Basic Math & Pre-Algebra For Dummies

Mark Zegarelli is the author of many successful For Dummies titles on math, logic, and test prep topics. He holds degrees in both English and math from Rutgers University and is the founder of SimpleStep Learning, an educational website (https://simplestep.co).

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