Basic Math & Pre-Algebra For Dummies

Introduction 1 About This Book 1 Foolish Assumptions 2 Icons Used in This Book 3 Beyond the Book 3 Where to Go from Here 4 Part 1: Getting Started with Basic Math and Pre-algebra 5 Chapter 1: Playing the Numbers Game 7 Inventing Numbers 8 Understanding Number Sequences 8 Evening the odds 9 Counting by threes, fours, fives, and so on 9 Getting square with square numbers 10 Composing yourself with composite numbers 11 Stepping out of the box with prime numbers 12 Multiplying quickly with exponents 13 Looking at the Number Line 14 Adding and subtracting on the number line 14 Getting a handle on nothing, or zero 15 Taking a negative turn: Negative numbers 16 Multiplying the possibilities 17 Dividing things up 17 Discovering the space in between: Fractions 19 Four Important Sets of Numbers 19 Counting on the counting numbers 20 Introducing integers 20 Staying rational 21 Getting real 21 Chapter 2: It’s All in the Fingers: Numbers and Digits 23 Knowing Your Place Value 23 Counting to ten and beyond 24 Telling placeholders from leading zeros 25 Reading long numbers 26 Close Enough for Rock ’n’ Roll: Rounding and Estimating 26 Rounding numbers 26 Estimating value to make problems easier 28 Table of Contents Chapter 3: The Big Four: Addition, Subtraction, Multiplication, and Division 29 Adding Things Up 30 In line: Adding larger numbers in columns 30 Carry on: Dealing with two-digit answers 31 Take It Away: Subtracting 31 Columns and stacks: Subtracting larger numbers 32 Can you spare a ten? Borrowing to subtract 33 Multiplying 35 Signs of the times 36 Memorizing the multiplication table 37 Double digits: Multiplying larger numbers 40 Doing Division Lickety-Split 42 Making short work of long division 42 Getting leftovers: Division with a remainder 45 Part 2: Getting a Handle on Whole Numbers 47 Chapter 4: Putting the Big Four Operations to Work 49 Knowing Properties of the Big Four Operations 50 Inverse operations 50 Commutative operations 51 Associative operations 52 Distribution to lighten the load 53 Doing Big Four Operations with Negative Numbers 53 Addition and subtraction with negative numbers 54 Multiplication and division with negative numbers 56 Understanding Units 57 Adding and subtracting units 57 Multiplying and dividing units 57 Understanding Inequalities 58 Doesn’t equal (≠) 58 Less than (<) and greater than (>) 59 Less than or equal to (≤) and greater than or equal to (≥) 59 Approximately equals (≈) 60 Moving Beyond the Big Four: Exponents, Square Roots, and Absolute Value 60 Understanding exponents 60 Discovering your roots 61 Figuring out absolute value 62 Chapter 5: A Question of Values: Evaluating Arithmetic Expressions 63 Seeking Equality for All: Equations 64 Basic Math & Pre-Algebra For Dummies, 2nd Edition Hey, it’s just an expression 64 Evaluating the situation 65 Putting the Three E’s together 66 Introducing Order of Operations 66 Applying order of operations to Big Four expressions 67 Using order of operations in expressions with exponents 70 Understanding order of precedence in expressions with parentheses 71 Chapter 6: Say What? Turning Words into Numbers 75 Dispelling Two Myths about Word Problems 76 Word problems aren’t always hard 76 Word problems are useful 76 Solving Basic Word Problems 77 Turning word problems into word equations 77 Plugging in numbers for words 80 Solving More-Complex Word Problems 82 When numbers get serious 82 Too much information 83 Putting it all together 85 Chapter 7: Divisibility 87 Knowing the Divisibility Tricks 87 Counting everyone in: Numbers you can divide everything by 88 In the end: Looking at the final digits 88 Add it up: Checking divisibility by adding up digits 90 Ups and downs: Divisibility by 11 91 Identifying Prime and Composite Numbers 92 Chapter 8: Fabulous Factors and Marvelous Multiples 95 Knowing Six Ways to Say the Same Thing 96 Connecting Factors and Multiples 97 Finding Fabulous Factors 97 Deciding when one number is a factor of another 97 Understanding factor pairs 98 Generating a number’s factors 98 Identifying prime factors 100 Finding the greatest common factor (GCF) 105 Making Marvelous Multiples 105 Generating multiples 105 Finding the least common multiple (LCM) 106 Table of Contents Part 3: Parts of the Whole: Fractions, Decimals, and Percents 109 Chapter 9: Fooling with Fractions 111 Slicing a Cake into Fractions 112 Knowing the Fraction Facts of Life 114 Telling the numerator from the denominator 114 Flipping for reciprocals 114 Using ones and zeros 115 Mixing things up 115 Knowing proper from improper 115 Increasing and Reducing Terms of Fractions 116 Increasing the terms of fractions 117 Reducing fractions to lowest terms 118 Converting between Improper Fractions and Mixed Numbers 120 Knowing the parts of a mixed number 121 Converting a mixed number to an improper fraction 121 Converting an improper fraction to a mixed number 122 Understanding Cross-multiplication 122 Making Sense of Ratios and Proportions 123 Chapter 10: Parting Ways: Fractions and the Big Four Operations 125 Multiplying and Dividing Fractions 126 Multiplying numerators and denominators straight across 126 Doing a flip to divide fractions 127 All Together Now: Adding Fractions 129 Finding the sum of fractions with the same denominator 129 Adding fractions with different denominators 130 Taking It Away: Subtracting Fractions 137 Subtracting fractions with the same denominator 137 Subtracting fractions with different denominators 138 Working Properly with Mixed Numbers 140 Multiplying and dividing mixed numbers 141 Adding and subtracting mixed numbers 142 Chapter 11: Dallying with Decimals 149 Understanding Basic Decimal Stuff 150 Counting dollars and decimals 150 Identifying the place value of decimals 152 Knowing the decimal facts of life 153 Basic Math & Pre-Algebra For Dummies, 2nd Edition Performing the Big Four with Decimals 157 Adding decimals 158 Subtracting decimals 159 Multiplying decimals 160 Dividing decimals 161 Converting between Decimals and Fractions 165 Making simple conversions 165 Changing decimals to fractions 165 Changing fractions to decimals 167 Chapter 12: Playing with Percents 171 Making Sense of Percents 171 Dealing with Percents Greater than 100% 172 Converting to and from Percents, Decimals, and Fractions 173 Going from percents to decimals 173 Changing decimals into percents 174 Switching from percents to fractions 174 Turning fractions into percents 175 Solving Percent Problems 175 Figuring out simple percent problems 176 Turning the problem around 177 Deciphering more-difficult percent problems 178 Putting All the Percent Problems Together 178 Identifying the three types of percent problems 179 Solving percent problems with equations 180 Chapter 13: Word Problems with Fractions, Decimals, and Percents 183 Adding and Subtracting Parts of the Whole in Word Problems 184 Sharing a pizza: Fractions 184 Buying by the pound: Decimals 185 Splitting the vote: Percents 185 Problems about Multiplying Fractions 186 Renegade grocery shopping: Buying less than they tell you to 186 Easy as pie: Working out what’s left on your plate 187 Multiplying Decimals and Percents in Word Problems 188 To the end: Figuring out how much money is left 188 Finding out how much you started with 189 Handling Percent Increases and Decreases in Word Problems 191 Raking in the dough: Finding salary increases 191 Earning interest on top of interest 192 Getting a deal: Calculating discounts 193 Table of Contents Part 4: Picturing and Measuring — Graphs, Measures, Stats, and Sets 195 Chapter 14: A Perfect Ten: Condensing Numbers with Scientific Notation 197 First Things First: Using Powers of Ten as Exponents 198 Counting zeros and writing exponents 198 Adding exponents to multiply 200 Working with Scientific Notation 200 Writing in scientific notation 201 Seeing why scientific notation works 202 Understanding order of magnitude 203 Multiplying with scientific notation 203 Chapter 15: How Much Have You Got? Weights and Measures 205 Examining Differences between the English and Metric Systems 206 Looking at the English system 206 Looking at the metric system 208 Estimating and Converting between the English and Metric Systems 210 Estimating in the metric system 211 Converting units of measurement 213 Chapter 16: Picture This: Basic Geometry 217 Getting on the Plane: Points, Lines, Angles, and Shapes 218 Making some points 218 Knowing your lines 218 Figuring the angles 219 Shaping things up 221 Closed Encounters: Shaping Up Your Understanding of 2-D Shapes 221 Polygons 222 Circles 224 Taking a Trip to Another Dimension: Solid Geometry 225 The many faces of polyhedrons 225 3-D shapes with curves 226 Measuring Shapes: Perimeter, Area, Surface Area, and Volume 227 2-D: Measuring on the flat 228 Spacing out: Measuring in three dimensions 235 Basic Math & Pre-Algebra For Dummies, 2nd Edition Chapter 17: Seeing Is Believing: Graphing as a Visual Tool 239 Looking at Three Important Graph Styles 240 Bar graph 240 Pie chart 241 Line graph 242 Using the xy-Graph 243 Plotting points on an xy-graph 244 Drawing lines on an xy-graph 245 Chapter 18: Solving Geometry and Measurement Word Problems 247 The Chain Gang: Solving Measurement Problems with Conversion Chains 248 Setting up a short chain 248 Working with more links 249 Pulling equations out of the text 250 Rounding off: Going for the short answer 251 Solving Geometry Word Problems 253 Working from words and images 253 Breaking out those sketching skills 255 Chapter 19: Figuring Your Chances: Statistics and Probability 259 Gathering Data Mathematically: Basic Statistics 260 Understanding differences between qualitative and quantitative data 260 Working with qualitative data 261 Working with quantitative data 264 Looking at Likelihoods: Basic Probability 266 Figuring the probability 267 Oh, the possibilities! Counting outcomes with multiple coins 268 Chapter 20: Setting Things Up with Basic Set Theory 271 Understanding Sets 272 Elementary, my dear: Considering what’s inside sets 273 Sets of numbers 275 Performing Operations on Sets 275 Union: Combined elements 276 Intersection: Elements in common 276 Relative complement: Subtraction (sorta) 277 Complement: Feeling left out 277 Table of Contents Part 5: the X-files: Introduction to Algebra 279 Chapter 21: Enter Mr X: Algebra and Algebraic Expressions 281 Seeing How X Marks the Spot 282 Expressing Yourself with Algebraic Expressions 282 Evaluating algebraic expressions 283 Coming to algebraic terms 285 Making the commute: Rearranging your terms 286 Identifying the coefficient and variable 287 Identifying like terms 288 Considering algebraic terms and the Big Four 289 Simplifying Algebraic Expressions 292 Combining like terms 292 Removing parentheses from an algebraic expression 293 Chapter 22: Unmasking Mr X: Algebraic Equations 299 Understanding Algebraic Equations 300 Using X in Equations 300 Choosing among four ways to solve algebraic equations 301 The Balancing Act: Solving for x 302 Striking a balance 303 Using the Balance Scale to Isolate X 304 Rearranging Equations and Isolating x 305 Rearranging terms on one side of an equation 305 Moving terms to the other side of the equals sign 306 Removing parentheses from equations 307 Cross-multiplying 309 Chapter 23: Putting Mr X to Work: Algebra Word Problems 311 Solving Algebra Word Problems in Five Steps 312 Declaring a variable 312 Setting up the equation 313 Solving the equation 314 Answering the question 314 Checking your work 315 Choosing Your Variable Wisely 315 Solving More-Complex Algebraic Problems 316 Charting four people 316 Crossing the finish line with five people 318 Basic Math & Pre-Algebra For Dummies, 2nd Edition Part 6: the Part of Tens 321 Chapter 24: Ten Little Math Demons That Trip People Up 323 Knowing the Multiplication Table 324 Adding and Subtracting Negative Numbers 324 Multiplying and Dividing Negative Numbers 325 Knowing the Difference between Factors and Multiples 325 Reducing Fractions to Lowest Terms 326 Adding and Subtracting Fractions 326 Multiplying and Dividing Fractions 327 Identifying Algebra’s Main Goal: Find X 327 Knowing Algebra’s Main Rule: Keep the Equation in Balance 328 Seeing Algebra’s Main Strategy: Isolate x 328 Chapter 25: Ten Important Number Sets to Know 329 Counting on Counting (or Natural) Numbers 330 Identifying Integers 330 Knowing the Rationale behind Rational Numbers 331 Making Sense of Irrational Numbers 331 Absorbing Algebraic Numbers 332 Moving through Transcendental Numbers 333 Getting Grounded in Real Numbers 333 Trying to Imagine Imaginary Numbers 333 Grasping the Complexity of Complex Numbers 334 Going beyond the Infinite with Transfinite Numbers 335 Index 337