Algebra Books

Book SynopsisThe goal of this book is to present local class field theory from the cohomo­ logical point of view, following the method inaugurated by Hochschild and developed by Artin-Tate. This theory is about extensions-primarily abelian-of "local" (i.e., complete for a discrete valuation) fields with finite residue field.Table of ContentsOne Local Fields (Basic Facts).- I Discrete Valuation Rings and Dedekind Domains.- II Completion.- Two Ramification.- III Discriminant and Different.- IV Ramification Groups.- V The Norm.- VI Artin Representation.- Three Group Cohomology.- VII Basic Facts.- VIII Cohomology of Finite Groups.- IX Theorems of Tate and Nakayama.- X Galois Cohomology.- XI Class Formations.- Four Local Class Field Theory.- XII Brauer Group of a Local Field.- XIII Local Class Field Theory.- XIV Local Symbols and Existence Theorem.- XV Ramification.- Supplementary Bibliography for the English Edition.

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Book SynopsisThis textbook provides an introduction to fundamental concepts of algebra at upper undergraduate to graduate level, covering the theory of rings, fields and modules, as well as the representation theory of finite groups.Throughout the book, the exposition relies on universal constructions, making systematic use of quotients and category theory — whose language is introduced in the first chapter. The book is divided into four parts. Parts I and II cover foundations of rings and modules, field theory and generalities on finite group representations, insisting on rings of polynomials and their ideals. Part III culminates in the structure theory of finitely generated modules over Dedekind domains and its applications to abelian groups, linear maps, and foundations of algebraic number theory. Part IV is an extensive study of linear representations of finite groups over fields of characteristic zero, including graded representations and graded characters as well as a final chapter on the Drinfeld–Lusztig double of a group algebra, appearing for the first time in a textbook at this level.Based on over twenty years of teaching various aspects of algebra, mainly at the École Normale Supérieure (Paris) and at Peking University, the book reflects the audiences of the author's courses. In particular, foundations of abstract algebra, like linear algebra and elementary group theory, are assumed of the reader. Each of the of four parts can be used for a course — with a little ad hoc complement on the language of categories. Thanks to its rich choice of topics, the book can also serve students as a reference throughout their studies, from undergraduate to advanced graduate level.Table of Contents1 Prerequisites and Preliminaries.- Part I Rings and Modules.- 2 Rings, Polynomials, Divisibility.- 3 Polynomial Rings in Several Indeterminates.- 4 More on Modules.- 5 On Representations of Finite Groups.- Part II Integral Domains, Polynomials, Fields.- 6 Prime and Maximal Ideals, Integral Domains.- 7 Fields, Division Rings.- Part III Finitely Generated Modules.- 8 Integrality, Noetherianity.- 9 Finitely Generated Projective Modules.- 10 Finitely Generated Modules Over Dedekind Domains.- 11 Complement on Dedekind Domains.- Part IV Characteristic Zero Linear Representations of Finite Groups.- 12 Monoidal Categories: An Introduction.- 13 Characteristic 0 Representations.- 14 Playing With the Base Field.- 15 Induction and Restriction: Some Applications to Finite Groups.- 16 Brauer’s Theorem and Some Applications.- 17 Graded Representations and Characters.- 18 The Drinfeld–Lusztig Double of a Group Algebra.

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Book SynopsisThis book provides a first introduction to the fundamental concepts of commutative algebra.

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Book Synopsis'Probably its most significant distinguishing feature is that this book is more algebraically oriented than most undergraduate number theory texts.'MAA ReviewsIntroduction to Number Theory is dedicated to concrete questions about integers, to place an emphasis on problem solving by students. When undertaking a first course in number theory, students enjoy actively engaging with the properties and relationships of numbers.The book begins with introductory material, including uniqueness of factorization of integers and polynomials. Subsequent topics explore quadratic reciprocity, Hensel's Lemma, p-adic powers series such as exp(px) and log(1+px), the Euclidean property of some quadratic rings, representation of integers as norms from quadratic rings, and Pell's equation via continued fractions.Throughout the five chapters and more than 100 exercises and solutions, readers gain the advantage of a number theory book that focuses on doing calculations. This textbook is a valuable resource for undergraduates or those with a background in university level mathematics.

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Book SynopsisThis book is based on a course for first-semester science students, held by the second author at the University of Zurich several times. Its goal is threefold: to have students learn a minimal working knowledge of linear algebra, acquire some computational skills, and familiarize them with mathematical language to make mathematical literature more accessible. Therefore, we give precise definitions, introduce helpful notations, and state any results carefully worded. We provide no proofs of these results but typically illustrate them with numerous examples. Additionally, for better understanding, we often give supporting arguments for why they are valid.Table of ContentsPart I Systems of linear equations.- 1 Introduction.- 2 Systems with two equations and two unknowns.- 3 Gaussian elimination.- Part II Matrices and related topics.- 4 Basic operations.- 5 Linear dependence, bases, coordinates.- 6 Determinants.- Part III Complex numbers.- 7 Complex numbers: definition and operations.- 8 The Fundamental Theorem of Algebra.- 9 Linear systems with complex coefficients.- Part IV Vector spaces and linear maps.- 10 Vector spaces and their linear subspaces.- 11 Linear maps.- 12 Inner products on K-vector spaces.- Part V Eigenvalues and eigenvectors.- 13 Eigenvalues and eigenvectors of C–linear maps.- 14 Eigenvalues and eigenvectors of R-linear maps.- 15 Quadratic forms on Rn.- Part VI Differential equations.- 16 Introduction.- 17 Linear ODEs with constant coefficients of first order.- 18 Linear ODEs with constant coefficients of higher order.- Appendix A Solutions.

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Book SynopsisTable of ContentsTable of Contents Introduction to Algebraic Expressions 1.1 Introduction to Algebra 1.2 The Commutative, Associative, and Distributive Laws 1.3 Fraction Notation 1.4 Positive and Negative Real Numbers 1.5 Addition of Real Numbers 1.6 Subtraction of Real Numbers 1.7 Multiplication and Division of Real Numbers 1.8 Exponential Notation and Order of Operations Equations, Inequalities, and Problem Solving 2.1 Solving Equations 2.2 Using the Principles Together 2.3 Formulas 2.4 Applications with Percent 2.5 Problem Solving 2.6 Solving Inequalities 2.7 Solving Applications with Inequalities Introduction to Graphing 3.1 Reading Graphs, Plotting Points, and Scaling Graphs 3.2 Graphing Linear Equations 3.3 Graphing and Intercepts 3.4 Rates 3.5 Slope 3.6 Slope-Intercept Form 3.7 Point-Slope Form Polynomials 4.1 Exponents and Their Properties 4.2 Polynomials 4.3 Addition and Subtraction of Polynomials 4.4 Multiplication of Polynomials 4.5 Special Products 4.6 Polynomials in Several Variables 4.7 Division of Polynomials 4.8 Negative Exponents and Scientific Notation Polynomials and Factoring 5.1 Introduction to Factoring 5.2 Factoring Trinomials of the Type x2 + bx + c 5.3 Factoring Trinomials of the Type ax2 + bx + c 5.4 Factoring Perfect-Square Trinomials and Difference of Squares 5.5 Factoring: A General Strategy 5.6 Solving Quadratic Equations by Factoring 5.7 Solving Applications Rational Expressions and Equations 6.1 Rational Expressions 6.2 Multiplication and Division 6.3 Addition, Subtraction, and Least Common Denominators 6.4 Addition and Subtraction with Unlike Denominators 6.5 Complex Rational Expressions 6.6 Rational Equations 6.7 Applications Using Rational Equations and Proportions Systems and More Graphing 7.1 Systems of Equations and Graphing 7.2 Systems of Equations and Substitution 7.3 Systems of Equations and Elimination 7.4 More Applications Using Systems 7.5 Linear Inequalities in Two Variables 7.6 Systems of Linear Inequalities 7.7 Direct Variation and Inverse Variation Radical Expressions and Equations 8.1 Introduction to Square Roots and Radical Expressions 8.2 Multiplying and Simplifying Radical Expressions 8.3 Quotients Involving Square Roots 8.4 Radical Expressions with Several Terms 8.5 Radical Equations 8.6 Applications Using Right Triangles 8.7 Higher Roots and Rational Exponents Quadratic Equations 9.1 Solving Quadratic Equations: The Principle of Square Roots 9.2 Solving Quadratic Equations: Completing the Square 9.3 The Quadratic Formula and Applications 9.4 Formulas 9.5 Complex Numbers as Solutions of Quadratic Equations 9.6 Graphs of Quadratic Functions 9.7 Functions Appendix A. Factoring Sums or Differences of Cubes Appendix B. Mean, Median, and Mode Appendix C. Sets

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Book SynopsisTable of ContentsTable of Contents R. Prealgebra Review R.1 Fractions R.2 Decimals and Percents Study Skills Using Your Math Text The Real Number System 1.1 Exponents, Order of Operations, and Inequality Study Skills Taking Lecture Notes 1.2 Variables, Expressions, and Equations Study Skills Reading Your Math Text 1.3 Real Numbers and the Number Line Study Skills Using Study Cards 1.4 Adding Real Numbers 1.5 Subtracting Real Numbers Study Skills Completing Your Homework 1.6 Multiplying and Dividing Real Numbers Summary Exercises Performing Operations with Real Numbers 1.7 Properties of Real Numbers 1.8 Simplifying Expressions Study Skills Reviewing a Chapter Equations, Inequalities, and Applications 2.1 The Addition Property of Equality Study Skills Managing Your Time 2.2 The Multiplication Property of Equality 2.3 More on Solving Linear Equations Study Skills Using Study Cards Revisited Summary Exercises Applying Methods for Solving Linear Equations 2.4 An Introduction to Applications of Linear Equations 2.5 Formulas and Additional Applications from Geometry 2.6 Ratio, Proportion, and Percent Summary Exercises Applying Problem-Solving Techniques 2.7 Solving Linear Inequalities Study Skills Taking Math Tests Graphs of Linear Equations and Inequalities in Two Variables 3.1 Linear Equations and Rectangular Coordinates Study Skills Analyzing Your Test Results 3.2 Graphing Linear Equations in Two Variables 3.3 The Slope of a Line Study Skills Preparing for Your Math Final Exam 3.4 Slope-Intercept Form of a Linear Equation 3.5 Point-Slope Form of a Linear Equation and Modeling Summary Exercises Applying Graphing and Equation-Writing Techniques for Lines 3.6 Graphing Linear Inequalities in Two Variables Systems of Linear Equations and Inequalities 4.1 Solving Systems of Linear Equations by Graphing 4.2 Solving Systems of Linear Equations by Substitution 4.3 Solving Systems of Linear Equations by Elimination Summary Exercises Applying Techniques for Solving Systems of Linear Equations 4.4 Applications of Linear Systems 4.5 Solving Systems of Linear Inequalities Exponents and Polynomials 5.1 The Product Rule and Power Rules for Exponents 5.2 Integer Exponents and the Quotient Rule Summary Exercises Applying the Rules for Exponents 5.3 An Application of Exponents: Scientific Notation 5.4 Adding and Subtracting Polynomials 5.5 Multiplying Polynomials 5.6 Special Products 5.7 Dividing a Polynomial by a Monomial 5.8 Dividing a Polynomial by a Polynomial Factoring and Applications 6.1 Greatest Common Factors; Factor by Grouping 6.2 Factoring Trinomials 6.3 Factoring Trinomials by Grouping 6.4 Factoring Trinomials Using the FOIL Method 6.5 Special Factoring Techniques Summary Exercises Recognizing and Applying Factoring Strategies 6.6 Solving Quadratic Equations Using the Zero-Factor Property 6.7 Applications of Quadratic Equations Rational Expressions and Applications 7.1 The Fundamental Property of Rational Expressions 7.2 Multiplying and Dividing Rational Expressions 7.3 Least Common Denominators 7.4 Adding and Subtracting Rational Expressions 7.5 Complex Fractions 7.6 Solving Equations with Rational Expressions Summary Exercises Simplifying Rational Expressions vs. Solving Rational Equations 7.7 Applications of Rational Expressions 7.8 Variation Roots and Radicals 8.1 Evaluating Roots 8.2 Multiplying, Dividing, and Simplifying Radicals 8.3 Adding and Subtracting Radicals 8.4 Rationalizing the Denominator 8.5 More Simplifying and Operations with Radicals Summary Exercises Applying Operations with Radicals 8.6 Solving Equations with Radicals Quadratic Equations 9.1 Solving Quadratic Equations by the Square Root Property 9.2 Solving Quadratic Equations by Completing the Square 9.3 Solving Quadratic Equations by the Quadratic Formula Summary Exercises Applying Methods for Solving Quadratic Equations 9.4 Graphing Quadratic Equations 9.5 Introduction to Functions Appendix: Factoring Sums and Differences of Cubes

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Book SynopsisTable of ContentsPreface CHAPTER 1 Whole Numbers 1.1 Reading and Writing Whole Numbers 1.2 Adding Whole Numbers 1.3 Subtracting Whole Numbers 1.4 Multiplying Whole Numbers 1.5 Dividing Whole Numbers 1.6 Long Division 1.7 Rounding Whole Numbers 1.8 Exponents, Roots, and Order of Operations 1.9 Reading Pictographs, Bar Graphs, and Line Graphs 1.10 Solving Application Problems CHAPTER 2 Multiplying and Dividing Fractions 2.1 Basics of Fractions 2.2 Mixed Numbers 2.3 Factors 2.4 Writing a Fraction in Lowest Terms 2.5 Multiplying Fractions 2.6 Applications of Multiplication 2.7 Dividing Fractions 2.8 Multiplying and Dividing Mixed Numbers CHAPTER 3 Adding and Subtracting Fractions 3.1 Adding and Subtracting Like Fractions 3.2 Least Common Multiples 3.3 Adding and Subtracting Unlike Fractions 3.4 Adding and Subtracting Mixed Numbers 3.5 Order Relations and the Order of Operations CHAPTER 4 Decimals 4.1 Reading and Writing Decimal Numbers 4.2 Rounding Decimal Numbers 4.3 Adding and Subtracting Decimal Numbers 4.4 Multiplying Decimal Numbers 4.5 Dividing Decimal Numbers 4.6 Fractions and Decimals CHAPTER 5 Ratio and Proportion 5.1 Ratios 5.2 Rates 5.3 Proportions 5.4 Solving Proportions 5.5 Solving Application Problems with Proportions CHAPTER 6 Percent 6.1 Basics of Percent 6.2 Percents and Fractions 6.3 Using the Percent Proportion and Identifying the Components in a Percent Problem 6.4 Using Proportions to Solve Percent Problems 6.5 Using the Percent Equation 6.6 Solving Application Problems with Percent 6.7 Simple Interest 6.8 Compound Interest CHAPTER 7 Geometry 8.1 Basic Geometric Terms 8.2 Angles and Their Relationships 8.3 Rectangles and Squares 8.4 Parallelograms and Trapezoids 8.5 Triangles 8.6 Circles 8.7 Volume 8.8 Pythagorean Theorem 8.9 Similar Triangles CHAPTER 8 Statistics 8.1 Circle Graphs 8.2 Bar Graphs and Line Graphs 8.3 Frequency Distributions and Histograms 8.4 Mean, Median, and Mode CHAPTER 9 The Real Number System 9.1 Exponents, Order of Operations, and Inequality 9.2 Variables, Expressions, and Equations 9.3 Real Numbers and the Number Line 9.4 Adding Real Numbers 9.5 Subtracting Real Numbers 9.6 Multiplying and Dividing Real Numbers 9.7 Properties of Real Numbers 9.8 Simplifying Expressions CHAPTER 10 Equations, Inequalities, and Applications 10.1 The Addition Property of Equality 10.2 The Multiplication Property of Equality 10.3 More on Solving Linear Equations 10.4 An Introduction to Applications of Linear Equations 10.5 Formulas and Additional Applications from Geometry 10.6 Solving Linear Inequalities CHAPTER 11 Graphs of Linear Equations and Inequalities in Two Variables 11.1 Linear Equations and Rectangular Coordinates 11.2 Graphing Linear Equations in Two Variables 11.3 The Slope of a Line 11.4 Slope-Intercept Form of a Linear Equation 11.5 Point-Slope Form of a Linear Equation and Modeling 11.6 Graphing Linear Inequalities in Two Variables CHAPTER 12 Exponents and Polynomials 12.1 The Product Rule and Power Rules for Exponents 12.2 Integer Exponents and the Quotient Rule 12.3 An Application of Exponents: Scientific Notation 12.4 Adding and Subtracting Polynomials 12.5 Multiplying Polynomials 12.6 Special Products 12.7 Dividing a Polynomial by a Monomial 12.8 Dividing a Polynomial by a Polynomial CHAPTER 13 Factoring and Applications 13.1 Greatest Common Factors; Factor by Grouping 13.2 Factoring Trinomials 13.3 Factoring Trinomials by Grouping 13.4 Factoring Trinomials Using the FOIL Method 13.5 Special Factoring Techniques 13.6 Solving Quadratic Equations Using the Zero-Factor Property 13.7 Applications of Quadratic Equations CHAPTER 14 Rational Expressions and Applications 14.1 The Fundamental Property of Rational Expressions 14.2 Multiplying and Dividing Rational Expressions 14.3 Least Common Denominators 14.4 Adding and Subtracting Rational Expressions 14.5 Complex Fractions 14.6 Solving Equations with Rational Expressions 14.7 Applications of Rational Expressions 14.8 Variation CHAPTER 15 Systems of Linear Equations and Inequalities 15.1 Solving Systems of Linear Equations by Graphing 15.2 Solving Systems of Linear Equations by Substitution 15.3 Solving Systems of Linear Equations by Elimination

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Book SynopsisTable of ContentsTable of Contents BASIC CONCEPTS OF ALGEBRA P.1 The Real Numbers and Their Properties P.2 Integer Exponents and Scientific Notation P.3 Polynomials P.4 Factoring Polynomials P.5 Rational Expressions P.6 Rational Exponents and Radicals Chapter P Review and Tests Review Exercises Practice Test EQUATIONS AND INEQUALITIES 1.1 Linear Equations in One Variable 1.2 Applications of Linear Equations: Modeling 1.3 Quadratic Equations 1.4 Complex Numbers: Quadratic Equations with Complex Solutions 1.5 Solving Other Types of Equations 1.6 Inequalities 1.7 Equations and Inequalities Involving Absolute Value Chapter 1 Review and Tests Review Exercises Practice Test A Practice Test B GRAPHS AND FUNCTIONS 2.1 The Coordinate Plane 2.2 Graphs of Equations 2.3 Lines 2.4 Functions 2.5 Properties of Functions 2.6 A Library of Functions 2.7 Transformations of Functions 2.8 Combining Functions; Composite Functions 2.9 Inverse Functions Chapter 2 Review and Tests Review Exercises Practice Test A Practice Test B Cumulative Review Exercises Chapters P—2 POLYNOMIAL AND RATIONAL FUNCTIONS 3.1 Quadratic Functions 3.2 Polynomial Functions 3.3 Dividing Polynomials 3.4 The Real Zeros of a Polynomial Function 3.5 The Complex Zeros of a Polynomial Function 3.6 Rational Functions 3.7 Variation Chapter 3 Review and Tests Review Exercises Practice Test A Practice Test B Cumulative Review Exercises Chapters P—3 EXPONENTIAL AND LOGARITHMIC FUNCTIONS 4.1 Exponential Functions 4.2 Logarithmic Functions 4.3 Rules of Logarithms 4.4 Exponential and Logarithmic Equations and Inequalities 4.5 Logarithmic Scales; Modeling Chapter 4 Review and Tests Review Exercises Practice Test A Practice Test B Cumulative Review Exercises Chapters P—4 TRIGONOMETRIC FUNCTIONS 5.1 Angles and Their Measure 5.2 Right-Triangle Trigonometry 5.3 Trigonometric Functions of Any Angle; The Unit Circle 5.4 Graphs of the Sine and Cosine Functions 5.5 Graphs of the Other Trigonometric Functions 5.6 Inverse Trigonometric Functions Chapter 5 Review and Tests Review Exercises Practice Test A Practice Test B Cumulative Review Exercises Chapters P—5 TRIGONOMETRIC IDENTITIES AND EQUATIONS 6.1 Trigonometric Identities 6.2 Sum and Difference Formulas 6.3 Double-Angle and Half-Angle Formulas 6.4 Product-to-Sum and Sum-to-Product Formulas 6.5 Trigonometric Equations I 6.6 Trigonometric Equations II Chapter 6 Review and Tests Review Exercises Practice Test A Practice Test B Cumulative Review Exercises Chapters P—6 APPLICATIONS OF TRIGONOMETRIC FUNCTIONS 7.1 The Law of Sines 7.2 The Law of Cosines 7.3 Areas of Polygons Using Trigonometry 7.4 Vectors 7.5 The Dot Product 7.6 Polar Coordinates 7.7 Polar Form of Complex Numbers; DeMoivre’s Theorem Chapter 7 Review and Tests Review Exercises Practice Test A Practice Test B Cumulative Review Exercises Chapters P—7 SYSTEMS OF EQUATIONS AND INEQUALITIES 8.1 Systems of Linear Equations in Two Variables 8.2 Systems of Linear Equations in Three Variables 8.3 Partial-Fraction Decomposition 8.4 Systems of Nonlinear Equations 8.5 Systems of Inequalities 8.6 Linear Programming Chapter 8 Review and Tests Review Exercises Practice Test A Practice Test B Cumulative Review Exercises Chapters P—8 MATRICES AND DETERMINANTS 9.1 Matrices and Systems of Equations 9.2 Matrix Algebra 9.3 The Matrix Inverse 9.4 Determinants and Cramer’s Rule Chapter 9 Review and Tests Review Exercises Practice Test A Practice Test B Cumulative Review Exercises Chapters P—9 CONIC SECTIONS 10.1 Conic Sections: Overview 10.2 The Parabola 10.3 The Ellipse 10.4 The Hyperbola Chapter 10 Review and Tests Review Exercises Practice Test A Practice Test B Cumulative Review Exercises Chapters P—10 FURTHER TOPICS IN ALGEBRA 11.1 Sequences and Series 11.2 Arithmetic Sequences; Partial Sums 11.3 Geometric Sequences and Series 11.4 Mathematical Induction 11.5 The Binomial Theorem 11.6 Counting Principles 11.7 Probability Chapter 11 Review and Tests Review Exercises Practice Test A Practice Test B Cumulative Review Exercises Chapters P—11 Answers to Selected Exercises Credits Index

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Book SynopsisTable of Contents1. Linear Functions, Equations, and Inequalities 1.1 Real Numbers and the Rectangular Coordinate System 1.2 Introduction to Relations and Functions Reviewing Basic Concepts (Sections 1.1 - 1.2) 1.3 Linear Functions 1.4 Equations of Lines and Linear Models Reviewing Basic Concepts (Sections 1.3 - 1.4) 1.5 Linear Equations and Inequalities Unifying Linear Functions 1.6 Applications of Linear Functions Reviewing Basic Concepts (Sections 1.5 - 1.6) Summary Review Exercises Test 2. Analysis of Graphs of Functions 2.1 Graphs of Basic Functions and Relations; Symmetry 2.2 Vertical and Horizontal Shifts of Graphs 2.3 Stretching, Shrinking, and Reflecting Graphs Reviewing Basic Concepts (Sections 2.1 - 2.3) 2.4 Absolute Value Functions Unifying Absolute Value Functions 2.5 Piecewise-Defined Functions 2.6 Operations and Composition Reviewing Basic Concepts (Sections 2.4 - 2.6) Summary Review Exercises Test 3. Quadratic Functions 3.1 Complex Numbers 3.2 Quadratic Functions and Graphs Reviewing Basic Concepts (Sections 3.1 - 3.2) 3.3 Quadratic Equations and Inequalities Unifying Quadratic Functions 3.4 Applications of Quadratic Functions and Models Reviewing Basic Concepts (Sections 3.3 - 3.4) Summary Review Exercises Test 4. Polynomial Functions of Higher Degree 4.1 Graphs of Polynomial Functions 4.2 Topics in the Theory of Polynomial Functions (I) Reviewing Basic Concepts (Sections 4.1 - 4.2) 4.3 Topics in the Theory of Polynomial Functions (II) 4.4 Polynomial Equations, Inequalities, Applications, and Models Reviewing Basic Concepts (Sections 4.3 - 4.4) Unifying Polynomial Functions Summary Review Exercises Test 5. Rational, Power, and Root Functions 5.1 Rational Functions and Graphs (I) 5.2 Rational Functions and Graphs (II) 5.3 Rational Equations, Inequalities, Models, and Applications Reviewing Basic Concepts (Sections 5.1 - 5.3) 5.4 Functions Defined by Powers and Roots 5.5 Equations, Inequalities, and Applications Involving Root Functions Reviewing Basic Concepts (Sections 5.4 - 5.5) Unifying Root Functions Summary Review Exercises Test 6. Inverse, Exponential, and Logarithmic Functions 6.1 Inverse Functions 6.2 Exponential Functions Unifying Exponential Functions 6.3 Logarithms and Their Properties Reviewing Basic Concepts (Sections 6.1 - 6.3) 6.4 Logarithmic Functions 6.5 Exponential and Logarithmic Equations and Inequalities Unifying Logarithmic Functions 6.6 Further Applications and Modeling with Exponential and Logarithmic Functions Reviewing Basic Concepts (Sections 6.4 - 6.6) Summary Exercises on Functions: Domains, Defining Equations, and Composition Summary Review Exercises Test 7. Systems and Matrices 7.1 Systems of Equations 7.2 Solution of Linear Systems in Three Variables 7.3 Solution of Linear Systems by Row Transformations Reviewing Basic Concepts (Sections 7.1 - 7.3) 7.4 Matrix Properties and Operations 7.5 Determinants and Cramer's Rule 7.6 Solution of Linear Systems by Matrix Inverses Reviewing Basic Concepts (Sections 7.4 - 7.6) 7.7 Systems of Inequalities and Linear Programming 7.8 Partial Fractions Reviewing Basic Concepts (Sections 7.7 - 7.8) Summary Review Exercises Test 8. Conic Sections, Nonlinear Systems, and Parametric Equations 8.1 Circles Revisited and Parabolas 8.2 Ellipses and Hyperbolas Reviewing Basic Concepts (Sections 8.1 - 8.2) 8.3 The Conic Sections and Nonlinear Systems 8.4 Introduction to Parametric Equations Reviewing Basic Concepts (Sections 8.3 - 8.4) Summary Review Exercises Test 9. Trigonometric Functions and Applications 9.1 Angles, Arcs, and Their Measures 9.2 Trigonometric Functions and Fundamental Identities 9.3 Right Triangles and Evaluating Trigonometric Functions 9.4 Applications of Right Triangles 9.5 The Circular Functions 9.6 Graphs of the Sine and Cosine Functions 9.7 Graphs of the Other Circular Functions 9.8 Harmonic Motion Summary Review Exercises Test 10. Trigonometric Identities and Equations 10.1 Trigonometric Identities 10.2 Sum and Difference Identities 10.3 Further Identities 10.4 The Inverse Circular Functions 10.5 Trigonometric Equations and Inequalities (I) 10.6 Trigonometric Equations and Inequalities (II) Unifying Trigonometric Functions Summary Review Exercises Test 11. Applications of Trigonometry and Vectors 11.1 The Law of Sines 11.2 The Law of Cosines and Area Formulas 11.3 Vectors and Their Applications 11.4 Trigonometric (Polar) Form of Complex Numbers 11.5 Powers and Roots of Complex Numbers 11.6 Polar Equations and Graphs 11.7 More Parametric Equations Summary Review Exercises Test 12. Further Topics in Algebra 12.1 Sequences and Series 12.2 Arithmetic Sequences and Series 12.3 Geometric Sequences and Series Reviewing Basic Concepts (Sections 12.1 - 12.3) 12.4 Counting Theory 12.5 The Binomial Theorem Reviewing Basic Concepts (Sections 12.4 -12.5) 12.6 Mathematical Induction 12.7 Probability Reviewing Basic Concepts (Sections 12.6 - 12.7) Summary Review Exercises Test R. Review: Basic Algebraic Concepts R.1 Review of Sets R.2 Review of Exponents and Polynomials R.3 Review of Factoring R.4 Review of Rational Expressions R.5 Review of Negative and Rational Exponents R.6 Review of Radicals Test Appendices A: Geometry Formulas B: Vectors in Space C: Polar Form of Conic Sections D: Rotation of Axes Instructor's Answers Answers to Selected Exercises (Note: In the AIE, Instructor's Answers replaces Answers to Selected Exercises.) Index

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Book SynopsisAbout our authors When John Hornsby enrolled as an undergraduate at Louisiana State University, he was uncertain whether he wanted to study mathematics, education or journalism. His ultimate decision was to become a teacher, but after 25 years of teaching at the high school and university levels and 15 years of writing mathematics textbooks, all 3 of his goals have been realized; his love for teaching and for mathematics is evident in his passion for working with students and fellow teachers as well. His specific professional interests are recreational mathematics, mathematics history and incorporating graphing calculators into the curriculum. John's personal life is busy as he devotes time to his family (wife Gwen, and sons Chris, Jack, and Josh). He has been a rabid baseball fan all of his life. John's other hobbies include numismatics (the study of coins) and record collecting. He loves the music of the 1960s and has an extensive collection of the recordTable of ContentsTable of Contents Linear Functions, Equations, and Inequalities 1.1 Real Numbers and the Rectangular Coordinate System 1.2 Introduction to Relations and Functions Reviewing Basic Concepts (Sections 1.1–1.2) 1.3 Linear Functions 1.4 Equations of Lines and Linear Models Reviewing Basic Concepts (Sections 1.3–1.4) 1.5 Linear Equations and Inequalities Unifying Linear Functions 1.6 Applications of Linear Functions Reviewing Basic Concepts (Sections 1.5–1.6) Summary Review Exercises Test Analysis of Graphs of Functions 2.1 Graphs of Basic Functions and Relations; Symmetry 2.2 Vertical and Horizontal Shifts of Graphs 2.3 Stretching, Shrinking, and Reflecting Graphs Reviewing Basic Concepts (Sections 2.1–2.3) 2.4 Absolute Value Functions Unifying Absolute Value Functions 2.5 Piecewise-Defined Functions 2.6 Operations and Composition Reviewing Basic Concepts (Sections 2.4–2.6) Summary Review Exercises Test Quadratic Functions 3.1 Complex Numbers 3.2 Quadratic Functions and Graphs Reviewing Basic Concepts (Sections 3.1–3.2) 3.3 Quadratic Equations and Inequalities Unifying Quadratic Functions 3.4 Applications of Quadratic Functions and Models Reviewing Basic Concepts (Sections 3.3–3.4) Summary Review Exercises Test Polynomial Functions of Higher Degree 4.1 Graphs of Polynomial Functions 4.2 Topics in the Theory of Polynomial Functions (I) Reviewing Basic Concepts (Sections 4.1–4.2) 4.3 Topics in the Theory of Polynomial Functions (II) 4.4 Polynomial Equations, Inequalities, Applications, and Models Reviewing Basic Concepts (Sections 4.3–4.4) Unifying Polynomial Functions Summary Review Exercises Test Rational, Power, and Root Functions 5.1 Rational Functions and Graphs (I) 5.2 Rational Functions and Graphs (II) 5.3 Rational Equations, Inequalities, Models, and Applications 5.4 Functions Defined by Powers and Roots 5.5 Equations, Inequalities, and Applications Involving Root Functions Reviewing Basic Concepts (Sections 5.4–5.5) Unifying Root Functions Summary Review Exercises Test Inverse, Exponential, and Logarithmic Functions 6.1 Inverse Functions 6.2 Exponential Functions 6.3 Logarithms and Their Properties 6.4 Logarithmic Functions 6.5 Exponential and Logarithmic Equations and Inequalities 6.6 Further Applications and Modeling with Exponential and Logarithmic Functions Reviewing Basic Concepts (Sections 6.4–6.6) Summary Review Exercises Test Systems and Matrices 7.1 Systems of Equations 7.2 Solution of Linear Systems in Three Variables 7.3 Solution of Linear Systems by Row Transformations Reviewing Basic Concepts (Sections 7.1–7.3) 7.4 Matrix Properties and Operations 7.5 Determinants and Cramer’s Rule 7.6 Solution of Linear Systems by Matrix Inverses Reviewing Basic Concepts (Sections 7.4–7.6) 7.7 Systems of Inequalities and Linear Programming 7.8 Partial Fractions Reviewing Basic Concepts (Sections 7.7–7.8) Summary Review Exercises Test Conic Sections, Nonlinear Systems, and Parametric Equations 8.1 Circles Revisited and Parabolas 8.2 Ellipses and Hyperbolas Reviewing Basic Concepts (Sections 8.1–8.2) 8.3 The Conic Sections and Nonlinear Systems 8.4 Introduction to Parametric Equations Reviewing Basic Concepts (Sections 8.3–8.4) Summary Review Exercises Test Further Topics in Algebra 9.1 Sequences and Series 9.2 Arithmetic Sequences and Series 9.3 Geometric Sequences and Series Reviewing Basic Concepts (Sections 9.1–9.3) 9.4 Counting Theory 9.5 The Binomial Theorem Reviewing Basic Concepts (Sections 9.4–9.5) 9.6 Mathematical Induction 9.7 Probability Reviewing Basic Concepts (Sections 9.6–9.7) Summary Review Exercises Test Review: Basic Algebraic Concepts R.2 Review of Exponents and Polynomials R.3 Review of Factoring R.4 Review of Rational Expressions R.5 Review of Negative and Rational Exponents R.6 Review of Radicals Test Appendix: Formulas from Geometry Instructor’s Answers Answers to Selected Exercises* Index *In the AIE, Instructor’s Answers replaces Answers to Selected Exercises.

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Book SynopsisAbout our author Elayn Martin-Gay has taught mathematics at the University of New Orleans for more than 25 years. Her numerous teaching awards include the local University Alumni Association's Award for Excellence in Teaching, and Outstanding Developmental Educator at University of New Orleans, presented by the Louisiana Association of Developmental Educators. Prior to writing textbooks, Elayn Martin-Gay developed an acclaimed series of lecture videos to support developmental mathematics students. These highly successful videos originally served as the foundation materials for her texts. Today, the videos are specific to each book in her series. She has also created Chapter Test Prep Videos to help students during their most teachable moment (as they prepare for a test) along with Instructor-to-Instructor videos that provide teaching tips, hints, and suggestions for every developmental mathematics course, including basic mathematics,Table of ContentsTable of Contents The Whole Numbers 1.1 Study Skill Tips for Success in Mathematics 1.2 Place Value, Names for Numbers, and Reading Tables 1.3 Adding and Subtracting Whole Numbers, and Perimeter 1.4 Rounding and Estimating 1.5 Multiplying Whole Numbers and Area 1.6 Dividing Whole Numbers Integrated Review–Operations on Whole Numbers 1.7 Exponents and Order of Operations 1.8 Introduction to Variables, Algebraic Expressions, and Equations Integers and Introduction to Solving Equations 2.1 Introduction to Integers 2.2 Adding Integers 2.3 Subtracting Integers 2.4 Multiplying and Dividing Integers Integrated Review–Integers 2.5 Order of Operations 2.6 Solving Equations: The Addition and Multiplication Properties Solving Equations and Problem Solving 3.1 Simplifying Algebraic Expressions 3.2 Solving Equations: Review of the Addition and Multiplication Properties Integrated Review–Expressions and Equations 3.3 Solving Linear Equations in One Variable 3.4 Linear Equations in One Variable and Problem Solving Fractions and Mixed Numbers 4.1 Introduction to Fractions and Mixed Numbers 4.2 Factors and Simplest Form 4.3 Multiplying and Dividing Fractions 4.4 Adding and Subtracting Like Fractions, Least Common Denominator, and Equivalent Fractions 4.5 Adding and Subtracting Unlike Fractions Integrated Review–Summary on Fractions and Operations on Fractions 4.6 Complex Fractions and Review of Order of Operations 4.7 Operations on Mixed Numbers 4.8 Solving Equations Containing Fractions Decimals 5.1 Introduction to Decimals 5.2 Adding and Subtracting Decimals 5.3 Multiplying Decimals and Circumference of a Circle 5.4 Dividing Decimals Integrated Review–Operations on Decimals 5.5 Fractions, Decimals, and Order of Operations 5.6 Solving Equations Containing Decimals 5.7 Decimal Applications: Mean, Median, and Mode Ratio, Proportion, and Triangle Applications 6.1 Ratios and Rates 6.2 Proportions Integrated Review–Ratio, Rate, and Proportion 6.3 Proportions and Problem Solving 6.4 Square Roots and the Pythagorean Theorem 6.5 Congruent and Similar Triangles Percent 7.1 Percents, Decimals, and Fractions 7.2 Solving Percent Problems with Equations 7.3 Solving Percent Problems with Proportions Integrated Review–Percent and Percent Problems 7.4 Applications of Percent 7.5 Percent and Problem Solving: Sales Tax, Commission, and Discount 7.6 Percent and Problem Solving: Interest Graphing and Introduction to Statistics and Probability 8.1 Pictographs, Bar Graphs, Histograms, Line Graphs, and Introduction to Statistics 8.2 Circle Graphs 8.3 The Rectangular Coordinate System and Paired Data Integrated Review–Reading Graphs 8.4 Graphing Linear Equations in Two Variables 8.5 Counting and Introduction to Probability Geometry and Measurement 9.1 Lines and Angles 9.2 Perimeter 9.3 Area, Volume, and Surface Area Integrated Review–Geometry Concepts 9.4 Linear Measurement 9.5 Weight and Mass 9.6 Capacity 9.7 Temperature and Conversions Between the U.S. and Metric Systems Exponents and Polynomials 10.1 Adding and Subtracting Polynomials 10.2 Multiplication Properties of Exponents Integrated Review–Operations on Polynomials 10.3 Multiplying Polynomials 10.4 Introduction to Factoring Polynomials Appendix A: Tables A.1 Tables of Geometric Figures A.2 Table of Percents, Decimals, and Fraction Equivalents A.3 Table on Finding Common Percents of a Number A.4 Table of Squares and Square Roots Appendix B: Quotient Rule and Negative Exponents Appendix C: Scientific Notation Appendix D: Geometric Formulas Student Resources Study Skills Builders Bigger Picture–Study Guide Outline Practice Final Exam

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Book SynopsisAbout our authors Allen Angel received his BS and MS in mathematics from SUNY at New Paltz. He completed additional graduate work at Rutgers University. He taught at Sullivan County Community College and Monroe Community College, where he served as chairperson of the Mathematics Department. He served as Assistant Director of the National Science Foundation at Rutgers University for the summers of 1967 - 1970. He was President of The New York State Mathematics Association of Two-Year Colleges (NYSMATYC). He also served as Northeast Vice President of the American Mathematics Association of Two-Year Colleges (AMATYC). Allen lives in Palm Harbor, Florida but spends his summers in Penfield, New York. He enjoys playing tennis and watching sports. He also enjoys traveling with his wife Kathy. Dennis Runde received his BS and MS in mathematics from the University of Wisconsin - Platteville and Milwaukee, respectively. He has a PhD in MathemaTable of ContentsTable of Contents Real Numbers 1.1 Study Skills for Success in Mathematics 1.2 Problem Solving 1.3 Fractions 1.4 The Real Number System 1.5 Inequalities and Absolute Value Mid-Chapter Test: Sections 1.1—1.5 1.6 Addition of Real Numbers 1.7 Subtraction of Real Numbers 1.8 Multiplication and Division of Real Numbers 1.9 Exponents, Parentheses, and the Order of Operations 1.10 Properties of the Real Number System Solving Linear Equations and Inequalities 2.1 Combining Like Terms 2.2 The Addition Property of Equality 2.3 The Multiplication Property of Equality 2.4 Solving Linear Equations with a Variable on Only One Side of the Equation Mid-Chapter Test: Sections 2.1—2.4 2.5 Solving Linear Equations with the Variable on Both Sides of the Equation 2.6 Formulas 2.7 Ratios and Proportions 2.8 Inequalities in One Variable Applications of Algebra 3.1 Changing Application Problems into Equations 3.2 Solving Application Problems Mid-Chapter Test: Sections 3.1—3.2 3.3 Geometric Problems 3.4 Motion, Money, and Mixture Problems Exponents and Polynomials 4.1 Exponents 4.2 Negative Exponents 4.3 Scientific Notation Mid-Chapter Test: Sections 4.1—4.3 4.4 Addition and Subtraction of Polynomials 4.5 Multiplication of Polynomials 4.6 Division of Polynomials Factoring 5.1 Factoring a Monomial from a Polynomial 5.2 Factoring by Grouping 5.3 Factoring Trinomials of the Form ax2 + bx + c, a = 1 5.4 Factoring Trinomials of the Form ax2 + bx + c, a ≠ 1 Mid-Chapter Test: Sections 5.1—5.4 5.5 Special Factoring Formulas and a General Review of Factoring 5.6 Solving Quadratic Equations Using Factoring 5.7 Applications of Quadratic Equations Rational Expressions and Equations 6.1 Simplifying Rational Expressions 6.2 Multiplication and Division of Rational Expressions 6.3 Addition and Subtraction of Rational Expressions with a Common Denominator and Finding the Least Common Denominator 6.4 Addition and Subtraction of Rational Expressions Mid-Chapter Test: Sections 6.1—6.4 6.5 Complex Fractions 6.6 Solving Rational Equations 6.7 Rational Equations: Applications and Problem Solving 6.8 Variation Graphing Linear Equations 7.1 The Cartesian Coordinate System and Linear Equations in Two Variables 7.2 Graphing Linear Equations 7.3 Slope of a Line Mid-Chapter Test: Sections 7.1—7.3 7.4 Slope-Intercept and Point-Slope Forms of a Linear Equation 7.5 Graphing Linear Inequalities 7.6 Functions Systems of Linear Equations 8.1 Solving Systems of Equations Graphically 8.2 Solving Systems of Equations by Substitution 8.3 Solving Systems of Equations by the Addition Method Mid-Chapter Test: Sections 8.1—8.3 8.4 Systems of Equations: Applications and Problem Solving 8.5 Solving Systems of Linear Inequalities Roots and Radicals 9.1 Evaluating Square Roots 9.2 Simplifying Square Roots 9.3 Adding, Subtracting, and Multiplying Square Roots 9.4 Dividing Square Roots Mid-Chapter Test: Sections 9.1—9.4 9.5 Solving Radical Equations 9.6 Radicals: Applications and Problem Solving 9.7 Higher Roots and Rational Exponents Quadratic Equations 10.1 The Square Root Property 10.2 Solving Quadratic Equations by Completing the Square 10.3 Solving Quadratic Equations by the Quadratic Formula Mid-Chapter Test: Sections 10.1—10.3 10.4 Graphing Quadratic Equations 10.5 Complex Numbers Appendix A: Review of Decimals and Percent Appendix B: Geometry

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Book SynopsisTable of ContentsTable of Contents Review R.1 Real Numbers R.2 Algebra Essentials R.3 Geometry Essentials R.4 Polynomials R.5 Factoring Polynomials R.6 Synthetic Division R.7 Rational Expressions R.8 nth Roots; Rational Exponents Equations and Inequalities 1.1 Linear Equations 1.2 Quadratic Equations 1.3 Complex Numbers; Quadratic Equations in the Complex Number System 1.4 Radical Equations; Equations Quadratic in Form; Factorable Equations 1.5 Solving Inequalities 1.6 Equations and Inequalities Involving Absolute Value 1.7 Problem Solving: Interest, Mixture, Uniform Motion, Constant Rate Job Applications Chapter 1 Review, Test, and Projects Graphs 2.1 The Distance and Midpoint Formulas 2.2 Graphs of Equations in Two Variables; Intercepts; Symmetry 2.3 Lines 2.4 Circles 2.5 Variation Chapter 2 Review, Test, and Projects Functions and Their Graphs 3.1 Functions 3.2 The Graph of a Function 3.3 Properties of Functions 3.4 Library of Functions; Piecewise-defined Functions 3.5 Graphing Techniques: Transformations 3.6 Mathematical Models: Building Functions Chapter 3 Review, Test, and Projects Linear and Quadratic Functions 4.1 Properties of Linear Functions and Linear Models 4.2 Building Linear Models from Data 4.3 Quadratic Functions and Their Properties 4.4 Build Quadratic Models from Verbal Descriptions and from Data 4.5 Inequalities Involving Quadratic Functions Chapter 4 Review, Test, and Projects Polynomial and Rational Functions 5.1 Polynomial Functions 5.2 Graphing Polynomials Functions; Models 5.3 Properties of Rational Functions 5.4 The Graph of a Rational Function 5.5 Polynomial and Rational Inequalities 5.6 The Real Zeros of a Polynomial Function Chapter 5 Review, Test, and Projects Exponential and Logarithmic Functions 6.1 Composite Functions 6.2 One-to-One Functions; Inverse Functions 6.3 Exponential Functions 6.4 Logarithmic Functions 6.5 Properties of Logarithms 6.6 Logarithmic and Exponential Equations 6.7 Financial Models 6.8 Exponential Growth and Decay Models; Newton’s Law; Logistic Growth and Decay Models 6.9 Building Exponential, Logarithmic, and Logistic Models from Data Chapter 6 Review, Test, and Projects Trigonometric Functions 7.1 Angles, Arc, Length, and Circular Motion 7.2 Right Triangle Trigonometry 7.3 Computing the Values of Trigonometric Functions of Acute Angles 7.4 Trigonometric Functions of Any Angle 7.5 Unit Circle Approach; Properties of the Trigonometric Functions 7.6 Graphs of the Sine and Cosine Functions 7.7 Graphs of the Tangent, Cotangent, Cosecant, and Secant Functions 7.8 Phase Shift; Sinusoidal Curve Fitting Chapter 7 Review, Test, and Projects Analytic Trigonometry 8.1 The Inverse Sine, Cosine, and Tangent Functions 8.2 The Inverse Trigonometric Functions (Continued) 8.3 Trigonometric Equations 8.4 Trigonometric Identities 8.5 Sum and Difference Formulas 8.6 Double-angle and Half-angle Formulas 8.7 Product-to-Sum and Sum-to-Product Formulas Chapter 8 Review, Test, and Projects Applications of Trigonometric Functions 9.1 Applications Involving Right Triangles 9.2 The Law of Sines 9.3 The Law of Cosines 9.4 Area of a Triangle 9.5 Simple Harmonic Motion; Damped Motion; Combining Waves Chapter 9 Review, Test, and Projects Polar Coordinates; Vectors 10.1 Polar Coordinates 10.2 Polar Equations and Graphs 10.3 The Complex Plane; De Moivre’s Theorem 10.4 Vectors 10.5 The Dot Product Chapter 10 Review, Test, and Projects Analytic Geometry 11.1 Conics 11.2 The Parabola 11.3 The Ellipse 11.4 The Hyperbola 11.5 Rotation of Axes; General Form of a Conic 11.6 Polar Equations of Conics 11.7 Plane Curves and Parametric Equations Chapter 11 Review, Test, and Projects Systems of Equations and Inequalities 12.1 Systems of Linear Equations: Substitution and Elimination 12.2 Systems of Linear Equations: Matrices 12.3 Systems of Linear Equations: Determinants 12.4 Matrix Algebra 12.5 Partial Fraction Decomposition 12.6 Systems of Nonlinear Equations 12.7 Systems of Inequalities 12.8 Linear Programming Chapter 12 Review, Test, and Projects Sequences; Induction; the Binomial Theorem 13.1 Sequences 13.2 Arithmetic Sequences 13.3 Geometric Sequences; Geometric Series 13.4 Mathematical Induction 13.5 The Binomial Theorem Chapter 13 Review, Test, and Projects Counting and Probability 14.1 Counting 14.2 Permutations and Combinations 14.3 Probability Chapter 14 Review, Test, and Projects Appendix: Graphing Utilities A.1 The Viewing Rectangle A.2 Using a Graphing Utility to Graph Equations A.3 Using a Graphing Utility to Locate Intercepts and Check for Symmetry A.4 Using a Graphing Utility to Solve Equations A.5 Square Screens A.6 Using a Graphing Utility to Graph Inequalities A.7 Using a Graphing Utility to Solve Systems of Linear Equations A.8 Using a Graphing Utility to Graph a Polar Equation A.9 Using a Graphing Utility to Graph Parametric Equations Answers Credits Index

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Book SynopsisAbout our author Elayn Martin-Gay has taught mathematics at the University of New Orleans for more than 25 years. Her numerous teaching awards include the local University Alumni Association's Award for Excellence in Teaching, and Outstanding Developmental Educator at University of New Orleans, presented by the Louisiana Association of Developmental Educators. Prior to writing textbooks, Elayn Martin-Gay developed an acclaimed series of lecture videos to support developmental mathematics students. These highly successful videos originally served as the foundation materials for her texts. Today, the videos are specific to each book in her series. She has also created Chapter Test Prep Videos to help students during their most teachable moment (as they prepare for a test) along with Instructor-to-Instructor videos that provide teaching tips, hints, and suggestions for every developmental mathematics course, including basic mathematics,Table of ContentsTable of Contents Chapter R. Prealgebra Review R.1 Factors and the Least Common Multiple R.2 Fractions R.3 Decimals and Percents R.4 Reading Pictographs and Bar, Line, and Circle Graphs Chapter 1. Real Numbers and Introduction to Algebra 1.1 Tips for Success in Mathematics 1.2 Symbols and Sets of Numbers 1.3 Exponents, Order of Operations, and Variable Expressions 1.4 Adding Real Numbers 1.5 Subtracting Real Numbers Integrated Review – Operations on Real Numbers 1.6 Multiplying and Dividing Real Numbers 1.7 Properties of Real Numbers 1.8 Simplifying Expressions Chapter 2. Equations, Inequalities, and Problem Solving 2.1 The Addition Property of Equality 2.2 The Multiplication Property of Equality 2.3 Further Solving Linear Equations Integrated Review – Solving Linear Equations 2.4 An Introduction to Problem Solving 2.5 Formulas and Problem Solving 2.6 Percent and Mixture Problem Solving 2.7 Solving Linear Inequalities and Problem Solving Chapter 3. Exponents and Polynomials 3.1 Exponents 3.2 Negative Exponents and Scientific Notation 3.3 Introduction to Polynomials 3.4 Adding and Subtracting Polynomials 3.5 Multiplying Polynomials 3.6 Special Products Integrated Review – Exponents and Operations on Polynomials 3.7 Dividing Polynomials Chapter 4. Factoring Polynomials 4.1 The Greatest Common Factor and Factoring by Grouping 4.2 Factoring Trinomials of the Form x2 bx + c 4.3 Factoring Trinomials of the Form ax2 + bx + c 4.4 Factoring Trinomials of the Form ax2 + bx + c by Grouping 4.5 Factoring Perfect Square Trinomials and the Difference of Two Squares Integrated Review – Choosing a Factoring Strategy 4.6 Solving Quadratic Equations by Factoring 4.7 Quadratic Equations and Problem Solving Chapter 5. Rational Expressions 5.1 Simplifying Rational Expressions 5.2 Multiplying and Dividing Rational Expressions 5.3 Adding and Subtracting Rational Expressions with the Same Denominator and Least Common Denominator 5.4 Adding and Subtracting Rational Expressions with Different Denominators 5.5 Solving Equations Containing Rational Expressions Integrated Review – Summary on Rational Expressions 5.6 Proportions and Problem Solving with Rational Equations 5.7 Simplifying Complex Fractions Chapter 6. Graphing Equations and Inequalities 6.1 The Rectangular Coordinate System 6.2 Graphing Linear Equations 6.3 Intercepts 6.4 Slope and Rate of Change 6.5 Equations of Lines Integrated Review – Summary on Linear Equations 6.6 Introduction to Functions 6.7. Graphing Linear Inequalities in two Variables Chapter 7. Systems of Equations 7.1 Solving systems of Linear Equations by Graphing 7.2 Solving Systems of Linear Equations by Substitution 7.3 Solving Systems of Linear Equations by Addition Integrated Review – Systems of Linear Equations 7.4 Systems of Linear Equations and Problem Solving Chapter 8. Roots and Radicals 8.1 Introduction to Radicals 8.2 Simplifying Radicals 8.3 Adding and Subtracting Radicals 8.4 Multiplying and Dividing Radicals Integrated Review – Simplifying Radicals 8.5 Solving Equations Containing Radicals 8.6 Radical Equations and Problem Solving 8.7 Direct and Inverse Variation Including Radical Applications Chapter 9. Quadratic Equations 9.1 Solving Quadratic Equations by the Square Root Property 9.2 Solving Quadratic Equations by Completing the Square 9.3 Solving Quadratic Equations by the Quadratic Formula Integrated Review – Summary of Solving Quadratic Equations 9.4 Graphing Quadratic Equations in Two Variables Appendices: Appendix A. Tables Appendix B. Factoring Sums and Differences of Cubes Appendix C. Mean, Median, Mode, Range, and Intro to Statistics Appendix D. Sets Appendix E. Review of Angles, Lines, and Special Triangles Appendix F. Interval Notation and Finding Domains and Ranges from Graphs

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Book SynopsisTable of ContentsR. Review R.1 Real Numbers R.2 Algebra Essentials R.3 Geometry Essentials R.4 Polynomials R.5 Factoring Polynomials R.6 Synthetic Division R.7 Rational Expressions R.8 nth Roots; Rational Exponents 1. Equations and Inequalities 1.1 Linear Equations 1.2 Quadratic Equations 1.3 Complex Numbers; Quadratic Equations in the Complex Number System 1.4 Radical Equations; Equations Quadratic in Form; Factorable Equations 1.5 Solving Inequalities 1.6 Equations and Inequalities Involving Absolute Value 1.7 Problem Solving: Interest, Mixture, Uniform Motion, and Constant Rate Job Applications 2. Graphs 2.1 The Distance and Midpoint Formulas 2.2 Graphs of Equations in Two Variables; Intercepts; Symmetry 2.3 Lines 2.4 Circles 2.5 Variation 3. Functions and Their Graphs 3.1 Functions 3.2 The Graph of a Function 3.3 Properties of Functions 3.4 Library of Functions; Piecewise-defined Functions 3.5 Graphing Techniques: Transformations 3.6 Mathematical Models: Building Functions 4. Linear and Quadratic Functions 4.1 Linear Functions and Their Properties 4.2 Building Linear Functions from Data 4.3 Quadratic Functions and Their Properties 4.4 Quadratic Models; Building Quadratic Functions from Data 4.5 Inequalities Involving Quadratic Functions 5. Polynomial and Rational Functions 5.1 Polynomial Functions and Models 5.2 Properties of Rational Functions 5.3 The Graph of a Rational Function 5.4 Polynomial and Rational Inequalities 5.5 The Real Zeros of a Polynomial Function 5.6 Complex Zeros: Fundamental Theorem of Algebra 6. Exponential and Logarithmic Functions 6.1 Composite Functions 6.2 One-to-One Functions; Inverse Functions 6.3 Exponential Functions 6.4 Logarithmic Functions 6.5 Properties of Logarithms 6.6 Logarithmic and Exponential Equations 6.7 Compound Interest 6.8 Exponential Growth and Decay Models; Newton's Law; Logistic Growth and Decay Models 6.9 Building Exponential, Logarithmic, and Logistic Functions from Data 7. Analytic Geometry 7.1 Conics 7.2 The Parabola 7.3 The Ellipse 7.4 The Hyperbola 8. Systems of Equations and Inequalities 8.1 Systems of Linear Equations: Substitution and Elimination 8.2 Systems of Linear Equations: Matrices 8.3 Systems of Linear Equations: Determinants 8.4 Matrix Algebra 8.5 Partial Fraction Decomposition 8.6 Systems of Nonlinear Equations 8.7 Systems of Inequalities 8.8 Linear Programming 9. Sequences; Induction; the Binomial Theorem 9.1 Sequences 9.2 Arithmetic Sequences 9.3 Geometric Sequences; Geometric Series 9.4 Mathematical Induction 9.5 The Binomial Theorem 10. Counting and Probability 10.1 Sets and Counting 10.2 Permutations and Combinations 10.3 Probability

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Book SynopsisTable of ContentsChapter 1: The Real Number System 1.1 Fractions 1.2 Exponents, Order of Operations, and Inequality Study Skills: Taking Lecture Notes 1.3 Variables, Expressions, and Equations 1.4 Real Numbers and the Number Line Study Skills: Tackling Your Homework 1.5 Adding and Subtracting Real Numbers Study Skills: Using Study Cards 1.6 Multiplying and Dividing Real Numbers Summary Exercises on Operations with Real Numbers 1.7 Properties of Real Numbers 1.8 Simplifying Expressions Study Skills: Reviewing a Chapter Chapter 2: Linear Equations and Inequalities in One Variable 2.1 The Addition Property of Equality 2.2 The Multiplication Property of Equality 2.3 More on Solving Linear Equations Summary Exercises on Solving Linear Equations Study Skills: Using Study Cards Revisited 2.4 An Introduction to Applications of Linear Equations 2.5 Formulas and Additional Applications from Geometry 2.6 Ratio, Proportion, and Percent 2.7 Further Applications of Linear Equations 2.8 Solving Linear Inequalities Study Skills: Taking Math Tests Chapter 3: Linear Equations and Inequalities in Two Variables; Functions 3.1 Linear Equations in Two Variables; The Rectangular Coordinate System Study Skills: Managing Your Time 3.2 Graphing Linear Equations in Two Variables 3.3 The Slope of a Line 3.4 Writing and Graphing Equations of Lines Summary Exercises on Linear Equations and Graphs 3.5 Graphing Linear Inequalities in Two Variables 3.6 Introduction to Functions Study Skills: Analyzing Your Test Results Chapter 4: Systems of Linear Equations and Inequalities 4.1 Solving Systems of Linear Equations by Graphing 4.2 Solving Systems of Linear Equations by Substitution 4.3 Solving Systems of Linear Equations by Elimination Summary Exercises on Solving Systems of Linear Equations 4.4 Applications of Linear Systems 4.5 Solving Systems of Linear Inequalities Chapter 5: Exponents and Polynomials 5.1 The Product Rule and Power Rules for Exponents 5.2 Integer Exponents and the Quotient Rule Summary Exercises on the Rules for Exponents 5.3 An Application of Exponents: Scientific Notation 5.4 Adding and Subtracting Polynomials; Graphing Simple Polynomials 5.5 Multiplying Polynomials 5.6 Special Products 5.7 Dividing Polynomials Chapter 6: Factoring and Applications 6.1 The Greatest Common Factor; Factoring by Grouping 6.2 Factoring Trinomials 6.3 More on Factoring Trinomials 6.4 Special Factoring Techniques Summary Exercises on Factoring 6.5 Solving Quadratic Equations by Factoring 6.6 Applications of Quadratic Equations Chapter 7: Rational Expressions and Applications 7.1 The Fundamental Property of Rational Expressions 7.2 Multiplying and Dividing Rational Expressions 7.3 Least Common Denominators 7.4 Adding and Subtracting Rational Expressions 7.5 Complex Fractions 7.6 Solving Equations with Rational Expressions Summary Exercises on Rational Expressions and Equations 7.7 Applications of Rational Expressions 7.8 Variation Study Skills: Preparing for Your Math Final Exam Chapter 8: Roots and Radicals 8.1 Evaluating Roots 8.2 Multiplying, Dividing, and Simplifying Radicals 8.3 Adding and Subtracting Radicals 8.4 Rationalizing the Denominator 8.5 More Simplifying and Operations with Radicals Summary Exercises on Ope

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Book SynopsisTrade Review"The authors give a clear and logically correct exposition of the foundations of linear algebra and rightly determinate the balance between the computational parts and the more theoretical parts." --K.Burian, zbMATH Open "A 35-page appendix gives brief descriptions of commands needed when using computational tools such as Derive, Maple, MATLAB, Mathematica and various graphing calculators." --J.D.Dixon, zbMATH OpenTable of Contents1. Vectors and Matrices 2. Systems of Linear Equations 3. Determinants and Eigenvalues 4. Finite Dimensional Vector Spaces 5. Linear Transformations 6. Orthogonality 7. Complex Vector Spaces and General Inner Products 8. Additional Applications 9. Numerical Methods Appendices A: Miscellaneous Proofs B: Functions C: Complex Numbers D: Elementary Matrices E: Answers to Selected Exercies

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Book SynopsisIn this well-written presentation, motivated by numerous examples and problems, the authors introduce the basic theory of braid groups, highlighting several definitions that show their equivalence;Trade ReviewFrom the reviews:"Details on … braid groups are carefully provided by Kassel and Turaev’s text Braid Groups. … Braid Groups is very well written. The proofs are detailed, clear, and complete. ... The text is to be praised for its level of detail. … For people … who want to understand current research in braid group related areas, Braid Groups is an excellent, in fact indispensable, text." (Scott Taylor, The Mathematical Association of America, October, 2008)"This is a very useful, carefully written book that will bring the reader up to date with some of the recent important advances in the study of the braid groups and their generalizations. It continues the tradition of these high quality graduate texts in mathematics. The book could easily be used as a text for a year course on braid groups for graduate students, one advantage being that the chapters are largely independent of each other." (Stephen P. Humphries, Mathematical Reviews, Issue 2009 e)“This book is a comprehensive introduction to the theory of braid groups. Assuming only a basic knowledge of topology and algebra, it is intended mainly for graduate and postdoctoral students.” (Hirokazu Nishimura, Zentralblatt MATH, Vol. 1208, 2011)“The book of Kassel and Turaev is a textbook … for graduate students and researchers. As such, it covers the basic material on braids, knots, and links … at a level which requires minimal background, yet moves rapidly to non-trivial topics. … It is a carefully planned and well-written book; the authors are true experts, and it fills a gap. … it will have many readers.” (Joan S. Birman, Bulletin of the American Mathematical Society, Vol. 48 (1), January, 2011)Table of ContentsBraids and Braid Groups.- Braids, Knots, and Links.- Homological Representations of the Braid Groups.- Symmetric Groups and Iwahori#x2013;Hecke Algebras.- Representations of the Iwahori#x2013;Hecke Algebras.- Garside Monoids and Braid Monoids.- An Order on the Braid Groups.- Presentations of SL(Z) and PSL(Z).- Fibrations and Homotopy Sequences.- The Birman#x2013;Murakami#x2013;Wenzl Algebras.- Left Self-Distributive Sets.

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Book SynopsisThis text provides a comprehensive treatment of representations on indefinite metric spaces, and their applications to the theory of *-derivations of C*-algebras.The book consists of two parts. The first studies the geometry of indefinite metric spaces (Krein and (Pi)(kappa)-spaces) and describes the theory of J-symmetric operator algebras and representations of *-algebras and groups on these spaces in a systematic form. For representations on (Pi)(kappa)-spaces, many significant new results are obtained; this establishes a possible approach to the general theory of representations.In the second part, different techniques of the theory of J-symmetric representations on Krein spaces are applied to the theory of *-derivations of C*-algebras implemented by skew-symmetric and dissipative operators. Various results are obtained, which establish a link between the deficiency indices of skew-symmetric operators implementing *-derivations of C*-algebras and dimensions of representationTable of ContentsPrefaceBackgroundSpaces with Indefinite MetricJ-Symmetric Operator Algebras and Representations on Krein SpacesNon-Degenerate J-Symmetric Operator Algebras on (Pi)(kappa)-SpacesJ-Symmetric Operator Algebras on (Pi)1-SpacesRepresentations of *-Algebras and Groups on (Pi)(kappa)-SpacesDerivations of C*-Algebras Implemented by Skew-Symmetric OperatorsBibliographyIndex

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Book SynopsisThis book contains accounts of talks held at a symposium in honor of John C. Moore in October 1983 at Princeton University, The work includes papers in classical homotopy theory, homological algebra, rational homotopy theory, algebraic K-theory of spaces, and other subjects.Table of Contents*Frontmatter, pg. i*Contents, pg. vii*Preface, pg. ix*I. Exponents in Homotopy Theory, pg. 1*II. The Exponent of a Moore Space, pg. 35*III. The Space of Maps of Moore Spaces Into Spheres, pg. 72*IV. The Adams Spectral Sequence of OMEGA2S3 and Brown Gitlet Spectra, pg. 101*V. Homotopy Groups of Some Mapping Telescopes, pg. 126*VI. Mapping Telescopes and K*-Localization, pg. 152*VII. The Geometric Realization of The Chromatic Resolution, pg. 168*VIII. Equivalences Between Homotopy Theories of Diagrams, pg. 180*IX. The Role of The Steenrod Algebra in The Mod 2 Cohomology of a Finite H-Space, pg. 206*X. Maps Between Classifying Spaces, pg. 228*XI. Generic Algebras and CW Complexes, pg. 247*XII. Deformation Theory and The Little Constructions of Cartan and Moore, pg. 322*XIII. Free (Z/2)3 - Actions on Finite Complexes, pg. 332*XIV. Equivariant Constructions of Nonequivariant Spectra, pg. 345*XV. A Decomposition of The Space of Generalized Morse Functions, pg. 365*XVI. Algebriac K-Theory of Spaces, Concordance, and Stable Homotopy Theory, pg. 392*XVII. The Map BSG --> A(*) --> QS , pg. 418*XVIII. Vector Bundles, Projective Modules and The K-Theory of Spheres, pg. 432*XIX. Limits Of Infinitesimal Group Cohomology, pg. 523*XX. Algebraic K-Theory of Group Scheme Actions, pg. 539*Backmatter, pg. 564

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Book SynopsisOffers an account of cocyclic Hadamard matrices and their applications in signal and data processing. This work translates physical applications into terms a pure mathematician will appreciate, and theoretical structures into ones an applied mathematician, computer scientist, or communications engineer can adapt and use.Trade Review"This book is a marvelous and timely contribution to a rapidly developing field, with new oflshoots into physics, engineering and algebra... Overall, the text gives an excellent introduction to Hadamard matrices, a masterful short survey of applications the field of communications, and a wild ride through these new algebraic tools and new combinatorial objects of study being spawned by this modern approach. Absolutely up-to-date and useful, his is a must-have text for all researchers in this field, and a must-read for aspiring researchers of Hadamard matrices, their generalizations, and their applications."--Robert Craigen, Mathematical ReviewsTable of ContentsPreface xi Chapter 1. Introduction 1 PART 1. HADAMARD MATRICES, THEIR APPLICATIONS AND GENERALISATIONS 7 Chapter 2. Hadamard Matrices 9 2.1 Classical Constructions 10 2.1.1 Sylvester Hadamard matrices 11 2.1.2 Paley Hadamard matrices 11 2.1.3 Hadamard designs 12 2.1.4Williamson Hadamard matrices 15 2.2 Equivalence Classes 16 2.3 The First Link: Group Developed Constructions 20 2.3.1 Menon Hadamard matrices 21 2.3.2 Ito Hadamard matrices 23 2.4 Towards the Hadamard Conjecture 25 Chapter 3. Applications in Signal Processing, Coding and Cryptography 27 3.1 Spectroscopy: Walsh-Hadamard Transforms 28 3.1.1 Signal analysis and synthesis 28 3.1.2 The Walsh-Hadamard Transform 29 3.1.3 The Fast Hadamard Transform 33 3.1.4 Hadamard spectroscopy 33 3.2 Error Correction: Hadamard Codes 35 3.2.1 Error-correcting codes 36 3.2.2 Hadamard codes 39 3.3 Signal Modulation and Separation: Hadamard Codes 43 3.3.1 CDMA for mobile, wireless and optical communications 45 3.3.2 3-D holographic memory for data storage and retrieval 47 3.4 Signal Correlation: Perfect Sequences and Arrays 48 3.4.1 Timing and synchronisation: Perfect binary sequences 49 3.4.2 Signal array correlation: Perfect binary arrays 50 3.5 Cryptography: Nonlinear Functions 53 3.5.1 Binary bent functions and maximally nonlinear functions 55 3.5.2 Perfect and almost perfect nonlinear functions 59 Chapter 4. Generalised Hadamard Matrices 62 4.1 Butson Matrices 63 4.2 Complex Hadamard Matrices 66 4.2.1 Quaternary complex Hadamard matrices 67 4.2.2 Unimodular complex Hadamard matrices 69 4.3 Generalised Hadamard Matrices 70 4.3.1 Generalised Hadamard matrix constructions 71 4.3.2 Generalised Hadamard matrices and Butson matrices 73 4.3.3 Generalised Hadamard matrices and class regular divisible designs 74 4.3.4 Group developed GH(w; v=w) and semiregular relative difference sets 75 4.4 Applications of Complex and Generalised Hadamard Matrices 78 4.4.1 Quaternary complex Hadamard transforms 78 4.4.2 Perfect quaternary sequences and arrays 79 4.4.3 Quaternary error-correcting codes 81 4.4.4 Generalised Hadamard matrices and Hadamard codes 83 4.5 Unification: Generalised Butson Hadamard Matrices and Transforms 84 4.5.1 The jacket matrix construction 85 4.5.2 The Generalised Hadamard Transform 90 Chapter 5. Higher Dimensional Hadamard Matrices 92 5.1 Classical Constructions 94 5.1.1 Boolean function construction for order 2 95 5.1.2 Product construction 97 5.1.3 Group developed construction 97 5.1.4 Perfect binary array construction 98 5.2 Equivalence Classes 99 5.3 Applications in Spectroscopy, Coding and Cryptography 100 5.3.1 Multidimensional Walsh Hadamard transforms 101 5.3.2 Error-correcting array codes 102 5.3.3 Cryptography: bent functions and the strict avalanche criterion 105 5.4 The Second Link: Cocyclic Construction 106 PART 2. COCYCLIC HADAMARD MATRICES 111 Chapter 6. Cocycles and Cocyclic Hadamard Matrices 113 6.1 Cocycles and Group Cohomology 114 6.2 Cocycles are Everywhere! 116 6.2.1 Examples of cocycles 116 6.2.2 New from old 117 6.2.3 Characteristic properties 119 6.2.4 Orthogonality and its inheritance 121 6.3 Computation of Cocycles 122 6.3.1 Algorithm 1-- abelian groups 124 6.3.2 Algorithm 2-- MAGMA implementation 126 6.3.3 Algorithm 3-- Homological perturbation 127 6.4 Cocyclic Hadamard Matrices 128 6.4.1 Sylvester Hadamard matrices 128 6.4.2 Menon Hadamard matrices 129 6.4.3Williamson Hadamard matrices 129 6.4.4 Ito Hadamard matrices 129 6.4.5 Generalisations of Ito Hadamard matrices 130 6.4.6 Numerical results 131 6.5 The Cocyclic Hadamard Conjecture 133 6.5.1 Noncocyclic Hadamard matrix constructions? 134 6.5.2 Status report--research problems in cocyclic Hadamard matrices 137 Chapter 7. The Five-fold Constellation 139 7.1 Factor Pairs and Extensions 139 7.2 Orthogonality for Factor Pairs 143 7.3 All the Cocyclic Generalised Hadamard Matrices 146 7.3.1 Cocyclic generalised Hadamard matrix constructions 149 7.4 The Five-fold Constellation 151 7.4.1 Restrictions on existence of cocyclic generalised Hadamard matrices 158 7.4.2 Two approaches 160 Chapter 8. Bundles and Shift Action 162 8.1 Bundles and the Five-fold Constellation 163 8.1.1 Equivalence of transversals 163 8.1.2 Bundles of factor pairs 165 8.2 Bundles of Functions--The Splitting Case 170 8.3 Bundles of Cocycles--The Central Case 174 8.3.1 Automorphism action versus shift action 174 8.3.2 A taxonomy for central semiregular RDSs 176 8.3.3 Bundles with trivial shift action--the multiplicative cocycles 178 8.4 Shift Action--The Central Case 181 8.4.1 Orbit structure for cyclic groups 184 8.4.2 Relationship between orbit structures in distinct cohomology classes 185 8.5 Shift Orbits--The Central Splitting Case 185 8.5.1 When C is an elementary abelian p-group 187 8.5.2 When C is an elementary abelian p-group and G is a p-group 188 Chapter 9. The Future: Novel Constructions and Applications 192 9.1 New Applications of Cocycles 192 9.1.1 Computation in Galois rings 192 9.1.2 Elliptic curve cryptosystems 195 9.1.3 Cocyclic codes 197 9.1.4 Cocyclic Butson matrices and codes 202 9.2 New Group Developed Generalised Hadamard Matrices 204 9.2.1 Group developed GH matrices and PN functions 204 9.2.2 PN functions and a theory of highly nonlinear functions 208 9.3 New Cocyclic Generalised Hadamard Matrices 212 9.3.1 Direct sum constructions 212 9.3.2 Multiplicative orthogonal cocycles and presemifields 216 9.3.3 Swing action 224 9.4 New Hadamard Codes 225 9.4.1 Class A cocyclic Hadamard codes 225 9.4.2 Class B cocyclic Hadamard codes 227 9.4.3 Class C cocyclic Hadamard codes 229 9.5 New Highly Nonlinear Functions 230 9.5.1 1-D differential uniformity 230 9.5.2 Differential 2-row uniformity and APN functions 233 9.5.3 2-D total differential uniformity 235 Bibliography 238 Index 259

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Book SynopsisWhat is algebra? For some, it is an abstract language of x's and y's. For mathematics majors and professional mathematicians, it is a world of axiomatically defined constructs like groups, rings, and fields. This title considers how these two seemingly different types of algebra evolved and how they relate.Trade Review"An excellent book; its accurate historical and pedagogical purpose offers an accessible read for historians and mathematicians."--Raffaele Pisano, Metascience "Well written and engaging with a wealth of useful material and a substantial bibliography for further reading, this book is a valuable resource for anyone with a serious interest in the history of algebra. With Taming the Unknown, Victor Katz and Karen Parshall have created a comprehensive synthesis of recent research on the subject, accessible to mathematicians, historians of mathematics and anyone involved in the teaching of algebra."--Adrian Rice, BSHM Bulletin "The authors have ... pitched their writing perfectly for their intended audience. The broad outline of the story is expressed in clear prose, combined with a judicious use of that other 'native tongue' of the college mathematics graduate, symbolic algebra... There is an extensive bibliography presenting the more detailed historical research that has been carried out... You could base a really nice third-year course on this book."--John Hannah, AestimatioTable of ContentsAcknowledgments xi 1 Prelude: What Is Algebra? 1 Why This Book? 3 Setting and Examining the Historical Parameters 4 The Task at Hand 10 2 Egypt and Mesopotamia 12 Proportions in Egypt 12 Geometrical Algebra in Mesopotamia 17 3 The Ancient Greek World 33 Geometrical Algebra in Euclid's Elements and Data 34 Geometrical Algebra in Apollonius's Conics 48 Archimedes and the Solution of a Cubic Equation 53 4 Later Alexandrian Developments 58 Diophantine Preliminaries 60 A Sampling from the Arithmetica: The First Three Greek Books 63 A Sampling from the Arithmetica: The Arabic Books 68 A Sampling from the Arithmetica: The Remaining Greek Books 73 The Reception and Transmission of the Arithmetica 77 5 Algebraic Thought in Ancient and Medieval China 81 Proportions and Linear Equations 82 Polynomial Equations 90 Indeterminate Analysis 98 The Chinese Remainder Problem 100 6 Algebraic Thought in Medieval India 105 Proportions and Linear Equations 107 Quadratic Equations 109 Indeterminate Equations 118 Linear Congruences and the Pulverizer 119 The Pell Equation 122 Sums of Series 126 7 Algebraic Thought in Medieval Islam 132 Quadratic Equations 137 Indeterminate Equations 153 The Algebra of Polynomials 158 The Solution of Cubic Equations 165 8 Transmission, Transplantation, and Diffusion in the Latin West 174 The Transplantation of Algebraic Thought in the Thirteenth Century 178 The Diffusion of Algebraic Thought on the Italian Peninsula and Its Environs from the Thirteenth Through the Fifteenth Centuries 190 The Diffusion of Algebraic Thought and the Development of Algebraic Notation outside of Italy 204 9 The Growth of Algebraic Thought in Sixteenth-Century Europe 214 Solutions of General Cubics and Quartics 215 Toward Algebra as a General Problem-Solving Technique 227 10 From Analytic Geometry to the Fundamental Theorem of Algebra 247 Thomas Harriot and the Structure of Equations 248 Pierre de Fermat and the Introduction to Plane and Solid Loci 253 Albert Girard and the Fundamental Theorem of Algebra 258 Rene Descartes and The Geometry 261 Johann Hudde and Jan de Witt, Two Commentators on The Geometry 271 Isaac Newton and the Arithmetica universalis 275 Colin Maclaurin's Treatise of Algebra 280 Leonhard Euler and the Fundamental Theorem of Algebra 283 11 Finding the Roots of Algebraic Equations 289 The Eighteenth-Century Quest to Solve Higher-Order Equations Algebraically 290 The Theory of Permutations 300 Determining Solvable Equations 303 The Work of Galois and Its Reception 310 The Many Roots of Group Theory 317 The Abstract Notion of a Group 328 12 Understanding Polynomial Equations in n Unknowns 335 Solving Systems of Linear Equations in n Unknowns 336 Linearly Transforming Homogeneous Polynomials in n Unknowns: Three Contexts 345 The Evolution of a Theory of Matrices and Linear Transformations 356 The Evolution of a Theory of Invariants 366 13 Understanding the Properties of "Numbers" 381 New Kinds of "Complex" Numbers 382 New Arithmetics for New "Complex" Numbers 388 What Is Algebra?: The British Debate 399 An "Algebra" of Vectors 408 A Theory of Algebras, Plural 415 14 The Emergence of Modern Algebra 427 Realizing New Algebraic Structures Axiomatically 430 The Structural Approach to Algebra 438 References 449 Index 477

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