem Solving
Summary and Review
Practice Test
Chapters 1–4 Cumulative Review Exercises

5. Fractions and Rational Expressions
5.1 Introduction to Fractions
5.2 Simplifying Fractions and Rational Expressions
5.3 Multiplying Fractions, Mixed Numbers, and Rational Expressions
5.4 Dividing Fractions, Mixed Numbers, and Rational Expressions
5.5 Least Common Multiple
5.6 Adding and Subtracting Fractions, Mixed Numbers, and Rational Expressions
5.7 Order of Operations; Evaluating and Simplifying Expressions
5.8 Solving Equations
Summary and Review
Practice Test
Chapters 1–5 Cumulative Review Exercises

6. Decimals
6.1 Introduction to Decimal Numbers
6.2 Adding and Subtracting Decimal Numbers
6.3 Multiplying Decimal Numbers; Exponents with Decimal Bases
6.4 Dividing Decimal Numbers; Square Roots with Decimals
6.5 Order of Operations and Applications in Geometry
6.6 Solving Equations and Problem Solving
Summary and Review
Practice Test
Chapters 1–6 Cumulative Review Exercises

7. Ratios, Proportions, and Measurement
7.1 Ratios, Probability, and Rates
7.2 Proportions 475
7.3 American Measurement; Time
7.4 Metric Measurement
7.5 Converting between Systems; Temperature
7.6 Applications and Problem Solving
Summary and Review
Practice Test
Chapters 1–7 Cumulative Review Exercises

8. Percents
8.1 Introduction to Percent
8.2 Solving Basic Percent Sentences
8.3 Solving Percent Problems (Portions)
8.4 Solving Problems Involving Percent of Increase or Decrease
8.5 Solving Problems Involving Interest
Summary and Review
Practice Test
Chapters 1–8 Cumulative Review Exercises

9. More with Geometry and Graphs
9.1 Points, Lines, and Angles
9.2 The Rectangular Coordinate System
9.3 Graphing Linear Equations
9.4 Applications with Graphing
Summary and Review
Practice Test
Chapters 1–9 Cumulative Review Exercises

Answers
Photo Credits
Glossary
Index
Index of Applications