Algebra Books

Book SynopsisAn engaging, accessible introduction into how numbers work and why we shouldn't be afraid of them, frommaths expertRachel Riley.Do you know your fractions from your percentages? Your adjacent to your hypotenuse? And who really knows how to do long division, anyway?Puzzled already? Don't blame youBut fret not! You won't be At Sixes and Sevens for long. In this brilliant, well-rounded guide, Countdown''s Rachel Riley will take you back to the very basics, allow you to revisit what you learnt at school (and may have promptly forgotten, *ahem*), build your understanding of maths from the get-go and provide you with the essential toolkit to gain confidence in your numerical abilities.Discover how to divide and conquer, make your decimal debut, become a pythagoras professional and so much more with these easy-to-learn tips and tricks. Packed full of working examples, fool-proof methods, quirky trivia and brainteasers to try from puzzle-pro Dr Gareth Moore, this book is an absolute must-read for anyone and everyone who ever thought maths was above' them. Because the truth is: you can do it. What's more, it can be pretty fun too!

Read more less

Book SynopsisTable of ContentsBasic Math and Pre-Algebra For Dummies, 2nd Edition Introduction 1 Part 1: Getting Started with Basic Math and Pre-Algebra 5 CHAPTER 1: Playing the Numbers Game 7 CHAPTER 2: It’s All in the Fingers: Numbers and Digits 23 CHAPTER 3: The Big Four: Addition, Subtraction, Multiplication, and Division 29 Part 2: Getting a Handle on Whole Numbers 47 CHAPTER 4: Putting the Big Four Operations to Work 49 CHAPTER 5: A Question of Values: Evaluating Arithmetic Expressions 63 CHAPTER 6: Say What? Turning Words into Numbers 75 CHAPTER 7: Divisibility 87 CHAPTER 8: Fabulous Factors and Marvelous Multiples 95 Part 3: Parts of the Whole: Fractions, Decimals, and Percents 109 CHAPTER 9: Fooling with Fractions 111 CHAPTER 10: Parting Ways: Fractions and the Big Four Operations 125 CHAPTER 11: Dallying with Decimals 149 CHAPTER 12: Playing with Percents 171 CHAPTER 13: Word Problems with Fractions, Decimals, and Percents 183 Part 4: Picturing and Measuring — Graphs, Measures, Stats, and Sets 195 CHAPTER 14: A Perfect Ten: Condensing Numbers with Scientific Notation 197 CHAPTER 15: How Much Have You Got? Weights and Measures 205 CHAPTER 16: Picture This: Basic Geometry 217 CHAPTER 17: Seeing Is Believing: Graphing as a Visual Tool 239 CHAPTER 18: Solving Geometry and Measurement Word Problems 247 CHAPTER 19: Figuring Your Chances: Statistics and Probability 259 CHAPTER 20: Setting Things Up with Basic Set Theory 271 Part 5: The X-Files: Introduction to Algebra 279 CHAPTER 21: Enter Mr X: Algebra and Algebraic Expressions 281 CHAPTER 22: Unmasking Mr X: Algebraic Equations 299 CHAPTER 23: Putting Mr X to Work: Algebra Word Problems 311 Part 6: The Part of Tens 321 CHAPTER 24: Ten Little Math Demons That Trip People Up 323 CHAPTER 25: Ten Important Number Sets to Know 329 Index 337 Basic Math and Pre-Algebra Workbook For Dummies, 3rd Edition Introduction 1 Part 1: Getting Started with Basic Math and Pre-Algebra 5 CHAPTER 1: We've Got Your Numbers 7 CHAPTER 2: Smooth Operators: Working with the Big Four Operations 23 CHAPTER 3: Getting Down with Negative Numbers 37 CHAPTER 4: It's Just an Expression 49 CHAPTER 5: Dividing Attention: Divisibility, Factors, and Multiples 69 Part 2: Slicing Things Up: Fractions, Decimals, and Percents 89 CHAPTER 6: Fractions Are a Piece of Cake 91 CHAPTER 7: Fractions and the Big Four 109 CHAPTER 8: Getting to the Point with Decimals 143 CHAPTER 9: Playing the Percentages 165 Part 3: A Giant Step Forward: Intermediate Topics 177 CHAPTER 10: Seeking a Higher Power through Scientific Notation 179 CHAPTER 11: Weighty Questions on Weights and Measures 189 CHAPTER 12: Shaping Up with Geometry 203 CHAPTER 13: Getting Graphic: Xy-Graphs 223 Part 4: The X Factor: Introducing Algebra 235 CHAPTER 14: Expressing Yourself with Algebraic Expressions 237 CHAPTER 15: Finding the Right Balance: Solving Algebraic Equations 259 Part 5: The Part of Tens 277 CHAPTER 16: Ten Alternative Numeral and Number Systems 279 CHAPTER 17: Ten Curious Types of Numbers 287 Index 293

Read more less

Book SynopsisAlgebra I For Dummies, 2nd Edition (9781119293576) was previously published as Algebra I For Dummies, 2nd Edition (9780470559642). While this version features a new Dummies cover and design, the content is the same as the prior release and should not be considered a new or updated product. Factor fearlessly, conquer the quadratic formula, and solve linear equations There''s no doubt that algebra can be easy to some while extremely challenging to others. If you''re vexed by variables,Algebra I For Dummies, 2nd Edition provides the plain-English, easy-to-follow guidance you need to get the right solution every time! Now with 25% new and revised content, this easy-to-understand reference not only explains algebra in terms you can understand, but it also gives you the necessary tools to solve complex problems with confidence. You''ll understand how to factor fearlessly, conquer the quadratic formula, and solve linear equations. Table of ContentsIntroduction 1 About This Book 1 Conventions Used in This Book 2 What You’re Not to Read 2 Foolish Assumptions 3 How This Book Is Organized 3 Part 1: Starting Off with the Basics 3 Part 2: Figuring Out Factoring 4 Part 3: Working Equations 4 Part 4: Applying Algebra 4 Part 5: The Part of Tens 5 Icons Used in This Book 5 Where to Go from Here 6 Part 1: Starting off with the Basics 7 Chapter 1: Assembling Your Tools 9 Beginning with the Basics: Numbers 10 Really real numbers 10 Counting on natural numbers 10 Wholly whole numbers 11 Integrating integers 12 Being reasonable: Rational numbers 12 Restraining irrational numbers 12 Picking out primes and composites 13 Speaking in Algebra 13 Taking Aim at Algebra Operations 14 Deciphering the symbols 14 Grouping 15 Defining relationships 16 Taking on algebraic tasks 16 Chapter 2: Assigning Signs: Positive and Negative Numbers 19 Showing Some Signs 20 Picking out positive numbers 20 Making the most of negative numbers 20 Comparing positives and negatives 21 Zeroing in on zero 22 Going In for Operations 22 Breaking into binary operations 22 Introducing non-binary operations 23 Operating with Signed Numbers 25 Adding like to like: Same-signed numbers 25 Adding different signs 26 Subtracting signed numbers 27 Multiplying and dividing signed numbers 29 Working with Nothing: Zero and Signed Numbers 31 Associating and Commuting with Expressions 31 Reordering operations: The commutative property 32 Associating expressions: The associative property 33 Chapter 3: Figuring Out Fractions and Dealing with Decimals 35 Pulling Numbers Apart and Piecing Them Back Together 36 Making your bow to proper fractions 36 Getting to know improper fractions 37 Mixing it up with mixed numbers 37 Following the Sterling Low-Fraction Diet 38 Inviting the loneliest number one 39 Figuring out equivalent fractions 40 Realizing why smaller or fewer is better 41 Preparing Fractions for Interactions 43 Finding common denominators 43 Working with improper fractions 45 Taking Fractions to Task 46 Adding and subtracting fractions 46 Multiplying fractions 47 Dividing fractions 50 Dealing with Decimals 51 Changing fractions to decimals 52 Changing decimals to fractions 53 Chapter 4: Exploring Exponents and Raising Radicals 55 Multiplying the Same Thing Over and Over and Over 55 Powering up exponential notation 56 Comparing with exponents 57 Taking notes on scientific notation 58 Exploring Exponential Expressions 60 Multiplying Exponents 65 Dividing and Conquering 66 Testing the Power of Zero 66 Working with Negative Exponents 67 Powers of Powers 68 Squaring Up to Square Roots 69 Chapter 5: Doing Operations in Order and Checking Your Answers 73 Ordering Operations 74 Gathering Terms with Grouping Symbols 76 Checking Your Answers 78 Making sense or cents or scents 79 Plugging in to get a charge of your answer 79 Curbing a Variable’s Versatility 80 Representing numbers with letters 81 Attaching factors and coefficients 82 Interpreting the operations 82 Doing the Math 83 Adding and subtracting variables 84 Adding and subtracting with powers 85 Multiplying and Dividing Variables 86 Multiplying variables 86 Dividing variables 87 Doing it all 88 Part 2: Figuring Out Factoring 91 Chapter 6: Working with Numbers in Their Prime 93 Beginning with the Basics 94 Composing Composite Numbers 95 Writing Prime Factorizations 96 Dividing while standing on your head 96 Getting to the root of primes with a tree 98 Wrapping your head around the rules of divisibility 99 Getting Down to the Prime Factor 100 Taking primes into account 100 Pulling out factors and leaving the rest 103 Chapter 7: Sharing the Fun: Distribution 107 Giving One to Each 108 Distributing first 109 Adding first 109 Distributing Signs 110 Distributing positives 110 Distributing negatives 111 Reversing the roles in distributing 112 Mixing It Up with Numbers and Variables 113 Negative exponents yielding fractional answers 115 Working with fractional powers 115 Distributing More Than One Term 117 Distributing binomials 117 Distributing trinomials 118 Multiplying a polynomial times another polynomial 119 Making Special Distributions 120 Recognizing the perfectly squared binomial 120 Spotting the sum and difference of the same two terms 121 Working out the difference and sum of two cubes 123 Chapter 8: Getting to First Base with Factoring 127 Factoring 127 Factoring out numbers 128 Factoring out variables 130 Unlocking combinations of numbers and variables 131 Changing factoring into a division problem 133 Grouping Terms 134 Chapter 9: Getting the Second Degree 139 The Standard Quadratic Expression 140 Reining in Big and Tiny Numbers 141 FOILing 142 FOILing basics 142 FOILed again, and again 143 Applying FOIL to a special product 146 UnFOILing 147 Unwrapping the FOILing package 148 Coming to the end of the FOIL roll 151 Making Factoring Choices 152 Combining unFOIL and the greatest common factor 153 Grouping and unFOILing in the same package 154 Chapter 10: Factoring Special Cases 157 Befitting Binomials 157 Factoring the difference of two perfect squares 158 Factoring the difference of perfect cubes 159 Factoring the sum of perfect cubes 162 Tinkering with Multiple Factoring Methods 163 Starting with binomials 163 Ending with binomials 164 Knowing When to Quit 165 Incorporating the Remainder Theorem 166 Synthesizing with synthetic division 166 Choosing numbers for synthetic division 167 Part 3: Working Equations 169 Chapter 11: Establishing Ground Rules for Solving Equations 171 Creating the Correct Setup for Solving Equations 172 Keeping Equations Balanced 172 Balancing with binary operations 173 Squaring both sides and suffering the consequences 174 Taking a root of both sides 175 Undoing an operation with its opposite 176 Solving with Reciprocals 176 Making a List and Checking It Twice 179 Doing a reality check 179 Thinking like a car mechanic when checking your work 180 Finding a Purpose 181 Chapter 12: Solving Linear Equations 183 Playing by the Rules 184 Solving Equations with Two Terms 184 Devising a method using division 185 Making the most of multiplication 186 Reciprocating the invitation 188 Extending the Number of Terms to Three 189 Eliminating the extra constant term 189 Vanquishing the extra variable term 190 Simplifying to Keep It Simple 191 Nesting isn’t for the birds 192 Distributing first 192 Multiplying or dividing before distributing 194 Featuring Fractions 196 Promoting practical proportions 196 Transforming fractional equations into proportions 198 Solving for Variables in Formulas 199 Chapter 13: Taking a Crack at Quadratic Equations 203 Squaring Up to Quadratics 204 Rooting Out Results from Quadratic Equations 206 Factoring for a Solution 208 Zeroing in on the multiplication property of zero 209 Assigning the greatest common factor and multiplication property of zero to solving quadratics 210 Solving Quadratics with Three Terms 211 Applying Quadratic Solutions 217 Figuring Out the Quadratic Formula 219 Imagining the Worst with Imaginary Numbers 221 Chapter 14: Distinguishing Equations with Distinctive Powers 223 Queuing Up to Cubic Equations 224 Solving perfectly cubed equations 224 Working with the not-so-perfectly cubed 225 Going for the greatest common factor 226 Grouping cubes 228 Solving cubics with integers 228 Working Quadratic-Like Equations 230 Rooting Out Radicals 234 Powering up both sides 235 Squaring both sides twice 237 Solving Synthetically 239 Chapter 15: Rectifying Inequalities 243 Translating between Inequality and Interval Notation 244 Intervening with interval notation 244 Grappling with graphing inequalities 246 Operating on Inequalities 247 Adding and subtracting inequalities 247 Multiplying and dividing inequalities 248 Solving Linear Inequalities 249 Working with More Than Two Expressions 250 Solving Quadratic and Rational Inequalities 252 Working without zeros 255 Dealing with more than two factors 255 Figuring out fractional inequalities 256 Working with Absolute-Value Inequalities 258 Working absolute-value equations 258 Working absolute-value inequalities 260 Part 4: Applying Algebra 263 Chapter 16: Taking Measure with Formulas 265 Measuring Up 265 Finding out how long: Units of length 266 Putting the Pythagorean theorem to work 267 Working around the perimeter 269 Spreading Out: Area Formulas 273 Laying out rectangles and squares 273 Tuning in triangles 274 Going around in circles 276 Pumping Up with Volume Formulas 276 Prying into prisms and boxes 277 Cycling cylinders 277 Scaling a pyramid 278 Pointing to cones 279 Rolling along with spheres 279 Chapter 17: Formulating for Profit and Pleasure 281 Going the Distance with Distance Formulas 282 Calculating Interest and Percent 283 Compounding interest formulas 284 Gauging taxes and discounts 286 Working Out the Combinations and Permutations 287 Counting down to factorials 288 Counting on combinations 288 Ordering up permutations 290 Chapter 18: Sorting Out Story Problems 291 Setting Up to Solve Story Problems 292 Working around Perimeter, Area, and Volume 294 Parading out perimeter and arranging area 294 Adjusting the area 295 Pumping up the volume 297 Making Up Mixtures 300 Mixing up solutions 301 Tossing in some solid mixtures 302 Investigating investments and interest 302 Going for the green: Money 304 Going the Distance 305 Figuring distance plus distance 306 Figuring distance and fuel 307 Going ’Round in Circles 307 Chapter 19: Going Visual: Graphing 311 Graphing Is Good 312 Grappling with Graphs 313 Making a point 314 Ordering pairs, or coordinating coordinates 315 Actually Graphing Points 316 Graphing Formulas and Equations 317 Lining up a linear equation 317 Going around in circles with a circular graph 318 Throwing an object into the air 319 Curling Up with Parabolas 321 Trying out the basic parabola 321 Putting the vertex on an axis 322 Sliding and multiplying 324 Chapter 20: Lining Up Graphs of Lines 327 Graphing a Line 327 Graphing the equation of a line 329 Investigating Intercepts 332 Sighting the Slope 333 Formulating slope 335 Combining slope and intercept 337 Getting to the slope-intercept form 337 Graphing with slope-intercept 338 Marking Parallel and Perpendicular Lines 339 Intersecting Lines 341 Graphing for intersections 341 Substituting to find intersections 342 Part 5: The Part of Tens 345 Chapter 21: The Ten Best Ways to Avoid Pitfalls 347 Keeping Track of the Middle Term 348 Distributing: One for You and One for Me 348 Breaking Up Fractions (Breaking Up Is Hard to Do) 348 Renovating Radicals 349 Order of Operations 349 Fractional Exponents 349 Multiplying Bases Together 350 A Power to a Power 350 Reducing for a Better Fit 351 Negative Exponents 351 Chapter 22: The Ten Most Famous Equations 353 Albert Einstein’s Theory of Relativity 353 The Pythagorean Theorem 354 The Value of e 354 Diameter and Circumference Related with Pi 354 Isaac Newton’s Formula for the Force of Gravity 355 Euler’s Identity 355 Fermat’s Last Theorem 356 Monthly Loan Payments 356 The Absolute-Value Inequality 356 The Quadratic Formula 357 Index 359

Read more less

Book Synopsis

Read more less

Book SynopsisThis Very Short Introduction invites readers to revisit algebra and appreciate the elegance and power of equations and inequalities. Offering a clear explanation of algebra through theory and example, Higgins shows how equations lead to complex numbers, matrices, groups, rings, and fields.Table of Contents1. Numbers and algebra ; 2. The laws of algebra ; 3. Linear equations and inequalities ; 4. Quadratic equations ; 5. The algebra of polynomials ; 6. Introduction to matrices ; 7. Matrices and groups ; 8. Determinants and matrices ; 9. Algebra and the arithmetic of remainders ; 10. Vector spaces ; Further Reading ; Index

Read more less

Book SynopsisBoost your chances of scoring higher at Algebra II Algebra II introduces students to complex algebra concepts in preparation for trigonometry and calculus. In this new edition of Algebra II Workbook For Dummies, high school and college students will work through the types of Algebra II problems they''ll see in class, including systems of equations, matrices, graphs, and conic sections. Plus, the book now comes with free 1-year access to chapter quizzes online! A recent report by ACT shows that over a quarter of ACT-tested 2012 high school graduates did not meet any of the four college readiness benchmarks in mathematics, English, reading, and science. Algebra II Workbook For Dummies presents tricky topics in plain English and short lessons, with examples and practice at every step to help students master the essentials, setting them up for success with each new lesson. Tracks to a typical Algebra II class Can be used as a suppTable of ContentsIntroduction About This Book 1 Foolish Assumptions 2 Icons Used in This Book 2 Beyond the Book 3 Where to Go from Here 3 Part 1: Getting Started With Algebra II 5 Chapter 1: Going Beyond Beginning Algebra 7 Good Citizenship: Following the Order of Operations and Other Properties 7 Specializing in Products and FOIL 10 Variables on the Side: Solving Linear Equations 11 Dealing with Linear Absolute Value Equations 12 Greater Math Skills: Equalizing Linear Inequalities 14 Answers to Problems on Going Beyond Beginning Algebra 16 Chapter 2: Handling Quadratic (and Quadratic-Like) Equations and Inequalities 21 Finding Reasonable Solutions with Radicals 22 UnFOILed Again! Successfully Factoring for Solutions 23 Your Bag of Tricks: Factoring Multiple Ways 25 Keeping Your Act Together: Factoring by Grouping 26 Resorting to the Quadratic Formula 27 Solving Quadratics by Completing the Square 29 Working with Quadratic-Like Equations 30 Checking Out Quadratic Inequalities 32 Answers to Problems on Quadratic (and Quadratic-Like) Equations and Inequalities 34 Chapter 3: Rooting Out the Rational, the Radical, and the Negative 43 Doing Away with Denominators with an LCD 44 Simplifying and Solving Proportions 46 Wrangling with Radicals 48 Changing Negative Attitudes toward Negative Exponents 49 Divided Powers: Solving Equations with Fractional Exponents 51 Answers to Problems on Rooting Out the Rational, the Radical, and the Negative 53 Chapter 4: Graphing for the Good Life 61 Coordinating Axes, Coordinates of Points, and Quadrants 62 Crossing the Line: Using Intercepts and Symmetry to Graph 64 Graphing Lines Using Slope-Intercept and Standard Forms 67 Graphing Basic Polynomial Curves 69 Grappling with Radical and Absolute Value Functions 71 Enter the Machines: Using a Graphing Calculator 73 Answers to Problems on Graphing for the Good Life 77 Part 2: Functions 89 Chapter 5: Formulating Functions 91 Evaluating Functions 91 Determining the Domain and Range of a Function 93 Recognizing Even, Odd, and One-to-One Functions 94 Composing Functions and Simplifying the Difference Quotient 96 Solving for Inverse Functions 99 Answers to Problems on Formulating Functions 101 Chapter 6: Specializing in Quadratic Functions 107 Finding Intercepts and the Vertex of a Parabola 108 Applying Quadratics to Real-Life Situations 109 Graphing Parabolas 111 Answers to Problems on Quadratic Functions 113 Chapter 7: Plugging in Polynomials 119 Finding Basic Polynomial Intercepts 120 Digging up More-Difficult Polynomial Roots with Factoring 122 Determining Where a Function Is Positive or Negative 123 Graphing Polynomials 125 Possible Roots and Where to Find Them: The Rational Root Theorem and Descartes’s Rule 127 Getting Real Results with Synthetic Division and the Remainder Theorem 130 Connecting the Factor Theorem with a Polynomial’s Roots 132 Answers to Problems on Plugging in Polynomials 134 Chapter 8: Acting Rationally with Functions 143 Determining Domain and Intercepts of Rational Functions 144 Introducing Vertical and Horizontal Asymptotes 145 Getting a New Slant with Oblique Asymptotes 147 Removing Discontinuities 148 Going the Limit: Limits at a Number and Infinity 149 Graphing Rational Functions 151 Answers to Problems on Rational Functions 156 Chapter 9: Exposing Exponential and Logarithmic Functions 163 Evaluating e-Expressions and Powers of e 164 Solving Exponential Equations 165 Making Cents: Applying Compound Interest and Continuous Compounding 167 Checking out the Properties of Logarithms 169 Presto-Chango: Expanding and Contracting Expressions with Log Functions 171 Solving Logarithmic Equations 173 They Ought to Be in Pictures: Graphing Exponential and Logarithmic Functions 175 Answers to Problems on Exponential and Logarithmic Functions 179 Part 3: Conics And Systems Of Equations 189 Chapter 10: Any Way You Slice It: Conic Sections 191 Putting Equations of Parabolas in Standard Form 192 Shaping Up: Determining the Focus and Directrix of a Parabola 194 Back to the Drawing Board: Sketching Parabolas 196 Writing the Equations of Circles and Ellipses in Standard Form 198 Determining Foci and Vertices of Ellipses 201 Rounding Out Your Sketches: Circles and Ellipses 203 Hyperbola: Standard Equations and Foci 205 Determining the Asymptotes and Intercepts of Hyperbolas 206 Sketching the Hyperbola 208 Answers to Problems on Conic Sections 211 Chapter 11: Solving Systems of Linear Equations 221 Solving Two Linear Equations Algebraically 221 Using Cramer’s Rule to Defeat Unruly Fractions 223 A Third Variable: Upping the Systems to Three Linear Equations 225 A Line by Any Other Name: Writing Generalized Solution Rules 227 Decomposing Fractions Using Systems 229 Answers to Problems on Systems of Equations 231 Chapter 12: Solving Systems of Nonlinear Equations and Inequalities 237 Finding the Intersections of Lines and Parabolas 237 Crossing Curves: Finding the Intersections of Parabolas and Circles 239 Appealing to a Higher Power: Dealing with Exponential Systems 240 Solving Systems of Inequalities 242 Answers to Problems on Solving Systems of Nonlinear Equations and Inequalities 245 Part 4: Other Good Stuff: Lists, Arrays, And Imaginary Numbers 251 Chapter 13: Getting More Complex with Imaginary Numbers 253 Simplifying Powers of i 254 Not Quite Brain Surgery: Doing Operations on Complex Numbers 255 “Dividing” Complex Numbers with a Conjugate 256 Solving Equations with Complex Solutions 257 Answers to Problems on Imaginary Numbers 259 Chapter 14: Getting Squared Away with Matrices 263 Describing Dimensions and Types of Matrices 263 Adding, Subtracting, and Doing Scalar Multiplication on Matrices 265 Trying Times: Multiplying Matrices by Each Other 267 The Search for Identity: Finding Inverse Matrices 268 Using Matrices to Solve Systems of Equations 272 Answers to Problems on Matrices 274 Chapter 15: Going Out of Sequence with Sequences and Series 279 Writing the Terms of a Sequence 279 Differences and Multipliers: Working with Special Sequences 282 Backtracking: Constructing Recursively Defined Sequences 283 Using Summation Notation 284 Finding Sums with Special Series 286 Answers to Problems on Sequences and Series 289 Chapter 16: Everything You Ever Wanted to Know about Sets and Counting 293 Writing the Elements of a Set from Rules or Patterns 294 Get Together: Combining Sets with Unions, Intersections, and Complements 295 Multiplication Countdowns: Simplifying Factorial Expressions 297 Checking Your Options: Using the Multiplication Property 298 Counting on Permutations When Order Matters 300 Mixing It Up with Combinations 301 Raising Binomials to Powers: Investigating the Binomial Theorem 303 Answers to Problems on Sets and Counting 304 Part 5: The Part Of Tens 309 Chapter 17: Basic Graphs 311 Putting Polynomials in Their Place 311 Lining Up Front and Center 312 Being Absolutely Sure with Absolute Value 313 Graphing Reciprocals of x and x2 .313 Rooting Out Square Root and Cube Root .314 Growing Exponentially with a Graph 315 Logging In on Logarithmic Graphing 316 Chapter 18: Ten Special Sequences and Their Sums 317 Adding as Easy as One, Two, Three 317 Summing Up the Squares 318 Finding the Sum of the Cubes 318 Not Being at Odds with Summing Odd Numbers 319 Evening Things Out by Adding Up Even Numbers 319 Adding Everything Arithmetic 319 Geometrically Speaking 320 Easing into a Sum for e 320 Signing In on the Sine 321 Powering Up on Powers of 2 322 Adding Up Fractions with Multiples for Denominators 322 Index 323

Read more less

Book SynopsisPractice your way to a great grade in Algebra I Algebra I: 1001 Practice Problems For Dummies gives you 1,001 opportunities to practice solving problems on all the major topics in Algebra Iin the book and online! Get extra help with tricky subjects, solidify what you've already learned, and get in-depth walk-throughs for every problem with this useful book. These practice problems and detailed answer explanations will get you solving for x in no-time, no matter what your skill level. Thanks to Dummies, you have a resource to you put key concepts into practice. Work through practice problems on all Algebra I topics covered in classStep through detailed solutions for every problem to build your understandingAccess practice questions online to study anywhere, any timeImprove your grade and up your study game with practice, practice, practiceThe material presented in Algebra I: 1001 Practice Problems For Dummies is an excellent resource for students, as well as parents and tutors looking to help supplement classroom instruction. Algebra I: 1001 Practice Problems For Dummies (9781119883470) was previously published as 1,001 Algebra I Practice Problems For Dummies (9781118446713). While this version features a new Dummies cover and design, the content is the same as the prior release and should not be considered a new or updated product.Table of ContentsIntroduction 1 Part 1: The Questions 5 Chapter 1: Signing on with Signed Numbers 7 Chapter 2: Recognizing Algebraic Properties and Notation 13 Chapter 3: Working with Fractions and Decimals 17 Chapter 4: Making Exponential Expressions and Operations More Compatible 23 Chapter 5: Raking in Radicals 29 Chapter 6: Creating More User-Friendly Algebraic Expressions 35 Chapter 7: Multiplying by One or More Terms 41 Chapter 8: Dividing Algebraic Expressions 47 Chapter 9: Factoring Basics 53 Chapter 10: Factoring Binomials 57 Chapter 11: Factoring Quadratic Trinomials 61 Chapter 12: Other Factoring Techniques 65 Chapter 13: Solving Linear Equations 69 Chapter 14: Taking on Quadratic Equations 73 Chapter 15: Solving Polynomials with Powers Three and Higher 79 Chapter 16: Reining in Radical and Absolute Value Equations 83 Chapter 17: Making Inequalities More Fair 87 Chapter 18: Using Established Formulas 93 Chapter 19: Using Formulas in Geometric Story Problems 101 Chapter 20: Tackling Traditional Story Problems 107 Chapter 21: Graphing Basics 113 Chapter 22: Using the Algebra of Lines 119 Chapter 23: Other Graphing Topics 123 Part 2: The Answers 127 Chapter 24: The Answers 129 Index 443

Read more less

Book SynopsisTable of Contents1. Linear Equations 1.1 Introduction to Linear Systems 1.2 Matrices, Vectors, and Gauss-Jordan Elimination 1.3 On the Solutions of Linear Systems; Matrix Algebra 2. Linear Transformations 2.1 Introduction to Linear Transformations and Their Inverses 2.2 Linear Transformations in Geometry 2.3 Matrix Products 2.4 The Inverse of a Linear Transformation 3. Subspaces of Rn and Their Dimensions 3.1 Image and Kernel of a Linear Transformation 3.2 Subspace of Rn; Bases and Linear Independence 3.3 The Dimension of a Subspace of Rn 3.4 Coordinates 4. Linear Spaces 4.1 Introduction to Linear Spaces 4.2 Linear Transformations and Isomorphisms 4.3 The Matrix of a Linear Transformation 5. Orthogonality and Least Squares 5.1 Orthogonal Projections and Orthonormal Bases 5.2 Gram-Schmidt Process and QR Factorization 5.3 Orthogonal Transformations and Orthogonal Matrices 5.4 Least Squares and Data Fitting 5.5 Inner Product Spaces 6. Determinants 6.1 Introduction to Determinants 6.2 Properties of the Determinant 6.3 Geometrical Interpretations of the Determinant; Cramer's Rule 7. Eigenvalues and Eigenvectors 7.1 Diagonalization 7.2 Finding the Eigenvalues of a Matrix 7.3 Finding the Eigenvectors of a Matrix 7.4 More on Dynamical Systems 7.5 Complex Eigenvalues 7.6 Stability 8. Symmetric Matrices and Quadratic Forms 8.1 Symmetric Matrices 8.2 Quadratic Forms 8.3 Singular Values Appendix A. Vectors Appendix B: Techniques of Proof Answers to Odd-numbered Exercises Subject Index Name Index

Read more less

Book Synopsis

Read more less

Book SynopsisAlgebra: A Complete Introduction is the most comprehensive yet easy-to-use introduction to using Algebra.Written by a leading expert, this book will help you if you are studying for an important exam or essay, or if you simply want to improve your knowledge.The book covers all the key areas of algebra including elementary operations, linear equations, formulae, simultaneous equations, quadratic equations, logarithms, variation, law and sequences. Everything you will need is here in this one book. Each chapter includes not only an explanation of the knowledge and skills you need, but also worked examples and test questions.

Read more less

Book SynopsisThe goal of this third edition of Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering is the same as previous editions: to provide a good foundation - and a joyful experience - for anyone whoâd like to learn about nonlinear dynamics and chaos from an applied perspective.The presentation stresses analytical methods, concrete examples, and geometric intuition. The theory is developed systematically, starting with first-order differential equations and their bifurcations, followed by phase plane analysis, limit cycles and their bifurcations, and culminating with the Lorenz equations, chaos, iterated maps, period doubling, renormalization, fractals, and strange attractors.The prerequisites are comfort with multivariable calculus and linear algebra, as well as a first course in physics. Ideas from probability, complex analysis, and Fourier analysis are invoked, but they're either worked out from scratch or can be safely skipped (or accepted on faith).Changes to this edition include substantial exercises about conceptual models of climate change, an updated treatment of the SIR model of epidemics, and amendments (based on recent research) about the Selkov model of oscillatory glycolysis. Equations, diagrams, and every word has been reconsidered and often revised. There are also about 50 new references, many of them from the recent literature.The most notable change is a new chapter. Chapter 13 is about the Kuramoto model.The Kuramoto model is an icon of nonlinear dynamics. Introduced in 1975 by the Japanese physicist Yoshiki Kuramoto, his elegant model is one of the rare examples of a high-dimensional nonlinear system that can be solved by elementary means.Students and teachers have embraced the book in the past, its general approach and framework continue to be sound.Table of ContentsChapter 1. Overview. Chapter 2. Flows on the Line. Chapter 3. Bifurcations. Chapter 4. Flows on the Circle. Chapter 5. Linear Systems. Chapter 6. Phase Plane. Chapter 7. Limit Cycles. Chapter 8. Bifurcations Revisited. Chapter 9. Lorenz Equations. Chapter 10. One-Dimensional Maps. Chapter 11. Fractals. Chapter 12. Strange Attractors. Chapter 13. Kuramoto Model.

Read more less

Book SynopsisEmphasizes a vectors approach and prepares students to make the transition from computational to theoretical mathematics. This book includes applications drawn from a variety of disciplines, which reinforce the fact that linear algebra is a valuable tool for modeling real-life problems.Table of Contents1. VECTORS. Introduction: The Racetrack Game. The Geometry and Algebra of Vectors. Length and Angle: The Dot Product. Exploration: Vectors and Geometry. Lines and Planes. Exploration: The Cross Product. Writing Project: Origins of the Dot Product and the Cross Product. Applications. 2. SYSTEMS OF LINEAR EQUATIONS. Introduction: Triviality. Introduction to Systems of Linear Equations. Direct Methods for Solving Linear Systems. Writing Project: A History of Gaussian Elimination. Explorations: Lies My Computer Told Me; Partial Pivoting; Counting Operations: An Introduction to the Analysis of Algorithms. Spanning Sets and Linear Independence. Applications. Vignette: The Global Positioning System. Iterative Methods for Solving Linear Systems. 3. MATRICES. Introduction: Matrices in Action. Matrix Operations. Matrix Algebra. The Inverse of a Matrix. The LU Factorization. Subspaces, Basis, Dimension, and Rank. Introduction to Linear Transformations. Vignette: Robotics. Applications. 4. EIGENVALUES AND EIGENVECTORS. Introduction: A Dynamical System on Graphs. Introduction to Eigenvalues and Eigenvectors. Determinants. Writing Project: Which Came First-the Matrix or the Determinant? Vignette: Lewis Carroll's Condensation Method. Exploration: Geometric Applications of Determinants. Eigenvalues and Eigenvectors of n x n Matrices. Writing Project: The History of Eigenvalues. Similarity and Diagonalization. Iterative Methods for Computing Eigenvalues. Applications and the Perron-Frobenius Theorem. Vignette: Ranking Sports Teams and Searching the Internet. 5. ORTHOGONALITY. Introduction: Shadows on a Wall. Orthogonality in Rn. Orthogonal Complements and Orthogonal Projections. The Gram-Schmidt Process and the QR Factorization. Explorations: The Modified QR Factorization; Approximating Eigenvalues with the QR Algorithm. Orthogonal Diagonalization of Symmetric Matrices. Applications. 6. VECTOR SPACES. Introduction: Fibonacci in (Vector) Space. Vector Spaces and Subspaces. Linear Independence, Basis, and Dimension. Writing Project: The Rise of Vector Spaces. Exploration: Magic Squares. Change of Basis. Linear Transformations. The Kernel and Range of a Linear Transformation. The Matrix of a Linear Transformation. Exploration: Tilings, Lattices and the Crystallographic Restriction. Applications. 7. DISTANCE AND APPROXIMATION. Introduction: Taxicab Geometry. Inner Product Spaces. Explorations: Vectors and Matrices with Complex Entries; Geometric Inequalities and Optimization Problems. Norms and Distance Functions. Least Squares Approximation. The Singular Value Decomposition. Vignette: Digital Image Compression. Applications. 8. CODES. (Online) Code Vectors. Vignette: The Codabar System. Error-Correcting Codes. Dual Codes. Linear Codes. The Minimum Distance of a Code. Appendix A: Mathematical Notation and Methods of Proof. Appendix B: Mathematical Induction. Appendix C: Complex Numbers. Appendix D: Polynomials.

Read more less

Book SynopsisThis book is a textbook for a semester-long or year-long introductory course in abstract algebra at the upper undergraduate or beginning graduate level.It treats set theory, group theory, ring and ideal theory, and field theory (including Galois theory), and culminates with a treatment of Dedekind rings, including rings of algebraic integers.In addition to treating standard topics, it contains material not often dealt with in books at this level. It provides a fresh perspective on the subjects it covers, with, in particular, distinctive treatments of factorization theory in integral domains and of Galois theory.As an introduction, it presupposes no prior knowledge of abstract algebra, but provides a well-motivated, clear, and rigorous treatment of the subject, illustrated by many examples. Written with an eye toward number theory, it contains numerous applications to number theory (including proofs of Fermat's theorem on sums of two squares and of the Law of Quadratic Reciprocity) and serves as an excellent basis for further study in algebra in general and number theory in particular.Each of its chapters concludes with a variety of exercises ranging from the straightforward to the challenging in order to reinforce students' knowledge of the subject. Some of these are particular examples that illustrate the theory while others are general results that develop the theory further.

Read more less

Book SynopsisSuitable for computational scientists and engineers in addition to researchers in numerical linear algebra community, this title includes an introduction to tensor computations and fresh sections on: discrete Poisson solvers; pseudospectra; structured linear equation problems; structured eigenvalue problems; and, polynomial eigenvalue problems.Trade ReviewProblems, solutions, and discussions of the formulas, methods and literature surrounding matrix computations make for a reference that is specific and well detailed: perfect for any college-level math collection appealing to engineers. Midwest Book Review Written for scientists and engineers, Matrix Computations, fourth edition provides comprehensive coverage of numerical linear algebra. Anyone whose work requires the solution to a matrix problem and an appreciation of mathematical properties will find this book to be an indispensable tool. MathWorksTable of ContentsPrefaceGlobal ReferencesOther BooksUseful URLsCommon NotationChapter 1. Matrix Multiplication1.1. Basic Algorithms and Notation1.2. Structure and Efficiency1.3. Block Matrices and Algorithms1.4. Fast Matrix-Vector Products1.5. Vectorization and Locality1.6. Parallel Matrix MultiplicationChapter 2. Matrix Analysis2.1. Basic Ideas from Linear Algebra2.2. Vector Norms2.3. Matrix Norms2.4. The Singular Value Decomposition2.5. Subspace Metrics2.6. The Sensitivity of Square Systems2.7. Finite Precision Matrix ComputationsChapter 3. General Linear Systems3.1. Triangular Systems3.2. The LU Factorization3.3. Roundoff Error in Gaussian Elimination3.4. Pivoting3.5. Improving and Estimating Accuracy3.6. Parallel LUChapter 4. Special Linear Systems4.1. Diagonal Dominance and Symmetry4.2. Positive Definite Systems4.3. Banded Systems4.4. Symmetric Indefinite Systems4.5. Block Tridiagonal Systems4.6. Vandermonde Systems4.7. Classical Methods for Toeplitz Systems4.8. Circulant and Discrete Poisson SystemsChapter 5. Orthogonalization and Least Squares5.1. Householder and Givens Transformations5.2. The QR Factorization5.3. The Full-Rank Least Squares Problem5.4. Other Orthogonal Factorizations5.5. The Rank-Deficient Least Squares Problem5.6. Square and Underdetermined SystemsChapter 6. Modified Least Squares Problems and Methods6.1. Weighting and Regularization6.2. Constrained Least Squares6.3. Total Least Squares6.4. Subspace Computations with the SVD6.5. Updating Matrix FactorizationsChapter 7. Unsymmetric Eigenvalue Problems7.1. Properties and Decompositions7.2. Perturbation Theory7.3. Power Iterations7.4. The Hessenberg and Real Schur Forms7.5. The Practical QR Algorithm7.6. Invariant Subspace Computations7.7. The Generalized Eigenvalue Problem7.8. Hamiltonian and Product Eigenvalue Problems7.9. PseudospectraChapter 8. Symmetric Eigenvalue Problems8.1. Properties and Decompositions8.2. Power Iterations8.3. The Symmetric QR Algorithm8.4. More Methods for Tridiagonal Problems8.5. Jacobi Methods8.6. Computing the SVD8.7. Generalized Eigenvalue Problems with SymmetryChapter 9. Functions of Matrices9.1. Eigenvalue Methods9.2. Approximation Methods9.3. The Matrix Exponential9.4. The Sign, Square Root, and Log of a MatrixChapter 10. Large Sparse Eigenvalue Problems10.1. The Symmetric Lanczos Process10.2. Lanczos, Quadrature, and Approximation10.3. Practical Lanczos Procedures10.4. Large Sparse SVD Frameworks10.5. Krylov Methods for Unsymmetric Problems10.6. Jacobi-Davidson and Related MethodsChapter 11. Large Sparse Linear System Problems11.1. Direct Methods11.2. The Classical Iterations11.3. The Conjugate Gradient Method11.4. Other Krylov Methods11.5. Preconditioning11.6. The Multigrid FrameworkChapter 12. Special Topics12.1. Linear Systems with Displacement Structure12.2. Structured-Rank Problems12.3. Kronecker Product Computations12.4. Tensor Unfoldings and Contractions12.5. Tensor Decompositions and IterationsIndex

Read more less

Book Synopsis

Read more less

Book SynopsisDonald E. Knuth is known throughout the world for his pioneering work on algorithms and programming techniques, for his invention of the Tex and Metafont systems for computer typesetting, and for his prolific and influential writing. Professor Emeritus of The Art of Computer Programming at Stanford University, he currently devotes full time to the completion of these fascicles and the seven volumes to which they belong.

Read more less

Book SynopsisA fine example of a great mathematician''s intellect and mathematical style, this classic on linear algebra is widely cited in the literature. The treatment is an ideal supplement to many traditional linear algebra texts and is accessible to undergraduates with some background in algebra. Extremely well-written and logical, with short and elegant proofs. ― MAA Reviews.

Read more less

Book SynopsisLinear Algebra for Data Science with Python provides an introduction to vectors and matrices within the context of data science. This book starts from the fundamentals of vectors and how vectors are used to model data, builds up to matrices and their operations, and then considers applications of matrices and vectors to data fitting, transforming time-series data into the frequency domain, and dimensionality reduction. This book uses a computational-first approach: the reader will learn how to use Python and the associated data-science libraries to work with and visualize vectors and matrices and their operations, as well as to import data to apply these techniques. Readers learn the basics of performing vector and matrix operations by hand but are also shown how to use several different Python libraries for performing these operations.Key Features: Teaches the most important concepts and techniques for working with multi-dimensional data using vectors and matrices. Introduces readers to the some of the most important Python libraries for working with data, including NumPy and and PyTorch. Examples using real data and engineering applications show the utility of the techniques covered in this book. Includes many color visualizations to illustrate mathematical operations involving vectors and matrices. Offers an accompanying website that provides a unique set of online, interactive tools to help the reader learn the material.

Read more less

a huge range and FREE tracked UK delivery on ALL orders.

Read more less

Book SynopsisMaster the fundamentals first for a smoother ride through math Basic Math & Pre-Algebra Workbook For Dummies is your ticket to finally getting a handle on math! Designed to help you strengthen your weak spots and pinpoint problem areas, this book provides hundreds of practice problems to help you get over the hump. Each section includes a brief review of key concepts and full explanations for every practice problem, so you''ll always know exactly where you went wrong. The companion website gives you access to quizzes for each chapter, so you can test your understanding and identify your sticking points before moving on to the next topic. You''ll brush up on the rules of basic operations, and then learn what to do when the numbers just won''t behavenegative numbers, inequalities, algebraic expressions, scientific notation, and other tricky situations will become second nature as you refresh what you know and learn what you missed. Each math class you taTable of ContentsIntroduction 1 About This Book 1 Foolish Assumptions 2 Icons Used in This Book 2 Beyond the Book 3 Where to Go from Here 3 Part 1: Getting Started with Basic Math and Pre-algebra 5 Chapter 1: We’ve Got Your Numbers 7 Getting in Place with Numbers and Digits 8 Rollover: Rounding Numbers Up and Down 10 Using the Number Line with the Big Four 12 The Column Lineup: Adding and Subtracting 14 Multiplying Multiple Digits 15 Cycling through Long Division 16 Solutions to We’ve Got Your Numbers 19 Chapter 2: Smooth Operators: Working with the Big Four Operations 23 Switching Things Up with Inverse Operations and the Commutative Property 24 Getting with the In-Group: Parentheses and the Associative Property 27 Becoming Unbalanced: Inequalities 29 Special Times: Powers and Square Roots 31 Answers to Problems in Smooth Operators 34 Chapter 3: Getting Down with Negative Numbers 37 Understanding Where Negative Numbers Come From 37 Sign-Switching: Understanding Negation and Absolute Value 39 Adding with Negative Numbers 40 Subtracting with Negative Numbers 42 Knowing Signs of the Times (And Division) for Negative Numbers 43 Answers to Problems in Getting Down with Negative Numbers 45 Chapter 4: It’s Just an Expression 49 Evaluating Expressions with Addition and Subtraction 50 Evaluating Expressions with Multiplication and Division 51 Making Sense of Mixed-Operator Expressions 52 Handling Powers Responsibly 53 Prioritizing Parentheses 55 Pulling Apart Parentheses and Powers 56 Figuring Out Nested Parentheses 58 Bringing It All Together: The Order of Operations 59 Solutions to It’s Just an Expression 61 Chapter 5: Dividing Attention: Divisibility, Factors, and Multiples 69 Checking for Leftovers: Divisibility Tests 70 Understanding Factors and Multiples 72 One Number, Indivisible: Identifying Prime (And Composite) Numbers 73 Generating a Number’s Factors 75 Decomposing a Number into Its Prime Factors 77 Finding the Greatest Common Factor 78 Generating the Multiples of a Number 80 Finding the Least Common Multiple 81 Solutions to Divisibility, Factors, and Multiples 83 Part 2: Slicing Things Up: Fractions, Decimals, and Percents 89 Chapter 6: Fractions Are a Piece of Cake 91 Getting Down the Basic Fraction Stuff 91 In Mixed Company: Converting between Mixed Numbers and Improper Fractions 94 Increasing and Reducing the Terms of Fractions 96 Comparing Fractions with Cross-Multiplication 99 Working with Ratios and Proportions 101 Solutions to Fractions Are a Piece of Cake 103 Chapter 7: Fractions and the Big Four 109 Multiplying Fractions: A Straight Shot 109 Flipping for Fraction Division 111 Reaching the Common Denominator: Adding Fractions 113 The Other Common Denominator: Subtracting Fractions 116 Multiplying and Dividing Mixed Numbers 118 Carried Away: Adding Mixed Numbers 120 Borrowing from the Whole: Subtracting Mixed Numbers 123 Solutions to Fractions and the Big Four 126 Chapter 8: Getting to the Point with Decimals 143 Getting in Place: Basic Decimal Stuff 143 Simple Decimal-Fraction Conversions 146 New Lineup: Adding and Subtracting Decimals 148 Counting Decimal Places: Multiplying Decimals 150 Points on the Move: Dividing Decimals 151 Decimals to Fractions 153 Fractions to Decimals 155 Solutions to Getting to the Point with Decimals 157 Chapter 9: Playing the Percentages 165 Converting Percents to Decimals 165 Changing Decimals to Percents 167 Switching from Percents to Fractions 168 Converting Fractions to Percents 169 Solving a Variety of Percent Problems Using Word Equations 171 Solutions to Playing the Percentages 173 Part 3: a Giant Step Forward: Intermediate Topics 177 Chapter 10: Seeking a Higher Power through Scientific Notation 179 On the Count of Zero: Understanding Powers of Ten 180 Exponential Arithmetic: Multiplying and Dividing Powers of Ten 182 Representing Numbers in Scientific Notation 183 Multiplying and Dividing with Scientific Notation 184 Answers to Problems in Seeking a Higher Power through Scientific Notation 186 Chapter 11: Weighty Questions on Weights and Measures 189 The Basics of the English System 190 Going International with the Metric System 192 Converting Between English and Metric Units 194 Answers to Problems in Weighty Questions on Weights and Measures 198 Chapter 12: Shaping Up with Geometry 203 Getting in Shape: Polygon (And Non-Polygon) Basics 204 Squaring Off with Quadrilaterals 204 Making a Triple Play with Triangles 208 Getting Around with Circle Measurements 212 Building Solid Measurement Skills 214 Answers to Problems in Shaping Up with Geometry 218 Chapter 13: Getting Graphic: Xy-Graphs 223 Getting the Point of the Xy-Graph 223 Drawing the Line on the Xy-Graph 227 Answers to Problems in Getting Graphic: Xy-Graphs 230 Part 4: the X Factor: Introducing Algebra 235 Chapter 14: Expressing Yourself with Algebraic Expressions 237 Plug It In: Evaluating Algebraic Expressions 238 Knowing the Terms of Separation 240 Adding and Subtracting Like Terms 242 Multiplying and Dividing Terms 243 Simplifying Expressions by Combining Like Terms 245 Simplifying Expressions with Parentheses 247 FOILing: Dealing with Two Sets of Parentheses 249 Answers to Problems in Expressing Yourself with Algebraic Expressions 251 Chapter 15: Finding the Right Balance: Solving Algebraic Equations 259 Solving Simple Algebraic Equations 259 Equality for All: Using the Balance Scale to Isolate X 262 Switching Sides: Rearranging Equations to Isolate X 264 Barring Fractions: Cross-Multiplying to Simplify Equations 266 Answers to Problems in Finding the Right Balance: Solving Algebraic Equations 268 Part 5: the Part of Tens 277 Chapter 16: Ten Alternative Numeral and Number Systems 279 Tally Marks 279 Bundled Tally Marks 280 Egyptian Numerals 280 Babylonian Numerals 281 Ancient Greek Numerals 281 Roman Numerals 282 Mayan Numerals 282 Base-2 (Binary) Numbers 283 Base-16 (Hexadecimal) Numbers 284 Prime-Based Numbers 285 Chapter 17: Ten Curious Types of Numbers 287 Square Numbers 287 Triangular Numbers 288 Cubic Numbers 289 Factorial Numbers 289 Powers of Two 290 Perfect Numbers 290 Amicable Numbers 291 Prime Numbers 291 Mersenne Primes 291 Fermat Primes 292 Index 293

Read more less

Book SynopsisA PRACTICAL GUIDE TO OPTIMIZATION PROBLEMS WITH DISCRETE OR INTEGER VARIABLES, REVISED AND UPDATED The revised second edition of Integer Programming explains in clear and simple terms how to construct custom-made algorithms or use existing commercial software to obtain optimal or near-optimal solutions for a variety of real-world problems. The second edition also includes information on the remarkable progress in the development of mixed integer programming solvers in the 22 years since the first edition of the book appeared. The updated text includes information on the most recent developments in the field such as the much improved preprocessing/presolving and the many new ideas for primal heuristics included in the solvers. The result has been a speed-up of several orders of magnitude. The other major change reflected in the text is the widespread use of decomposition algorithms, in particular column generation (branch-(cut)-and-price) and Benders' decompositiTable of ContentsPreface to the Second Edition xii Preface to the First Edition xiii Abbreviations and Notation xvii About the Companion Website xix 1 Formulations 1 1.1 Introduction 1 1.2 What Is an Integer Program? 3 1.3 Formulating IPs and BIPs 5 1.4 The Combinatorial Explosion 8 1.5 Mixed Integer Formulations 9 1.6 Alternative Formulations 12 1.7 Good and Ideal Formulations 15 1.8 Notes 18 1.9 Exercises 19 2 Optimality, Relaxation, and Bounds 25 2.1 Optimality and Relaxation 25 2.2 Linear Programming Relaxations 27 2.3 Combinatorial Relaxations 28 2.4 Lagrangian Relaxation 29 2.5 Duality 30 2.6 Linear Programming and Polyhedra 32 2.7 Primal Bounds: Greedy and Local Search 34 2.8 Notes 38 2.9 Exercises 38 3 Well-Solved Problems 43 3.1 Properties of Easy Problems 43 3.2 IPs with Totally Unimodular Matrices 44 3.3 Minimum Cost Network Flows 46 3.4 Special Minimum Cost Flows 48 3.4.1 Shortest Path 48 3.4.2 Maximum s − t Flow 49 3.5 Optimal Trees 50 3.6 Submodularity and Matroids 54 3.7 Two Harder Network Flow Problems 57 3.8 Notes 59 3.9 Exercises 60 4 Matchings and Assignments 63 4.1 Augmenting Paths and Optimality 63 4.2 Bipartite Maximum Cardinality Matching 65 4.3 The Assignment Problem 67 4.4 Matchings in Nonbipartite Graphs 73 4.5 Notes 74 4.6 Exercises 75 5 Dynamic Programming 79 5.1 Some Motivation: Shortest Paths 79 5.2 Uncapacitated Lot-Sizing 80 5.3 An Optimal Subtree of a Tree 83 5.4 Knapsack Problems 84 5.4.1 0–1 Knapsack Problems 85 5.4.2 Integer Knapsack Problems 86 5.5 The Cutting Stock Problem 89 5.6 Notes 91 5.7 Exercises 92 6 Complexity and Problem Reductions 95 6.1 Complexity 95 6.2 Decision Problems, and Classes NP and P 96 6.3 Polynomial Reduction and the Class NPC 98 6.4 Consequences of P =NP orP ≠NP 103 6.5 Optimization and Separation 104 6.6 The Complexity of Extended Formulations 105 6.7 Worst-Case Analysis of Heuristics 106 6.8 Notes 109 6.9 Exercises 110 7 Branch and Bound 113 7.1 Divide and Conquer 113 7.2 Implicit Enumeration 114 7.3 Branch and Bound: an Example 116 7.4 LP-Based Branch and Bound 120 7.5 Using a Branch-and-Bound/Cut System 123 7.6 Preprocessing or Presolve 129 7.7 Notes 134 7.8 Exercises 135 8 Cutting Plane Algorithms 139 8.1 Introduction 139 8.2 Some Simple Valid Inequalities 140 8.3 Valid Inequalities 143 8.4 A Priori Addition of Constraints 147 8.5 Automatic Reformulation or Cutting Plane Algorithms 149 8.6 Gomory’s Fractional Cutting Plane Algorithm 150 8.7 Mixed Integer Cuts 153 8.7.1 The Basic Mixed Integer Inequality 153 8.7.2 The Mixed Integer Rounding (MIR) Inequality 155 8.7.3 The Gomory Mixed Integer Cut 155 8.7.4 Split Cuts 156 8.8 Disjunctive Inequalities and Lift-and-Project 158 8.9 Notes 161 8.10 Exercises 162 9 Strong Valid Inequalities 167 9.1 Introduction 167 9.2 Strong Inequalities 168 9.3 0–1 Knapsack Inequalities 175 9.3.1 Cover Inequalities 175 9.3.2 Strengthening Cover Inequalities 176 9.3.3 Separation for Cover Inequalities 178 9.4 Mixed 0–1 Inequalities 179 9.4.1 Flow Cover Inequalities 179 9.4.2 Separation for Flow Cover Inequalities 181 9.5 The Optimal Subtour Problem 183 9.5.1 Separation for Generalized Subtour Constraints 183 9.6 Branch-and-Cut 186 9.7 Notes 189 9.8 Exercises 190 10 Lagrangian Duality 195 10.1 Lagrangian Relaxation 195 10.2 The Strength of the Lagrangian Dual 200 10.3 Solving the Lagrangian Dual 202 10.4 Lagrangian Heuristics 205 10.5 Choosing a Lagrangian Dual 207 10.6 Notes 209 10.7 Exercises 210 11 Column (and Row) Generation Algorithms 213 11.1 Introduction 213 11.2 The Dantzig–Wolfe Reformulation of an IP 215 11.3 Solving the LP Master Problem: Column Generation 216 11.4 Solving the Master Problem: Branch-and-Price 219 11.5 Problem Variants 222 11.5.1 Handling Multiple Subproblems 222 11.5.2 Partitioning/Packing Problems with Additional Variables 223 11.5.3 Partitioning/Packing Problems with Identical Subsets 224 11.6 Computational Issues 225 11.7 Branch-Cut-and-Price: An Example 226 11.7.1 A Capacitated Vehicle Routing Problem 226 11.7.2 Solving the Subproblems 229 11.7.3 The Load Formulation 230 11.8 Notes 231 11.9 Exercises 232 12 Benders’ Algorithm 235 12.1 Introduction 235 12.2 Benders’ Reformulation 236 12.3 Benders’ with Multiple Subproblems 240 12.4 Solving the Linear Programming Subproblems 242 12.5 Integer Subproblems: Basic Algorithms 244 12.5.1 Branching in the (x, 𝜂, y)-Space 244 12.5.2 Branching in (x, 𝜂)-Space and “No-Good” Cuts 246 12.6 Notes 247 12.7 Exercises 248 13 Primal Heuristics 251 13.1 Introduction 251 13.2 Greedy and Local Search Revisited 252 13.3 Improved Local Search Heuristics 255 13.3.1 Tabu Search 255 13.3.2 Simulated Annealing 256 13.3.3 Genetic Algorithms 257 13.4 Heuristics Inside MIP Solvers 259 13.4.1 Construction Heuristics 259 13.4.2 Improvement Heuristics 261 13.5 User-Defined MIP heuristics 262 13.6 Notes 265 13.7 Exercises 266 14 From Theory to Solutions 269 14.1 Introduction 269 14.2 Software for Solving Integer Programs 269 14.3 How Do We Find an Improved Formulation? 272 14.4 Multi-item Single Machine Lot-Sizing 277 14.5 A Multiplexer Assignment Problem 282 14.6 Integer Programming and Machine Learning 285 14.7 Notes 287 14.8 Exercises 287 References 291 Index 311

Read more less

Book SynopsisTable of ContentsIntroduction 1 About This Book 1 Foolish Assumptions 2 Icons Used in This Book 2 Beyond the Book 3 Where to Go from Here 3 Book 1: Starting Out With Numbers and Properties 5 Chapter 1: Assembling Your Tools: Number Systems 7 Identifying Numbers by Name 8 Realizing real numbers 8 Counting on natural numbers 8 Whittling out whole numbers 8 Integrating integers 9 Being reasonable: Rational numbers 9 Restraining irrational numbers 9 Picking out primes and composites 10 Zero: It’s Complicated 10 Imagining imaginary numbers 10 Coping with complex numbers 10 Placing Numbers on the Number Line 12 Speaking in Algebra 13 Being precise with words 13 Describing the size of an expression 15 Relating operations with symbols 15 Taking Aim at Algebra-Speak 17 Herding numbers with grouping symbols 17 Defining relationships 19 Taking on algebraic tasks 20 Practice Questions Answers and Explanations 22 Whaddya Know? Chapter 1 Quiz 23 Answers to Chapter 1 Quiz 25 Chapter 2: Deciphering Signs in Expressions 27 Assigning Numbers Their Place 27 Using the number line 28 Comparing positives and negatives with symbols 29 Zeroing in on Zero 31 Going in for Operations 32 Sorting out types of operations 32 Tackling the Basic Binary Operations 35 Adding signed numbers 35 Making a Difference with Signed Numbers 38 Multiplying Signed Numbers 40 Dividing Signed Numbers 41 Working with Nothing: Zero and Signed Numbers 42 Practice Questions Answers and Explanations 44 Whaddya Know? Chapter 2 Quiz 47 Answers to Chapter 2 Quiz 48 Chapter 3: Incorporating Algebraic Properties 49 Getting a Grip on Grouping Symbols 49 Spreading, Grouping, and Changing the Order 52 Distributing the wealth 52 Making Associations Work 53 Computing by Commuting 55 Relating Inverses and Identities 56 Investigating Inverses 56 Identifying Identities 58 Working with Factorial 59 Applying the Greatest Integer Function 60 Practice Question Answers and Explanations 62 Whaddya Know? Chapter 3 Quiz 66 Answers to Chapter 3 Quiz 67 Chapter 4: Coordinating Fractions and Decimals 69 Converting Improper Fractions and Mixed Numbers 70 Finding Fraction Equivalences 72 Rewriting fractions 72 Determining lowest terms 74 Making Proportional Statements 75 Finding Common Denominators 77 Creating common denominators from multiples of factors 78 Using the box method 80 Applying Fractional Operations 81 Adding and subtracting fractions 81 Multiplying and dividing fractions 83 Simplifying Complex Fractions 86 Performing Operations with Decimals 88 Changing Fractions to Decimals and Vice Versa 89 Making fractions become decimals 90 Rounding decimals 91 Writing decimals as equivalent fractions 91 Practice Question Answers and Explanations 94 Whaddya Know? Chapter 4 Quiz 101 Answers to Chapter 4 Quiz 103 Book 2: Operating On Operations 105 Chapter 5: Taming Rampaging Radicals 107 Simplifying Radical Terms 108 Working through Radical Expressions 109 Recognizing perfect square terms 109 Rewriting radical terms 110 Rationalizing Fractions 113 Managing Radicals as Exponential Terms 114 Using Fractional Exponents 116 Making the switch to fractional exponents 117 Simplifying expressions with exponents 118 Estimating Answers 119 Practice Questions Answers and Explanations 120 Whaddya Know? Chapter 5 Quiz 124 Answers to Chapter 5 Quiz 125 Chapter 6: Exploring Exponents 127 Powering up with Exponential Notation 127 Using Negative Exponents 130 Multiplying and Dividing Exponentials 132 Multiplying the same base 132 Multiplying the same power 133 Dividing with exponents 135 Raising Powers to Powers 137 Testing the Power of Zero 139 Writing Numbers with Scientific Notation 140 Practice Questions Answers and Explanations 143 Whaddya Know? Chapter 6 Quiz 145 Answers to Chapter 6 Quiz 146 Book 3: Making Things Simple by Simplifying 149 Chapter 7: Simplifying Algebraic Expressions 151 Addressing the Order of Operations 152 Adding and Subtracting Like Terms 154 Multiplying and Dividing Algebraically 157 Dealing with factors 157 Diving into dividing 158 Gathering Terms with Grouping Symbols 160 Evaluating Expressions 163 Checking Your Answers 165 Seeing if it makes sense 165 Plugging in values 166 Practice Questions Answers and Explanations 168 Whaddya Know? Chapter 7 Quiz 171 Answers to Chapter 7 Quiz 172 Chapter 8: Working with Numbers in Their Prime 175 Beginning with the Basics 176 Composing Composite Numbers 178 Writing Prime Factorizations 178 Dividing while standing on your head 178 Getting to the root of primes with a tree 180 Wrapping your head around the rules of divisibility 182 Making Use of a Prime Factor 185 Taking primes into account 185 Pulling out factors and leaving the rest 187 Practice Questions Answers and Explanations 190 Whaddya Know? Chapter 8 Quiz 194 Answers to Chapter 8 Quiz 195 Chapter 9: Specializing in Multiplication Matters 197 Distributing One Factor Over Many 198 Distributing Signs 200 Mixing It up with Numbers and Variables 201 Negative exponents yielding fractional answers 203 Working with Fractional Powers 205 Distributing More Than One Term 207 Distributing binomials 207 Distributing trinomials 209 Curses, Foiled Again — Or Not 210 Squaring Binomials 211 Multiplying the Sum and Difference of the Same Two Terms 212 Powering Up Binomials 213 Cubing binomials 213 Raising Binomials to Higher Powers 215 Creating the Sum and Difference of Cubes 217 Multiplying Conjugates 218 Practice Questions Answers and Explanations 220 Whaddya Know? Chapter 9 Quiz 225 Answers to Chapter 9 Quiz 226 Chapter 10: Dividing the Long Way to Simplify Algebraic Expressions 229 Dividing by a Monomial 229 Dividing by a Binomial 231 Dividing by Polynomials with More Terms 233 Simplifying Division Synthetically 235 Practice Questions Answers and Explanations 237 Whaddya Know? Chapter 10 Quiz 240 Answers to Chapter 10 Quiz 241 Book 4: Factoring 243 Chapter 11: Figuring on Factoring 245 Factoring out the Greatest Common Factor 245 Factoring out numbers 246 Factoring out variables 249 Unlocking combinations of numbers and variables 252 Using the Box Method 255 Changing Factoring into a Division Problem 257 Reducing Algebraic Fractions 258 Practice Questions Answers and Explanations 260 Whaddya Know? Chapter 11 Quiz 262 Answers to Chapter 11 Quiz 264 Chapter 12: Taking the Bite out of Binomial Factoring 267 Reining in Big and Tiny Numbers 268 Factoring the Difference of Squares 269 Factoring Differences and Sums of Cubes 271 Making Factoring a Multiple Mission 274 Practice Questions Answers and Explanations 277 Whaddya Know? Chapter 12 Quiz 279 Answers to Chapter 12 Quiz 280 Chapter 13: Factoring Trinomials and Special Polynomials 281 Recognizing the Standard Quadratic Expression 281 Focusing First on the Greatest Common Factor 283 Unwrapping the FOILing Package 284 The opening to unFOIL 284 Coming to the end of the FOIL roll 287 Factoring Quadratic-Like Trinomials 290 Factoring Trinomials Using More Than One Method 291 Factoring by Grouping 293 Putting All the Factoring Together and Making Factoring Choices 297 Combining unFOIL and the GCF 297 Grouping and unFOILing in the same package 298 Incorporating the Remainder Theorem 301 Synthesizing with synthetic division 302 Choosing numbers for synthetic division 303 Practice Questions Answers and Explanations 305 Whaddya Know? Chapter 13 Quiz 310 Answers to Chapter 13 Quiz 311 Book 5: Solving Linear And Polynomial Equations 313 Chapter 14: Establishing Ground Rules for Solving Equations 315 Creating the Correct Setup for Solving Equations 316 Setting up equations for further action 316 Making plans for solving equations 316 Keeping Equations Balanced 318 Balancing with binary operations 318 Squaring both sides and suffering the consequences 320 Taking a root of both sides 322 Solving with Reciprocals 323 Making a List and Checking It Twice 324 Doing a reality check 324 Thinking like a car mechanic when checking your work 325 Practice Problems Answers and Explanations 328 Whaddya Know? Chapter 14 Quiz 330 Answers to Chapter 14 Quiz 331 Chapter 15: Lining Up Linear Equations 333 Playing by the Rules 334 Using the Addition/Subtraction Property 334 Using the Multiplication/Division Property 336 Devising a method using division 336 Making the most of multiplication 337 Reciprocating the invitation 338 Putting Several Operations Together 339 Solving Linear Equations with Grouping Symbols 343 Nesting isn’t for the birds 343 Distributing first 343 Multiplying or dividing before distributing 346 Working with Proportions 349 Using the rules for proportions 349 Transforming fractional equations into proportions 351 Solving for Variables in Formulas 352 Practice Questions Answers and Explanations 355 Whaddya Know? Chapter 15 Quiz 363 Answers to Chapter 15 Quiz 364 Chapter 16: Muscling Up to Quadratic Equations 367 Using the Square-Root Rule 368 Factoring for a Solution 370 Zeroing in on the multiplication property of zero 370 Assigning the greatest common factor and multiplication property of zero to solving quadratics 372 Solving Quadratics with Three Terms 373 Using the Quadratic Formula 379 Completing the Square 383 Imagining the Worst with Imaginary Numbers 384 Practice Problems Answers and Explanations 387 Whaddya Know? Chapter 16 Quiz 392 Answers to Chapter 16 Quiz 393 Book 6: Dealing With Non-Polynomial Equations and Inequalities 395 Chapter 17: Yielding to Higher Powers 397 Queuing Up to Cubic Equations 397 Solving perfectly cubed equations 398 Working with the not-so-perfectly cubed 400 Going for the greatest common factor 401 Grouping cubes 404 Solving cubics with integers 405 Determining How Many Possible Roots 407 Applying the Rational Root Theorem 408 Using the Factor/Root Theorem 410 Solving by Factoring 411 Solving Powers That Are Quadratic-Like 412 Solving Synthetically 416 Practice Questions Answers and Explanations 420 Whaddya Know? Chapter 17 Quiz 426 Answers to Chapter 17 Quiz 427 Chapter 18: Reeling in Radical and Absolute Value Equations 429 Raising Both Sides to Solve Radical Equations 430 Powering up by squaring both sides 430 Raising to higher powers 433 Doubling the Fun with Radical Equations .435 Solving Absolute Value Equations 437 Making linear absolute value equations absolutely wonderful 437 Factoring absolute value equations for solutions 440 Checking for Absolute Value Extraneous Roots 441 Practice Questions Answers and Explanations 443 Whaddya Know? Chapter 18 Quiz 447 Answers to Chapter 18 Quiz 448 Chapter 19: Getting Even with Inequalities 449 Defining the Inequality Notation 450 Pointing in the right direction 450 Grappling with graphing inequalities 450 Using the Rules to Work on Inequality Statements 451 Rewriting Inequalities Using Interval Notation 453 Solving Linear Inequalities 455 Solving Quadratic Inequalities 457 Dealing with Polynomial and Rational Inequalities 460 Solving Absolute-Value Inequalities 463 Solving Complex Inequalities 465 Practice Questions Answers and Explanations 467 Whaddya Know? Chapter 19 Quiz 471 Answers to Chapter 19 Quiz 472 Book 7: Evaluating Formulas and Story Problems 475 Chapter 20: Facing Up to Formulas 477 Working with Formulas 477 Measuring Up 479 Finding out how long: Units of length 479 Putting the Pythagorean Theorem to work 482 Deciphering Perimeter, Area, and Volume 484 Using perimeter formulas to get around 484 Squaring off with area formulas 486 Soaring with Heron’s formula 488 Working with volume formulas 490 Getting Interested in Using Percent 492 Compounding interest formulas 492 Gauging taxes and discounts 495 Working out the Combinations and Permutations 497 Counting down to factorials 498 Counting on combinations 498 Ordering up permutations 500 Practice Questions Answers and Explanations 502 Whaddya Know? Chapter 20 Quiz 505 Answers to Chapter 20 Quiz 507 Chapter 21: Making Formulas Work in Basic Story Problems 509 Setting Up to Solve Story Problems 509 Applying the Pythagorean Theorem 511 Using Geometry to Solve Story Problems 513 Working around Perimeter, Area, and Volume 515 Parading out perimeter and arranging area 515 Adjusting the area 517 Pumping up the volume 519 Going ’Round in Circles 523 Putting Distance, Rate, and Time in a Formula 525 Going the distance with the distance-rate-time formula 525 Figuring distance plus distance 527 Equating distances 528 Figuring distance and fuel 529 Counting on Interest and Percent 530 Practice Questions Answers and Explanations 532 Whaddya Know? Chapter 21 Quiz 537 Answers to Chapter 21 Quiz 539 Chapter 22: Relating Values in Story Problems 541 Tackling Age Problems 542 Tackling Consecutive Integer Problems 543 Working Together on Work Problems 545 Throwing an Object into the Air 547 Practice Questions Answers and Explanations 550 Whaddya Know? Chapter 22 Quiz 553 Answers to Chapter 22 Quiz 554 Chapter 23: Measuring Up with Quality and Quantity Story Problems 555 Achieving the Right Blend with Mixture Problems 556 Concocting the Correct Solution 100% of the Time 559 Dealing with Money Problems 561 Investigating investments and interest 561 Going for the green: Money 564 Practice Questions Answers and Explanations 567 Whaddya Know? Chapter 23 Quiz 571 Answers to Chapter 23 Quiz 572 Book 8: Getting a Grip On Graphing 573 Chapter 24: Getting a Handle on Graphing 575 Thickening the Plot with Points 575 Interpreting ordered pairs 576 Actually Graphing Points 577 Sectioning Off by Quadrants 578 Graphing Lines 579 Using points to lay out lines 579 Going with the horizontal and vertical 581 Graphing Lines Using Intercepts 583 Computing Slopes of Lines 585 Sighting the slope 585 Formulating slope 587 Graphing with the Slope-Intercept Form 590 Changing to the Slope-Intercept Form 591 Writing Equations of Lines 592 Given a point and a slope 593 Given two points 593 Picking on Parallel and Perpendicular Lines 594 Finding Distances between Points 595 Finding Midpoints of Segments 597 Practice Questions Answers and Explanations 598 Whaddya Know? Chapter 24 Quiz 602 Answers to Chapter 24 Quiz 604 Chapter 25: Extending the Graphing Horizon 605 Finding the Intersections of Lines 605 Graphing for intersections 606 Substituting to find intersections 607 Graphing Parabolas and Circles 608 Curling Up with Parabolas 609 Trying out the basic parabola 609 Putting the vertex on an axis 610 Going around in circles with a circular graph 612 Plotting and Plugging in Polynomial Graphs 614 Investigating Graphs of Inequality Functions 618 Taking on Absolute-Value Function Graphs 620 Graphing with Transformations 621 Sliding and multiplying 622 Practice Questions Answers and Explanations 625 Whaddya Know? Chapter 25 Quiz 632 Answers to Chapter 25 Quiz 633 Chapter 26: Coordinating Systems of Equations and Graphing 639 Defining Solutions of Systems of Equations 639 Solving Systems of Linear Equations 641 Using elimination 641 Using substitution 642 Introducing intersections of lines 643 Solving Systems Involving Non-Linear Equations 644 Taking on Systems of Three Linear Equations 647 Practice Problems Answers and Explanations 648 Whaddya Know? Chapter 26 Quiz 651 Answers to Chapter 26 Quiz 652 Index 655

Read more less

Book SynopsisThis innovative undergraduate textbook approaches number theory through the lens of abstract algebra. Written in an engaging and whimsical style, this text will introduce students to rings, groups, fields, and other algebraic structures as they discover the key concepts of elementary number theory. Inquiry-based learning (IBL) appears throughout the chapters, allowing students to develop insights for upcoming sections while simultaneously strengthening their understanding of previously covered topics. The text is organized around three core themes: the notion of what a “number” is, and the premise that it takes familiarity with a large variety of number systems to fully explore number theory; the use of Diophantine equations as catalysts for introducing and developing structural ideas; and the role of abstract algebra in number theory, in particular the extent to which it provides the Fundamental Theorem of Arithmetic for various new number systems. Other aspects of modern number theory – including the study of elliptic curves, the analogs between integer and polynomial arithmetic, p-adic arithmetic, and relationships between the spectra of primes in various rings – are included in smaller but persistent threads woven through chapters and exercise sets.Each chapter concludes with exercises organized in four categories: Calculations and Informal Proofs, Formal Proofs, Computation and Experimentation, and General Number Theory Awareness. IBL “Exploration” worksheets appear in many sections, some of which involve numerical investigations. To assist students who may not have experience with programming languages, Python worksheets are available on the book’s website. The final chapter provides five additional IBL explorations that reinforce and expand what students have learned, and can be used as starting points for independent projects. The topics covered in these explorations are public key cryptography, Lagrange’s four-square theorem, units and Pell’s Equation, various cases of the solution to Fermat’s Last Theorem, and a peek into other deeper mysteries of algebraic number theory.Students should have a basic familiarity with complex numbers, matrix algebra, vector spaces, and proof techniques, as well as a spirit of adventure to explore the “numberverse.”Table of ContentsPreface.- What is a Number?- A Quick Survey of the Last Two Millenia.- Number Theory in $\mathcal{Z}$ Beginning.- Number Theory in the Mod-n Era.- Gaussian Number Theory: $\mathcal{Z}[i]$ of the Storm.- Number Theory: From Where We $\mathcal{R}$ to across the $mathcal{C}$.- Cyclotomic Number Theory: Roots and Reciprocity. Number Theory Unleashed: Release $\mathcal{Z}_p$!- The Adventure Continues.- Appendix: Number Systems.

Read more less

Book SynopsisDieses Buch will dem Leser eine Einführung in wichtige Techniken und Methoden der heutigen reellen Algebra und Geometrie vermitteln. An Voraussetzungen werden dabei nur Grundkenntnisse der Algebra erwartet, so daß das Buch für Studenten mittlerer Semester geeignet ist.Das erste Kapitel enthält zunächst grundlegende Fakten über angeordnete Körper und ihre reellen Abschlüsse und behandelt dann verschiedene Methoden zur Bestimmung der Anzahl reeller Nullstellen von Polynomen. Das zweite Kapitel befaßt sich mit reellen Stellen und gipfelt in Artins Lösung des 17. Hilbertschen Problems. Kapitel III schließlich ist dem noch jungen Begriff des reellen Spektrums und seinen Anwendungen gewidmet."Neben dem 1987 erschienenen "Géometrie algébrique réelle" von J. Bochnak-M. Coste- M. Roy stellt die vorliegende Monographie das erste Lehrbuch auf diesem Gebiet dar... Damit liegt eine sehr empfehlenswerte Einführung...vor..." (H. Mitsch, Monatshefte für Mathematik 3/111, 1991)Trade Review“More than 30 years after its initial publication, the present textbook is still a very valuable source for results in real algebra. It can serve as a textbook for a university course, but also experts will benefit from the nice account of concepts and results. It’s great that the book is available again, in particular in an English translation for an international audience.” (Tim Netzer, zbMATH 1505.13001, 2023)Table of Contents1 Ordered fields and their real closures.- 2 Convex valuation rings and real places.- 3 The real spectrum.- 4 Recent developments.

Read more less

Book SynopsisThis textbook provides an accessible account of the history of abstract algebra, tracing a range of topics in modern algebra and number theory back to their modest presence in the seventeenth and eighteenth centuries, and exploring the impact of ideas on the development of the subject. Beginning with Gauss’s theory of numbers and Galois’s ideas, the book progresses to Dedekind and Kronecker, Jordan and Klein, Steinitz, Hilbert, and Emmy Noether. Approaching mathematical topics from a historical perspective, the author explores quadratic forms, quadratic reciprocity, Fermat’s Last Theorem, cyclotomy, quintic equations, Galois theory, commutative rings, abstract fields, ideal theory, invariant theory, and group theory. Readers will learn what Galois accomplished, how difficult the proofs of his theorems were, and how important Camille Jordan and Felix Klein were in the eventual acceptance of Galois’s approach to the solution of equations. The book also describes the relationship between Kummer’s ideal numbers and Dedekind’s ideals, and discusses why Dedekind felt his solution to the divisor problem was better than Kummer’s. Designed for a course in the history of modern algebra, this book is aimed at undergraduate students with an introductory background in algebra but will also appeal to researchers with a general interest in the topic. With exercises at the end of each chapter and appendices providing material difficult to find elsewhere, this book is self-contained and therefore suitable for self-study.Trade Review“This volume is well written and nicely complements other works on the history of algebra. It can be recommended to all mathematicians and students of mathematics who want to understand how algebra turned into the rather abstract field it is today.” (C. Baxa, Monatshefte für Mathematik, Vol. 201 (4), August, 2023)“The book under review is an excellent contribution to the history of abstract algebra and the beginnings of algebraic number theory. I recommend it to everyone interested in the history of mathematics.” (Franz Lemmermeyer, zbMATH 1411.01005, 2019)“This is a nice book to have around; it reflects careful scholarship and is filled with interesting material. … there is much to like about this book. It is quite detailed, contains a lot of information, is meticulously researched, and has an extensive bibliography. Anyone interested in the history of mathematics, or abstract algebra, will want to make the acquaintance of this book.” (Mark Hunacek, MAA Reviews, June 24, 2019)Table of ContentsIntroduction.- 1 Simple quadratic forms.- 2 Fermat’s Last Theorem.- 3 Lagrange’s theory of quadratic forms.- 4 Gauss’s Disquisitiones Arithmeticae.- 5 Cyclotomy.- 6 Two of Gauss’s proofs of quadratic reciprocity.- 7 Dirichlet’s Lectures.- 8 Is the quintic unsolvable?.- 9 The unsolvability of the quintic.- 10 Galois’s theory.- 11 After Galois – Introduction.- 12 Revision and first assignment.- 13 Jordan’s Traité.- 14 Jordan and Klein.- 15 What is ‘Galois theory’?.- 16 Algebraic number theory: cyclotomy.- 17 Dedekind’s first theory of ideals.- 18 Dedekind’s later theory of ideals.- 19 Quadratic forms and ideals.- 20 Kronecker’s algebraic number theory.- 21 Revision and second assignment.- 22 Algebra at the end of the 19th century.- 23 The concept of an abstract field.- 24 Ideal theory.- 25 Invariant theory.- 26 Hilbert’s Zahlbericht.- 27 The rise of modern algebra – group theory.- 28 Emmy Noether.- 29 From Weber to van der Waerden.- 30 Revision and final assignment.- A Polynomial equations in the 18th Century.- B Gauss and composition of forms.- C Gauss on quadratic reciprocity.- D From Jordan’s Traité.- E Klein’s Erlanger Programm.- F From Dedekind’s 11th supplement.- G Subgroups of S4 and S5.- H Curves.- I Resultants.- Bibliography.- Index.

Read more less

Book SynopsisThis is a basic introduction to modern algebra, providing a solid understanding of the axiomatic treatment of groups and then rings, aiming to promote a feeling for the evolutionary and historical development of the subject. It includes problems and fully worked solutions, enabling readers to master the subject rather than simply observing it.Table of Contents1 Sets and Mappings.- 2 The Integers.- 3 Introduction to Rings.- 4 Introduction to Groups.- 5 Rings.- 6 Topics in Group Theory.- Hints to Solutions.- Suggestions for Further Study.

Read more less

Book SynopsisThis book provides a comprehensive knowledge of linear algebra for graduate and undergraduate courses. As a self-contained text, it aims at covering all important areas of the subject, including algebraic structures, matrices and systems of linear equations, vector spaces, linear transformations, dual and inner product spaces, canonical, bilinear, quadratic, sesquilinear, Hermitian forms of operators and tensor products of vector spaces with their algebras. The last three chapters focus on empowering readers to pursue interdisciplinary applications of linear algebra in numerical methods, analytical geometry and in solving linear system of differential equations. A rich collection of examples and exercises are present at the end of each section to enhance the conceptual understanding of readers. Basic knowledge of various notions, such as sets, relations, mappings, etc., has been pre-assumed.Table of Contents1. Algebraic Structures2. Matrices and Systems of Linear Equations3. Vector Spaces4. Linear Transformations5. Dual Spaces6. Inner Product Spaces7. Canonical Forms of an Operator8. Bilinear and Quadratic Forms9. Sesquilinear and Hermitian Forms10. Applications of Linear Algebra to Numerical Methods11. Affine and Euclidean Spaces and the Applications of Linear Algebra to Geometry12. Ordinary differential equations and linear systems of ordinary differential equations

Read more less

Book SynopsisWidely acclaimed algebra text. This book is designed to give the reader insight into the power and beauty that accrues from a rich interplay between different areas of mathematics.Table of ContentsPreface. Preliminaries. PART I: GROUP THEORY. Chapter 1. Introduction to Groups. Chapter 2. Subgroups. Chapter 3. Quotient Group and Homomorphisms. Chapter 4. Group Actions. Chapter 5. Direct and Semidirect Products and Abelian Groups. Chapter 6. Further Topics in Group Theory. PART II: RING THEORY. Chapter 7. Introduction to Rings. Chapter 8. Euclidean Domains, Principal Ideal Domains and Unique Factorization Domains. Chapter 9. Polynomial Rings. PART III: MODULES AND VECTOR SPACES. Chapter 10. Introduction to Module Theory. Chapter 11. Vector Spaces. Chapter 12. Modules over Principal Ideal Domains. PART IV: FIELD THEORY AND GALOIS THEORY. Chapter 13. Field Theory. Chapter 14. Galois Theory. PART V: AN INTRODUCTION TO COMMUTATIVE RINGS, ALGEBRAIC GEOMETRY, AND HOMOLOGICAL ALGEBRA. Chapter 15. Commutative Rings and Algebraic Geometry. Chapter 16. Artinian Rings, Discrete Valuation Rings, and Dedekind Domains. Chapter 17. Introduction to Homological Algebra and Group Cohomology. PART VI: INTRODUCTION TO THE REPRESENTATION THEORY OF FINITE GROUPS. Chapter 18. Representation Theory and Character Theory. Chapter 19. Examples and Applications of Character Theory. Appendix I: Cartesian Products and Zorn's Lemma. Appendix II: Category Theory. Index.

Read more less

Book SynopsisHow fast can you calculate? Would you like to be faster? This book presents the time honored tricks and tips of calculation, from a fresh perspective, to boost the speed at which you can add whether a couple of numbers, or columns so long an accountant may faint. Find out how to subtract, multiply, divide, and find square roots more quickly.Trade ReviewIf you think mental arithmetic is out of date in the 21st century, think again. This engaging book is about insight and interestingness beyond the simple utility of quicker calculations. The general style is original and characterful, and makes the book distinctive. * Prasenjit Saha, University of Zurich *This book is about very elementary concepts that ought to be read by sophisticated people who appreciate that elementary does not mean trivial. The author's erudite scholarship shines in the prose, along with just the right level of dry wit. It's serious stuff he's writing about (without numbers and arithmetic, our modern world simply vanishes into the ancient past where numbers were limited to none, one, and many), but in such a way that the reader does not slowly nod-off into a coma. * Paul J. Nahin, University of New Hampshire *Lipscombe's book is unusual, being, as it is, an expansive view of a small subject. The text he presents here is excellent, and is a model of everything a writer strives for: concision, simplicity, directness, accuracy, and surprise. * Don S. Lemons, Bethel College, Kansas *Table of ContentsPreface Introduction Challenge 1: Arithmetical Advice 2: Speedier Sums and Subtractions Interlude I: The Magic of 111,111 3: Accounting for Taste -- Adding Columns Quickly Interlude II: Checking, Check Digits, and Casting out Nines 4: Quicker Quotients and Pleasing Products -- Multiply and Divide by Specific Numbers Interlude III: Doomsday 5: Calculations with Constraints -- Multiply and Divide by Numbers with Specific Properties Interlude IV: Multicultural Multiplication 6: Super Powers -- Calculate Squares, Square Roots, Cube Roots, and More 7: Close-Enough Calculations -- Quick and Accurate Approximations Interlude V: Approximating the Number of Space Aliens 8: Multiplying Irrationally The Grand Finale Further Reading Appendix I: Calculating Doomsday Appendix II: The Squares from 1 to 100

Read more less

Book SynopsisAn Introduction to the Theory of Numbers by G.H. Hardy and E. M. Wright is found on the reading list of virtually all elementary number theory courses and is widely regarded as the primary and classic text in elementary number theory. Developed under the guidance of D.R. Heath-Brown this Sixth Edition of An Introduction to the Theory of Numbers has been extensively revised and updated to guide today''s students through the key milestones and developments in number theory. Updates include a chapter by J.H. Silverman on one of the most important developments in number theory -- modular elliptic curves and their role in the proof of Fermat''s Last Theorem -- a foreword by A. Wiles, and comprehensively updated end-of-chapter notes detailing the key developments in number theory. Suggestions for further reading are also included for the more avid readerThe text retains the style and clarity of previous editions making it highly suitable for undergraduates in mathematics from the first year upwards as well as an essential reference for all number theorists.Trade ReviewReview from previous edition Mathematicians of all kinds will find the book pleasant and stimulating reading, and even experts on the theory of numbers will find that the authors have something new to say on many of the topics they have selected... Each chapter is a model of clear exposition, and the notes at the ends of the chapters, with the references and suggestions for further reading, are invaluable. * Nature *This fascinating book... gives a full, vivid and exciting account of its subject, as far as this can be done without using too much advanced theory. * Mathematical Gazette *...an important reference work... which is certain to continue its long and successful life... * Mathematical Reviews *...remains invaluable as a first course on the subject, and as a source of food for thought for anyone wishing to strike out on his own. * Matyc Journal *Table of ContentsPREFACE TO THE SIXTH EDITION; PREFACE TO THE FIFTH EDITION; APPENDIX; LIST OF BOOKS; INDEX OF SPECIAL SYMBOLS AND WORDS; INDEX OF NAMES; GENERAL INDEX

Read more less

Book SynopsisAn introduction to category theory as a rigorous, flexible, and coherent modeling language that can be used across the sciences.Category theory was invented in the 1940s to unify and synthesize different areas in mathematics, and it has proven remarkably successful in enabling powerful communication between disparate fields and subfields within mathematics. This book shows that category theory can be useful outside of mathematics as a rigorous, flexible, and coherent modeling language throughout the sciences. Information is inherently dynamic; the same ideas can be organized and reorganized in countless ways, and the ability to translate between such organizational structures is becoming increasingly important in the sciences. Category theory offers a unifying framework for information modeling that can facilitate the translation of knowledge between disciplines.Written in an engaging and straightforward style, and assuming little background in mathematics, the book is

Read more less

Book SynopsisThis set featuresLinear Algebra and Its Applications, Second Edition (978-0-471-75156-4)Linear Algebra and Its Applications, Second Edition presents linear algebra as the theory and practice of linear spaces and linear maps with a unique focus on the analytical aspects as well as the numerous applications of the subject. In addition to thorough coverage of linear equations, matrices, vector spaces, game theory, and numerical analysis, the Second Edition features student-friendly additions that enhance the book''s accessibility, including expanded topical coverage in the early chapters, additional exercises, and solutions to selected problems.Beginning chapters are devoted to the abstract structure of finite dimensional vector spaces, and subsequent chapters address convexity and the duality theorem as well as describe the basics of normed linear spaces and linear maps between normed spaces.Further updates and revisions have been iTrade Review"...an informative and useful book, distinguished by its blend of theory and applications, which fulfills its goals admirably." (MAA Review March 2008)Table of ContentsPreface. Preface to the First Edition. 1. Fundamentals. 2. Duality. 3. Linear Mappings. 4. Matrices. 5. Determinant and Trace. 6. Spectral Theory. 7. Euclidean Structure. 8. Spectral Theory of Self-Adjoint Mappings. 9. Calculus of Vector- and Matrix-Valued Functions. 10. Matrix Inequalities. 11. Kinematics and Dynamics. 12. Convexity. 13. The Duality Theorem. 14. Normed Linear Spaces. 15. Linear Mappings Between Normed Linear Spaces. 16. Positive Matrices. 17. How to Solve Systems of Linear Equations. 18. How to Calculate the Eigenvalues of Self-Adjoint Matrices. 19. Solutions. Bibliography. Appendix 1. Special Determinants. Appendix 2. The Pfaffian. Appendix 3. Symplectic Matrices. Appendix 4. Tensor Product. Appendix 5. Lattices. Appendix 6. Fast Matrix Multiplication. Appendix 7. Gershgorin's Theorem. Appendix 8. The Multiplicity of Eigenvalues. Appendix 9. The Fast Fourier Transform. Appendix 10. The Spectral Radius. Appendix 11. The Lorentz Group. Appendix 12. Compactness of the Unit Ball. Appendix 13. A Characterization of Commutators. Appendix 14. Liapunov's Theorem. Appendix 15. The Jordan Canonical Form. Appendix 16. Numerical Range. Index.

Read more less

Book Synopsis

Read more less

Book SynopsisAccessible but rigorous, this outstanding text encompasses all of the topics covered by a typical course in elementary abstract algebra. Its easy-to-read treatment offers an intuitive approach, featuring informal discussions followed by thematically arranged exercises. This second edition features additional exercises to improve student familiarity with applications. 1990 edition.

Read more less

Book SynopsisOver 300 unusual problems, ranging from easy to difficult, involving equations and inequalities, Diophantine equations, number theory, quadratic equations, logarithms, more. Detailed solutions, as well as brief answers, for all problems are provided.

Read more less

Book SynopsisA thorough first course in linear algebra, this two-part treatment begins with the basic theory of vector spaces and linear maps, including dimension, determinants, eigenvalues, and eigenvectors. The second section addresses more advanced topics such as the study of canonical forms for matrices. Ample examples, applications, and exercises appear throughout the text. 1992 edition.

Read more less

Book SynopsisDirected at advanced undergraduates and graduate students of mathematics, this classic overview explores affine and projective geometry, symplectic and orthogonal geometry, the general linear group, and the structure of symplectic and orthogonal groups. Ample evidence of the author''s ability to penetrate a subject and to present material in a particularly elegant manner. ? Bulletin of the American Mathematical Society.

Read more less

Book SynopsisOne of the first books to approach Lie groups from the global point of view, this introductory treatment was the standard text on the subject for many years. Topics include the classical linear groups, topological groups, manifolds, analytic groups, differential calculus of Cartan, and compact Lie groups and their representations. The basic reference on Lie groups. — Bulletin of the American Mathematical Society.

Read more less

Book SynopsisIntended for advanced undergraduates and graduate students in mathematics, this precise, fundamental study pays particular attention to field theory and the ideal theory of Noetherian rings and Dedekind domains. 1958 edition.

Read more less

Book SynopsisRichard Stanley's two-volume basic introduction to enumerative combinatorics has become the standard guide to the topic for students and experts alike. This updated edition provides the only comprehensive high-level treatment of enumerative combinatorics, including the theory of symmetric functions, with over 150 new exercises and solutions.Trade Review'This is one of the great books; readable, deep and full of gems. It brings algebraic combinatorics to life. I teach out of it and feel that if I can get my students to 'touch Stanley' I have given them a gift for life.' Persi Diaconis, Stanford University'It is wonderful to celebrate the completion of the second edition of Richard Stanley's Enumerative Combinatorics, one of the finest mathematical works of all time. He has added nearly 200 exercises, together with their answers, to what was already a uniquely masterful summary of a vast and beautiful theory. When paired with the second edition of Volume 1, his two classic volumes will surely be a timeless treasure for generations to come.' Donald E. Knuth, Stanford University'An updated classic with a mesmerizing array of interconnected examples. Through Stanley's masterful exposition, the current and future generations of mathematicians will learn the inherent beauty and pleasures of enumeration.' June Huh, Princeton University'I have used Richard Stanley's books on Enumerative Combinatorics numerous times for the combinatorics classes I have taught. This new edition contains many new exercises, which will no doubt be extremely useful for the next generation of combinatorialists.' Anne Schilling, University of California, Davis'Richard Stanley's Enumerative Combinatorics, in two volumes, is an essential reference for researchers and graduate students in the field of enumeration. Volume 2, newly revised, includes comprehensive coverage of composition and inversion of generating functions, exponential and algebraic generating functions, and symmetric functions. The treatment of symmetric functions is especially noteworthy for its thoroughness and accessibility. Engaging problems and solutions, and detailed historical notes, add to the value of this book. It provides an excellent introduction to the subject for beginners while also offering advanced researchers new insights and perspectives.' Ira Gessel, Brandeis UniversityTable of ContentsPreface to Second Edition; Preface; 5. Trees and the Composition of Generating Functions; 6. Algebraic Generating Functions; 7. Symmetric Functions; Appendices: References; Index.

Read more less

Book SynopsisStudying abstract algebra can be an adventure of awe-inspiring discovery. The subject need not be watered down nor should it be presented as if all students will become mathematics instructors. This is a beautiful, profound, and useful field which is part of the shared language of many areas both within and outside of mathematics. To begin this journey of discovery, some experience with mathematical reasoning is beneficial. This text takes a fairly rigorous approach to its subject, and expects the reader to understand and create proofs as well as examples throughout.The book follows a single arc, starting from humble beginnings with arithmetic and high-school algebra, gradually introducing abstract structures and concepts, and culminating with Niels Henrik Abel and Evariste Galoisâ achievement in understanding how we canâand cannotârepresent the roots of polynomials. The mathematically experienced reader may recognize a bias toward commutative algebra and fondness for number theory. The presentation includes the following features: Exercises are designed to support and extend the material in the chapter, as well as prepare for the succeeding chapters. The text can be used for a one, two, or three-term course. Each new topic is motivated with a question. A collection of projects appears in Chapter 23. Abstract algebra is indeed a deep subject; it can transform not only the way one thinks about mathematics, but the way that one thinksâperiod. This book is offered as a manual to a new way of thinking. The authorâs aim is to instill the desire to understand the material, to encourage more discovery, and to develop an appreciation of the subject for its own sake.

Read more less

Book SynopsisCONTEMPORARY ABSTRACT ALGEBRA, ELEVENTH EDITION is intended for a course whose main purpose is to enable students to do computations and write proofs. This text stresses the importance of obtaining a solid introduction to the traditional topics, while at the same time presenting abstract algebra as a contemporary and very much active subject which is currently being used by working physicists, chemists, and computer scientists.For more than three decades, this classic text has been widely appreciated by instructors and students alike. The book offers an enjoyable read and conveys and develops enthusiasm for the beauty of the topics presented. It is comprehensive, lively, and engaging.The author presents the concepts and methodologies of used by working mathematicians, computer scientists, physicists, and chemists. Students will learn how to do computations and to write proofs. A unique feature of the book are exercises that build the skill of gener

Read more less

Book SynopsisStop algebra from being a mathematical gatekeeper. With rich math tasks, all students can succeed.Every teacher strives to make instruction effective and interesting, yet traditional methods of teaching algebra are not working for many students! That's a problem. But the answer isn't to supplement the curriculum with random tasks.Classroom Ready-Rich Math Tasks for Grades 6-12equips you with a cohesive solution--50+ mathematical tasks that are rich, research-based, standards-aligned, and classroom-tested. The tasks: Are organized into learning progressions that help all students make the leap from arithmetic to algebra Offer students interesting mathematics problems to think about and solve so math is investigative, interactive, and engaging Provide opportunities for you to connect new content to prior knowledge or focus on an underdeveloped concept Engage students in conceptual understanding, procedural practice, and problem solving through critical thinking and application Come with downloadable planning tools, student resource pages, and extension questions Include additional support for students who may be struggling Every learner deserves opportunities to engage in meaningful, rigorous mathematics. And every teacher can develop mathematical thinking and reasoning abilities in students. Part of the bestselling series spanning elementary and middle school,Classroom-Ready Rich Algebra Tasks, Grades 6-12is a powerful add-on to any core mathematics program at your school.Trade ReviewIn this astounding book, Dougherty and Venenciano skillfully illustrate tasks that develop students’ reasoning, problem-solving, and communication skills. They thoughtfully include guiding and reflective questions along with suggestions for differentiation and extensions of each task. The wide range of topics makes this perfect for any algebra course! -- Laura Ashley YoungBring the love of learning mathematics back into your classroom by implementing effective mathematical practices, building numeracy skills in your students, and providing rich algebra tasks that are engaging. Dougherty and Venenciano have done some great work to support educators that will enhance students’ learning and grit. -- Katie MajeresDougherty and Venenciano provoke us to reframe our thinking from, "can my students get the correct answer for this task?" to "how can I use this task to help further my students’ understanding of the mathematics?" The authors provide a vision, tools, and practical advice for strategically using engaging tasks to enhance students’ understanding of mathematical ideas and processes. -- Dewey GottliebThis is a treasure trove of amazing tasks and supporting materials! Framing the tasks in an overview of research-grounded practices promotes fidelity and provides access for all students. Tasks presented in Part 2 spell out the mathematical topics, content and practice alignments, watch-fors, anticipated solutions, prompts, and post-task notes that allow them to be efficiently integrated. -- Jessica IvyIt’s important to implement rich tasks that support students’ math proficiency, drawing from their strengths and assets. It can be difficult locating or creating math tasks that are both rich and assetbased. I am excited to use tasks from this book in my work at the university with preservice teachers, in professional development with in-service teachers, and working with grades 6–12 students. -- Jonathan D. BostieClassroom-Ready Rich Algebra Tasks is a must-have resource for all algebra teachers. Dougherty and Venenciano outline how to implement these tasks in your classroom and give numerous examples that could be used today. After more than a decade of teaching algebra and supporting algebra teachers, this is a resource I wish I had when I started. -- Nick DaviesTable of ContentsForeword by Kevin Dykema Preface Acknowledgements About the Authors Part 1: GETTING STARTED WITH RICH ALGEBRA TASKS Chapter 1: RICH ALGEBRA TASKS: WHAT ARE THEY, WHY ARE THEY VALUABLE, AND HOW DO I PLAN FOR IMPLEMENTATION? Chapter 2: LAYING THE GROUNDWORK FOR TEACHING WITH RICH ALGEBRA TASKS Chapter 3: IMPLEMENTING RICH ALGEBRA TASK LESSONS Chapter 4: TASKS TO ESTABLISH MATHEMATICAL COMMUNITY Part 2: RICH ALGEBRA TASKS Chapter 5: RATIONAL NUMBER TASKS Chapter 6: EXPRESSIONS TASKS Chapter 7: EQUATIONS TASKS Chapter 8: LINEAR AND NON-LINEAR RELATIONSHIP TASKS Chapter 9: SYSTEMS OF EQUATIONS TASKS Chapter 10: POLYNOMIAL AND RATIONAL EXPRESSIONS AND EQUATIONS TASKS Chapter 11: Your Turn Appendix A References

Read more less

Book SynopsisFrom rings to modules to groups to fields, this undergraduate introduction to abstract algebra follows an unconventional path. The text emphasizes a modern perspective on the subject, with gentle mentions of the unifying categorical principles underlying the various constructions and the role of universal properties. A key feature is the treatment of modules, including a proof of the classification theorem for finitely generated modules over Euclidean domains. Noetherian modules and some of the language of exact complexes are introduced. In addition, standard topics - such as the Chinese Remainder Theorem, the Gauss Lemma, the Sylow Theorems, simplicity of alternating groups, standard results on field extensions, and the Fundamental Theorem of Galois Theory - are all treated in detail. Students will appreciate the text''s conversational style, 400+ exercises, an appendix with complete solutions to around 150 of the main text problems, and an appendix with general background on basic loTrade Review'I highly recommend this excellent textbook for undergraduate algebra classes. The book features the 'rings-first' approach which I personally strongly prefer to the traditional 'groups-first' in an undergraduate class. The exposition is highly conceptual, mathematically neat and rigorous. A slight bias toward algebraic geometry and number theory is another advantage. The textbook can be fully covered within two semesters of undergraduate algebra providing the students with a rock solid foundation for further education in any algebra-related area of math: the breadth and depth of the coverage of all standard undergraduate algebra ideas and topics are just right.' Pavel Guerzhoy, University of Hawaii at Manoa'Aluffi's Algebra: Notes from the Underground will modernize the teaching of undergraduate algebra, much like his earlier book Algebra: Chapter 0 did for graduate algebra. The book emphasizes the aspects and mindset of the subject most relevant to current researchers in mathematics. The standard material covered in undergraduate algebra is very well-presented and is supplemented with numerous interesting applications to other fields such as algebraic geometry and number theory. The book maintains a conversational tone throughout and teaches the reader universal properties and the modern categorical viewpoint by infusion. This is an excellent book for a year-long undergraduate course in algebra and will bridge the gap between undergraduate and graduate courses on the subject. A wealth of well-chosen exercises and detailed solutions make this book an invaluable resource for self-study.' Izzet Coskun, University of Illinois at Chicago'Algebra: Notes from the Underground is the modern introduction to abstract algebra that mathematics has been missing. Paolo Aluffi invites students to fall in love with rings, modules, abelian groups, groups, and fields through a beautifully written tour that is equally suited for self-study or for a two-semester undergraduate course.' Emily Riehl, Johns Hopkins University'For anyone planning to teach an undergraduate abstract algebra class anytime soon I strongly recommend the book under review. I have nothing but praise for this book. … What really sets this text apart is Aluffi's writing style. His book has a remarkable narrative drive, and he is constantly reminding the reader of the big picture and tossing out tantalizing hints of what lies ahead.' John J. Watkins, MathSciNetTable of ContentsPart I. Rings: 1. The integers; 2. Modular arithmetic; 3. Rings; 4. The category of rings; 5. Canonical decomposition, quotients, and isomorphism theorems; 6. Integral domains; 7. Polynomial rings and factorization; Part II. Modules: 8. Modules and abelian groups; 9. Modules over integral domains; 10. Abelian groups; Part III. Groups: 11. Groups – preliminaries; 12. Basic results on finite groups; Part IV. Fields: 13. Field extensions; 14. Normal and separable extensions, and splitting fields; 15. Galois theory.

Read more less

Book SynopsisTable of ContentsPreface xiii Part One — Matrices 1 Basic properties of vectors and matrices 3 1 Introduction 3 2 Sets 3 3 Matrices: addition and multiplication 4 4 The transpose of a matrix 6 5 Square matrices 6 6 Linear forms and quadratic forms 7 7 The rank of a matrix 9 8 The inverse 10 9 The determinant 10 10 The trace 11 11 Partitioned matrices 12 12 Complex matrices 14 13 Eigenvalues and eigenvectors 14 14 Schur’s decomposition theorem 17 15 The Jordan decomposition 18 16 The singular-value decomposition 20 17 Further results concerning eigenvalues 20 18 Positive (semi)definite matrices 23 19 Three further results for positive definite matrices 25 20 A useful result 26 21 Symmetric matrix functions 27 Miscellaneous exercises 28 Bibliographical notes 30 2 Kronecker products, vec operator, and Moore-Penrose inverse 31 1 Introduction 31 2 The Kronecker product 31 3 Eigenvalues of a Kronecker product 33 4 The vec operator 34 5 The Moore-Penrose (MP) inverse 36 6 Existence and uniqueness of the MP inverse 37 7 Some properties of the MP inverse 38 8 Further properties 39 9 The solution of linear equation systems 41 Miscellaneous exercises 43 Bibliographical notes 45 3 Miscellaneous matrix results 47 1 Introduction 47 2 The adjoint matrix 47 3 Proof of Theorem 3.1 49 4 Bordered determinants 51 5 The matrix equation AX = 0 51 6 The Hadamard product 52 7 The commutation matrix Kmn 54 8 The duplication matrix Dn 56 9 Relationship between Dn+1 and Dn, I 58 10 Relationship between Dn+1 and Dn, II 59 11 Conditions for a quadratic form to be positive (negative) subject to linear constraints 60 12 Necessary and sufficient conditions for r(A : B) = r(A) + r(B) 63 13 The bordered Gramian matrix 65 14 The equations X1A + X2B′ = G1,X1B = G2 67 Miscellaneous exercises 69 Bibliographical notes 70 Part Two — Differentials: the theory 4 Mathematical preliminaries 73 1 Introduction 73 2 Interior points and accumulation points 73 3 Open and closed sets 75 4 The Bolzano-Weierstrass theorem 77 5 Functions 78 6 The limit of a function 79 7 Continuous functions and compactness 80 8 Convex sets 81 9 Convex and concave functions 83 Bibliographical notes 86 5 Differentials and differentiability 87 1 Introduction 87 2 Continuity 88 3 Differentiability and linear approximation 90 4 The differential of a vector function 91 5 Uniqueness of the differential 93 6 Continuity of differentiable functions 94 7 Partial derivatives 95 8 The first identification theorem 96 9 Existence of the differential, I 97 10 Existence of the differential, II 99 11 Continuous differentiability 100 12 The chain rule 100 13 Cauchy invariance 102 14 The mean-value theorem for real-valued functions 103 15 Differentiable matrix functions 104 16 Some remarks on notation 106 17 Complex differentiation 108 Miscellaneous exercises 110 Bibliographical notes 110 6 The second differential 111 1 Introduction 111 2 Second-order partial derivatives 111 3 The Hessian matrix 112 4 Twice differentiability and second-order approximation, I 113 5 Definition of twice differentiability 114 6 The second differential 115 7 Symmetry of the Hessian matrix 117 8 The second identification theorem 119 9 Twice differentiability and second-order approximation, II 119 10 Chain rule for Hessian matrices 121 11 The analog for second differentials 123 12 Taylor’s theorem for real-valued functions 124 13 Higher-order differentials 125 14 Real analytic functions 125 15 Twice differentiable matrix functions 126 Bibliographical notes 127 7 Static optimization 129 1 Introduction 129 2 Unconstrained optimization 130 3 The existence of absolute extrema 131 4 Necessary conditions for a local minimum 132 5 Sufficient conditions for a local minimum: first-derivative test 134 6 Sufficient conditions for a local minimum: second-derivative test 136 7 Characterization of differentiable convex functions 138 8 Characterization of twice differentiable convex functions 141 9 Sufficient conditions for an absolute minimum 142 10 Monotonic transformations 143 11 Optimization subject to constraints 144 12 Necessary conditions for a local minimum under constraints 145 13 Sufficient conditions for a local minimum under constraints 149 14 Sufficient conditions for an absolute minimum under constraints 154 15 A note on constraints in matrix form 155 16 Economic interpretation of Lagrange multipliers 155 Appendix: the implicit function theorem 157 Bibliographical notes 159 Part Three — Differentials: the practice 8 Some important differentials 163 1 Introduction 163 2 Fundamental rules of differential calculus 163 3 The differential of a determinant 165 4 The differential of an inverse 168 5 Differential of the Moore-Penrose inverse 169 6 The differential of the adjoint matrix 172 7 On differentiating eigenvalues and eigenvectors 174 8 The continuity of eigenprojections 176 9 The differential of eigenvalues and eigenvectors: symmetric case 180 10 Two alternative expressions for dλ 183 11 Second differential of the eigenvalue function 185 Miscellaneous exercises 186 Bibliographical notes 189 9 First-order differentials and Jacobian matrices 191 1 Introduction 191 2 Classification 192 3 Derisatives 192 4 Derivatives 194 5 Identification of Jacobian matrices 196 6 The first identification table 197 7 Partitioning of the derivative 197 8 Scalar functions of a scalar 198 9 Scalar functions of a vector 198 10 Scalar functions of a matrix, I: trace 199 11 Scalar functions of a matrix, II: determinant 201 12 Scalar functions of a matrix, III: eigenvalue 202 13 Two examples of vector functions 203 14 Matrix functions 204 15 Kronecker products 206 16 Some other problems 208 17 Jacobians of transformations 209 Bibliographical notes 210 10 Second-order differentials and Hessian matrices 211 1 Introduction 211 2 The second identification table 211 3 Linear and quadratic forms 212 4 A useful theorem 213 5 The determinant function 214 6 The eigenvalue function 215 7 Other examples 215 8 Composite functions 217 9 The eigenvector function 218 10 Hessian of matrix functions, I 219 11 Hessian of matrix functions, II 219 Miscellaneous exercises 220 Part Four — Inequalities 11 Inequalities 225 1 Introduction 225 2 The Cauchy-Schwarz inequality 226 3 Matrix analogs of the Cauchy-Schwarz inequality 227 4 The theorem of the arithmetic and geometric means 228 5 The Rayleigh quotient 230 6 Concavity of λ1 and convexity of λn 232 7 Variational description of eigenvalues 232 8 Fischer’s min-max theorem 234 9 Monotonicity of the eigenvalues 236 10 The Poincar´e separation theorem 236 11 Two corollaries of Poincar´e’s theorem 237 12 Further consequences of the Poincar´e theorem 238 13 Multiplicative version 239 14 The maximum of a bilinear form 241 15 Hadamard’s inequality 242 16 An interlude: Karamata’s inequality 242 17 Karamata’s inequality and eigenvalues 244 18 An inequality concerning positive semidefinite matrices 245 19 A representation theorem for ( ∑api )1/p 246 20 A representation theorem for (trAp)1/p 247 21 Hölder’s inequality 248 22 Concavity of log|A| 250 23 Minkowski’s inequality 251 24 Quasilinear representation of |A|1/n 253 25 Minkowski’s determinant theorem 255 26 Weighted means of order p 256 27 Schlömilch’s inequality 258 28 Curvature properties of Mp(x, a) 259 29 Least squares 260 30 Generalized least squares 261 31 Restricted least squares 262 32 Restricted least squares: matrix version 264 Miscellaneous exercises 265 Bibliographical notes 269 Part Five — The linear model 12 Statistical preliminaries 273 1 Introduction 273 2 The cumulative distribution function 273 3 The joint density function 274 4 Expectations 274 5 Variance and covariance 275 6 Independence of two random variables 277 7 Independence of n random variables 279 8 Sampling 279 9 The one-dimensional normal distribution 279 10 The multivariate normal distribution 280 11 Estimation 282 Miscellaneous exercises 282 Bibliographical notes 283 13 The linear regression model 285 1 Introduction 285 2 Affine minimum-trace unbiased estimation 286 3 The Gauss-Markov theorem 287 4 The method of least squares 290 5 Aitken’s theorem 291 6 Multicollinearity 293 7 Estimable functions 295 8 Linear constraints: the case M(R′) ⊂M(X′) 296 9 Linear constraints: the general case 300 10 Linear constraints: the case M(R′) ∩M(X′) = {0} 302 11 A singular variance matrix: the case M(X) ⊂M(V ) 304 12 A singular variance matrix: the case r(X′V +X) = r(X) 305 13 A singular variance matrix: the general case, I 307 14 Explicit and implicit linear constraints 307 15 The general linear model, I 310 16 A singular variance matrix: the general case, II 311 17 The general linear model, II 314 18 Generalized least squares 315 19 Restricted least squares 316 Miscellaneous exercises 318 Bibliographical notes 319 14 Further topics in the linear model 321 1 Introduction 321 2 Best quadratic unbiased estimation of σ2 322 3 The best quadratic and positive unbiased estimator of σ2 322 4 The best quadratic unbiased estimator of σ2 324 5 Best quadratic invariant estimation of σ2 326 6 The best quadratic and positive invariant estimator of σ2 327 7 The best quadratic invariant estimator of σ2 329 8 Best quadratic unbiased estimation: multivariate normal case 330 9 Bounds for the bias of the least-squares estimator of σ2, I 332 10 Bounds for the bias of the least-squares estimator of σ2, II 333 11 The prediction of disturbances 335 12 Best linear unbiased predictors with scalar variance matrix 336 13 Best linear unbiased predictors with fixed variance matrix, I 338 14 Best linear unbiased predictors with fixed variance matrix, II 340 15 Local sensitivity of the posterior mean 341 16 Local sensitivity of the posterior precision 342 Bibliographical notes 344 Part Six — Applications to maximum likelihood estimation 15 Maximum likelihood estimation 347 1 Introduction 347 2 The method of maximum likelihood (ML) 347 3 ML estimation of the multivariate normal distribution 348 4 Symmetry: implicit versus explicit treatment 350 5 The treatment of positive definiteness 351 6 The information matrix 352 7 ML estimation of the multivariate normal distribution: distinct means 354 8 The multivariate linear regression model 354 9 The errors-in-variables model 357 10 The nonlinear regression model with normal errors 359 11 Special case: functional independence of mean and variance parameters 361 12 Generalization of Theorem 15.6 362 Miscellaneous exercises 364 Bibliographical notes 365 16 Simultaneous equations 367 1 Introduction 367 2 The simultaneous equations model 367 3 The identification problem 369 4 Identification with linear constraints on B and Γ only 371 5 Identification with linear constraints on B, Γ, and ∑ 371 6 Nonlinear constraints 373 7 FIML: the information matrix (general case) 374 8 FIML: asymptotic variance matrix (special case) 376 9 LIML: first-order conditions 378 10 LIML: information matrix 381 11 LIML: asymptotic variance matrix 383 Bibliographical notes 388 17 Topics in psychometrics 389 1 Introduction 389 2 Population principal components 390 3 Optimality of principal components 391 4 A related result 392 5 Sample principal components 393 6 Optimality of sample principal components 395 7 One-mode component analysis 395 8 One-mode component analysis and sample principal components 398 9 Two-mode component analysis 399 10 Multimode component analysis 400 11 Factor analysis 404 12 A zigzag routine 407 13 A Newton-Raphson routine 408 14 Kaiser’s varimax method 412 15 Canonical correlations and variates in the population 414 16 Correspondence analysis 417 17 Linear discriminant analysis 418 Bibliographical notes 419 Part Seven — Summary 18 Matrix calculus: the essentials 423 1 Introduction 423 2 Differentials 424 3 Vector calculus 426 4 Optimization 429 5 Least squares 431 6 Matrix calculus 432 7 Interlude on linear and quadratic forms 434 8 The second differential 434 9 Chain rule for second differentials 436 10 Four examples 438 11 The Kronecker product and vec operator 439 12 Identification 441 13 The commutation matrix 442 14 From second differential to Hessian 443 15 Symmetry and the duplication matrix 444 16 Maximum likelihood 445 Further reading 448 Bibliography 449 Index of symbols 467 Subject index 471

Read more less

Book SynopsisTable of ContentsIntroduction 1 Part 1: The Questions 5 Chapter 1: The Big Four Operations 7 Chapter 2: Less than Zero: Working with Negative Numbers 11 Chapter 3: You’ve Got the Power: Powers and Roots 17 Chapter 4: Following Orders: Order of Operations 23 Chapter 5: Big Four Word Problems 29 Chapter 6: Divided We Stand 35 Chapter 7: Factors and Multiples 43 Chapter 8: Word Problems about Factors and Multiples 49 Chapter 9: Fractions 53 Chapter 10: Decimals 63 Chapter 11: Percents 69 Chapter 12: Ratios and Proportions 75 Chapter 13: Word Problems for Fractions, Decimals, and Percents 79 Chapter 14: Scientific Notation 87 Chapter 15: Weights and Measures 91 Chapter 16: Geometry 97 Chapter 17: Graphing 109 Chapter 18: Statistics and Probability 115 Chapter 19: Set Theory 123 Chapter 20: Algebraic Expressions 127 Chapter 21: Solving Algebraic Equations 133 Chapter 22: Solving Algebra Word Problems 139 Part 2: The Answers 143 Chapter 23: Answers 145 Index 407

Read more less