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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Serie: Wiley Self-Teaching Guides
- Sprache: Englisch
The most practical, complete, and accessible guide for understanding algebra If you want to make sense of algebra, check out Practical Algebra: A Self-Teaching Guide. Written by two experienced classroom teachers, this Third Edition is completely revised to align with the Common Core Algebra I math standards used in many states. You'll get an overview of solving linear and quadratic equations, using ratios and proportions, decoding word problems, graphing and interpreting functions, modeling the real world with statistics, and other concepts found in today's algebra courses. This book also contains a brief review of pre-algebra topics, including arithmetic and fractions. It has concrete strategies that help diverse students to succeed, such as: * over 500 images and tables that illustrate important concepts * over 200 model examples with complete solutions * almost 1,500 exercises with answers so you can monitor your progress Practical Algebra emphasizes making connections to what you already know and what you'll learn in the future. You'll learn to see algebra as a logical and consistent system of ideas and see how it connects to other mathematical topics. This book makes math more accessible by treating it as a language. It has tips for pronouncing and using mathematical notation, a glossary of commonly used terms in algebra, and a glossary of symbols. Along the way, you'll discover how different cultures around the world over thousands of years developed many of the mathematical ideas we use today. Since students nowadays can use a variety of tools to handle complex modeling tasks, this book contains technology tips that apply no matter what device you're using. It also describes strategies for avoiding common mistakes that students make. By working through Practical Algebra, you'll learn straightforward techniques for solving problems, and understand why these techniques work so you'll retain what you've learned. You (or your students) will come away with better scores on algebra tests and a greater confidence in your ability to do math.
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Table of Contents
COVER
TITLE PAGE
COPYRIGHT
ACKNOWLEDGMENTS
INTRODUCTION
1 BASIC CONCEPTS
1.1 Addition, Subtraction, Multiplication, Division
1.2 Order of Operations
1.3 Sets and Properties of Numbers
CHAPTER 1 TEST
CHAPTER 1 SOLUTIONS
CHAPTER 1 TEST SOLUTIONS
2 FRACTIONS
2.1 Basic Operations
2.2 Simplifying Fractions
CHAPTER 2 TEST
CHAPTER 2 SOLUTIONS
CHAPTER 2 TEST SOLUTIONS
3 LINEAR EQUATIONS
3.1 Solving One-Step Equations
3.2 Solving Two-Step Equations
3.3 Translating Between Words and Symbols
3.4 Solving Word Problems
3.5 Equations with Like Terms
3.6 Equations with Variables on Both Sides
3.7 Equations with Parentheses
3.8 Using Tables to Solve Word Problems Involving Values
3.9 Transforming Formulas
CHAPTER 3 TEST
CHAPTER 3 SOLUTIONS
CHAPTER 3 TEST SOLUTIONS
4 RATIOS AND PROPORTIONS
4.1 Expressing Ratios in Simplest Form
4.2 Using Ratios in the Real World
4.3 Proportions and Equations with Fractions
4.4 Converting Units
4.5 Percents
CHAPTER 4 TEST
CHAPTER 4 SOLUTIONS
CHAPTER 4 TEST SOLUTIONS
5 LINEAR INEQUALITIES
5.1 Basic Principles of Solving Inequalities
5.2 Representing Inequalities
5.3 Solving Linear Inequalities
5.4 Compound Inequalities
5.5 Word Problems with Inequalities
CHAPTER 5 TEST
CHAPTER 5 SOLUTIONS
CHAPTER 5 TEST SOLUTIONS
6 FUNCTIONS AND GRAPHS WITH TWO VARIABLES
6.1 Functions and Function Notation
6.2 Introduction to Graphing
6.3 Characteristics of Graphs
6.4 Evaluating Functions from Equations and Graphs
CHAPTER 6 TEST
CHAPTER 6 SOLUTIONS
CHAPTER 6 TEST SOLUTIONS
7 LINEAR FUNCTIONS AND THEIR GRAPHS
7.1 Introduction to Linear Functions
7.2 Slope Formula
7.3 Determining If an Ordered Pair Is a Solution to an Equation
7.4 Writing the Equation of a Line
7.5 Solving Systems of Linear Equations by Graphing
7.6 Solving Systems of Linear Equations by Substitution
7.7 Solving Systems of Linear Equations by Elimination
7.8 Solving Systems of Linear Inequalities
7.9 Word Problems with Systems of Linear Equations and Inequalities
CHAPTER 7 TEST
REVIEW TEST 1: CHAPTERS 1–7
CHAPTER 7 SOLUTIONS
CHAPTER 7 TEST SOLUTIONS
REVIEW TEST 1 SOLUTIONS
8 OPERATIONS WITH POLYNOMIALS
8.1 Adding and Subtracting Polynomials
8.2 Multiplying Monomials: Rules of Exponents
8.3 Dividing Monomials: Rules of Exponents
8.4 Multiplying Polynomials
8.5 Dividing Polynomials
CHAPTER 8 TEST
CHAPTER 8 SOLUTIONS
CHAPTER 8 TEST SOLUTIONS
9 QUADRATIC FUNCTIONS
9.1 Factoring a Monomial from a Polynomial
9.2 Factoring by Grouping
9.3 Factoring Trinomials
9.4 Special Cases of Factoring
9.5 Solving Quadratic Equations by Factoring
9.6 Radical Expressions
9.7 Solving Quadratic Equations by Completing the Square
9.8 Solving Quadratic Equations Using the Quadratic Formula
9.9 Graphing Quadratic Functions
9.10 Solving Quadratic Equations by Graphing
9.11 Solving Quadratic Equations by the Mean-Product Method
9.12 Solving Quadratic-Linear Systems
9.13 Using Quadratic Equations to Solve Word Problems
CHAPTER 9 TEST
CHAPTER 9 SOLUTIONS
CHAPTER 9 TEST SOLUTIONS
10 EXPONENTIAL FUNCTIONS
10.1 Graphing Exponential Functions
10.2 Using Exponential Functions to Solve Word Problems
CHAPTER 10 TEST
CHAPTER 10 SOLUTIONS
CHAPTER 10 TEST SOLUTIONS
11 SEQUENCES
11.1 Writing Recursive Formulas for Sequences
11.2 Writing Explicit Formulas for Arithmetic and Geometric Sequences
11.3 Modeling with Sequences
CHAPTER 11 TEST
CHAPTER 11 SOLUTIONS
CHAPTER 11 TEST SOLUTIONS
12 SUMMARY OF FUNCTIONS
12.1 Cubic, Square Root, and Cube Root Functions
12.2 Piecewise Functions
12.3 Transformations of Functions
12.4 Average Rate of Change of Functions
12.5 Comparing Functions
CHAPTER 12 TEST
CHAPTER 12 SOLUTIONS
CHAPTER 12 TEST SOLUTIONS
13 STATISTICS
13.1 Two-Way Tables
13.2 Dotplots and Histograms
13.3 Shape, Center, and Spread
13.4 Scatterplots and Regression
CHAPTER 13 TEST
REVIEW TEST 2: CHAPTERS 8–13
CHAPTER 13 SOLUTIONS
CHAPTER 13 TEST SOLUTIONS
REVIEW TEST 2 SOLUTIONS
FORMULAS
Linear Equations
Exponents
Radicals
Quadratic Equations
Exponential Equations
Sequences
Functions
Statistics
GLOSSARY OF MATHEMATICAL SYMBOLS
Basic Operations
Ratios and Proportions
Polynomial and Radical Expressions
Equations and Inequalities
Functions and Graphs
Statistics
GLOSSARY OF MATHEMATICAL TERMS
ABOUT THE AUTHORS
INDEX
END USER LICENSE AGREEMENT
List of Tables
Chapter 1
Table 1.1 Operations with signed numbers.
Table 1.2 Properties of Numbers
Chapter 2
Table 2.1 Operations with fractions.
Chapter 3
Table 3.1 Translating words into symbols.
Table 3.2 How to solve word problems.
Table 3.3 Examples and non-examples of like terms.
Table 3.4 Variables on 1 vs. 2 sides of an equation.
Table 3.5 Relating rate to distance and time.
Chapter 4
Table 4.1 Students in a 3:2 ratio.
Table 4.2 375 students in a 3:2 ratio.
Table 4.3 Problems with ratios or proportions.
Table 4.4 Place value.
Table 4.5 Problems with percentage.
Chapter 5
Table 5.1 Representing inequalities.
Table 5.2 Operations with inequalities.
Table 5.3 Inequalities in words.
Chapter 6
Table 6.1 Functional and non-functional vending machines.
Table 6.2 Characteristics of functions.
Chapter 7
Table 7.1 Types of slope.
Table 7.2 How to write the equation of a line.
Table 7.3 We can eliminate
u
or
v
.
Table 7.4 Comparing solution methods for systems of equations.
Table 7.5 Comparing systems of equations and inequalities.
Chapter 8
Table 8.1 Polynomials.
Table 8.2 Rules of exponents.
Chapter 9
Table 9.1 Vocabulary for multiplication, division, and factoring.
Table 9.2 Perfect squares.
Table 9.3 Examples and non-examples of radicals in simplest form.
Table 9.4 Operations with radicals.
Table 9.5 Examples of completing the square.
Table 9.6 Properties determined from equations of parabolas.
Table 9.7 Methods for Solving Quadratic Equations.
Chapter 10
Table 10.1 Number of grains of wheat per square.
Chapter 11
Table 11.1 Comparing subscript notation and function notation.
Table 11.2 Explicit formula for an arithmetic sequence.
Table 11.3 Explicit formula for a geometric sequence.
Table 11.4 Comparing recursive and explicit formulas.
Chapter 12
Table 12.1 Comparing quadratic and square root functions.
Table 12.2 Comparing cubic and cube root functions.
Table 12.3 Parent functions.
Table 12.4 Summary of transformation.
Chapter 13
Table 13.1 Frequency table.
Table 13.2 Measures of shape.
Table 13.3 Measures of center.
Table 13.4 Measures of spread.
Table 13.5 Hours studied vs. test grade.
Table 13.6 Form of a bivariate relationship.
Table 13.7 Direction of a bivariate relationship.
Table 13.8 Table with exponential growth.
Table 13.9 Scatterplots and residual plots.
List of Illustrations
Chapter 1
Figure 1.1 Number line
Figure 1.2 Terms associated with division.
Figure 1.3 Order of operations.
Figure 1.4 Sets of numbers.
Chapter 2
Figure 2.1 Dividing a rectangular pizza into eighths.
Figure 2.2 Numerator and denominator.
Figure 2.3 Adding fractions.
Figure 2.4 Multiplying fractions.
Figure 2.5 Adding unlike denominators.
Chapter 3
Figure 3.1 How many forks and how many plates?
Figure 3.2 Group similar items.
Figure 3.3 Vocabulary for like terms.
Figure 3.4 How many chocolate squares?
Figure 3.5 Distributive property.
Figure 3.6 Calculating coin value.
Chapter 4
Figure 4.1 Number line showing a 6:4 ratio.
Figure 4.2 5 students in a 3:2 ratio.
Figure 4.3 10 students in a 3:2 ratio.
Figure 4.4 15 students in a 3:2 ratio.
Figure 4.5 30 students in a 3:2 ratio.
Figure 4.6 Means and extremes in a proportion.
Figure 4.7 3%.
Chapter 5
Figure 5.1 5 is greater than 3.
Figure 5.2
x
< 10.
Figure 5.3
x
= 10.
Figure 5.4
x
≤ 10.
Figure 5.5 Interval notation for
x
≤ 10.
Figure 5.6 Set-builder notation for
x
≤ 10.
Figure 5.7 Graphing an intersection.
Chapter 6
Figure 6.1 Vocabulary for relations.
Figure 6.2 Function as a table, diagram, or set of ordered pairs.
Figure 6.3 Non-function as a table, diagram, or set of ordered pairs.
Figure 6.4 Vocabulary for the coordinate plane.
Figure 6.5 Graph of a table vs. graph of an equation.
Chapter 7
Figure 7.1 Describe the patterns in the tables.
Figure 7.2 Constant difference between consecutive
y
-values.
Figure 7.3 Three views of a function.
Figure 7.4 Slope of a line.
Figure 7.5 Point-slope form of the equation of a line.
Figure 7.6 System of linear equations.
Figure 7.7 Some systems aren't solved easily by graphing.
Figure 7.8
x
is less than 10.
Figure 7.9
x
is less than 10 on the coordinate plane.
Chapter 8
Figure 8.1 Multiplication as a rectangle.
Figure 8.2 Multiplication as addition of smaller rectangles.
Figure 8.3 Multiplication with the area model.
Chapter 9
Figure 9.1 Find the missing factor or product.
Figure 9.2 Find the missing factors.
Figure 9.3 Factoring binomials with a common binomial factor.
Figure 9.4 Factor by grouping.
Figure 9.5 Examples of multiplying binomials.
Figure 9.6 Factoring with the
ac
method.
Figure 9.7 Vocabulary for radicals.
Figure 9.8 Subsets of the real numbers.
Figure 9.9 Table and graph of a quadratic function.
Figure 9.10 Vocabulary for parabolas.
Figure 9.11 Mistakes to avoid when graphing parabolas.
Figure 9.12 Mean-product method.
Chapter 10
Figure 10.1 Equation of an exponential function.
Figure 10.2 Equation of an exponential function with a rate.
Chapter 11
Figure 11.1 Recursive formula for a sequence.
Figure 11.2 Comparing sequences and equations.
Chapter 12
Figure 12.1 Vocabulary for cubic functions.
Figure 12.2 Leading coefficients of cubic polynomials.
Figure 12.3 Piecewise function.
Figure 12.4 Vertical shift.
Figure 12.5 Horizontal shift.
Figure 12.6 Vertical stretch or compression.
Figure 12.7 Comparing the rate of change of linear and nonlinear functions....
Figure 12.8 Average rate of change.
Chapter 13
Figure 13.1 Bar graph.
Figure 13.2 Two-way table.
Figure 13.3 Joint frequency in a two-way table.
Figure 13.4 Bar graph vs. histogram.
Figure 13.5 The meaning of the mean.
Figure 13.6 Scatterplot of hours studied vs. test grade.
Figure 13.7 Strength of a bivariate relationship: correlation coefficient.
Figure 13.8 Residuals.
Guide
Cover
Table of Contents
Title Page
Copyright
Acknowledgments
Introduction
Begin Reading
Formulas
Glossary of Mathematical Symbols
Glossary of Mathematical Terms
About the Authors
Index
End User License Agreement
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Wiley Self-Teaching Guides teach practical skills from accounting to astronomy, management to mathematics. Look for them at your local bookstore.
Other Science and Math Wiley Self-Teaching Guides:
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Astronomy: A Self-Teaching Guide, by Dinah L. Moche
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All the Math You'll Ever Need: A Self-Teaching Guide, by Steve Slavin
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Practical Algebra: A Self-Teaching Guide, Second Edition, by Peter H. Selby and Steve Slavin
Quick Algebra Review: A Self-Teaching Guide, by Peter H. Selby and Steve Slavin
Quick Arithmetic: A Self-Teaching Guide, by Robert A. Carman and Marilyn J. Carman
Quick Business Math: A Self-Teaching Guide, by Steve Slavin
Quick Calculus: A Self-Teaching Guide, Second Edition, by Daniel Kleppner and Norman Ramsey
Statistics: A Self-Teaching Guide, by Donald Koosis
Practical Algebra
A Self-Teaching Guide
Third Edition
Bobson Wong
Larisa Bukalov
Steve Slavin
Copyright © 2022 by Jossey-Bass. All rights reserved.
Published by Jossey-BassA Wiley Brand111 River StHoboken, New Jersey 07030www.josseybass.com
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Cover Image and Design: Paul McCarthy
ACKNOWLEDGMENTS
Writing a book is hard. Writing a book while teaching full-time during a pandemic is even harder. Fortunately, many people helped make this edition of Practical Algebra a reality. Our students' mathematical struggles and joys over the years inspired us to write this book. Conversations with our colleagues at Bayside High School and Math for America helped us develop many of the ideas and techniques we describe. Bayside students Juliana Campopiano and Queena Yue helped us proofread the text. The team at Desmos designed a powerful online graphing tool that we used to create the graphs in this book. The staff at John Wiley & Sons (especially Pete Gaughan, Christine O'Connor, Riley Harding, Julie Kerr, and Mackenzie Thompson) have been especially patient and supportive. Larry Ferlazzo introduced us to publishing math books, opening up countless opportunities. Finally, our spouses and children deserve special mention for tolerating our conversations about this book, peppering us with mathematical questions over the years, and helping to keep our work in perspective.
INTRODUCTION
What is algebra? You may associate it with solving equations such as 2x + 7 = 19. However, both the history of algebra and the way that it's taught today show that algebra is much more. For thousands of years, people solved algebraic problems without symbols such as x and +. By the 9th century, people including the Persian mathematician Muḥammad ibn Mūsā al-Khwārizmī had popularized the idea of using an algorithm (a set of well-defined instructions) to determine unknown quantities. In fact, the word algebra comes from the Arab word al-jabr, meaning “the reduction,” from the title of al-Khwārizmī's most famous mathematical text, Kitāb al-jabr wa al-muqābalah. Symbolic notation didn't become widespread until European mathematicians such as François Viète and René Descartes developed them in the 16th and 17th centuries. Nowadays, algebra courses include not just equations but also functions (the special rules that define mathematical relationships) and real-world modeling with statistics. In short, today's algebra students must know how to understand word problems, make and interpret graphs, create and solve equations, and draw appropriate conclusions from data.
Not surprisingly, algebra makes many people nervous. Maybe you recall endless drills and elaborate procedures from years ago. Perhaps you're a middle school or high school student who's intimidated by the high level of abstract reasoning that's required. If so, you're not alone. We understand how you feel! For many years, we've taught all levels of high school math, so we have a lot of experience working with diverse learners. This book contains concrete strategies that help our students succeed. We strongly believe that people can get better at math if they have access to the right tools.
We wrote this book as a general introduction to algebra. We assume that you're familiar with basic arithmetic (adding, subtracting, multiplying, and dividing numbers) and fractions. If you're not comfortable with these topics, don't worry—we briefly review them in Chapters 1 and 2. Even if you are comfortable with them, we suggest that you look through these chapters anyway. We explain why these ideas work and how they're related to the algebraic ideas we discuss later on.
Each chapter in this book is divided into sections, with model examples and tips. At the end of each section, you'll find several exercises to help you practice and apply your skills. These exercises include what we call Questions to Think About (open-ended questions designed to help you think about important concepts) as well as dozens of word problems. Each chapter has a test with multiple-choice and open-ended questions. The solutions to all exercises and chapter tests are located at the end of each chapter.
As you work through this book, you'll see some important ideas about algebra that we emphasize:
Algebra is a language.
We believe that many people find algebra intimidating because the words and symbols we use, such as polynomial,
a
n
, and
f
(
x
), literally look like a different language. In addition, we don't just
write
math, we also
read
and
speak
it. In the Reading and Writing Tips, we discuss how to write and pronounce mathematical symbols as well as how to use them in context. We also include a glossary of mathematical terms and symbols in the back of the book.
Algebra should make sense.
We believe that algebra should be taught in a way that makes sense. In our experience, part of the reason why so many people suffer from math anxiety is that they see it as a collection of disjointed and confusing tricks. Throughout this book, we use techniques (such as the area model for multiplication) that relate to other mathematical topics, such as geometry and statistics. By making these connections, you can extend what you learned in one situation to another context, which will strengthen your mathematical skills and boost your confidence!
Algebra requires pictures.
As we taught during the pandemic, we had to adjust our instruction. We couldn't be with our students in person, so they often had to teach themselves more independently. Incorporating graphs, tables, diagrams, and other images into our teaching helped our students make sense of math. Since this book is a
self-teaching
guide, we've included many visual strategies throughout this book.
Algebra requires technology.
Calculators, computers, and other technology aren't just shortcuts for menial computations. They are now required for today's complex modeling tasks. Using technology helps us to see patterns more efficiently. Since each of these tools has vastly different user instructions, we don't include specific instructions for each device. Instead, we include Technology Tips that apply
no matter what device you're using
.
Algebra is a human endeavor.
We believe that algebra should not be perceived as a set of rigid rules developed by a select group of people. In fact, as we note throughout this book, many mathematical concepts were developed in different cultures around the world over thousands of years. (We mention some of the more interesting stories in the Did You Know? callouts.) In addition, we recognize that making mistakes is a natural part of doing math. In the Watch Out! callouts, we point out many of the common errors that we've seen students make over the years so that you can avoid them!
We hope that as you work through this book, you'll find that algebra can be less intimidating and more meaningful than you originally thought.
— Bobson Wong and Larisa Bukalov
1BASIC CONCEPTS
In this chapter, we review some of the concepts that students are typically expected to know before learning algebra. Although we don't have the space to fully develop these concepts, we point out some common mistakes and other important points that you should keep in mind. Even if you think that you know these topics, we recommend that you work through this chapter.
1.1 Addition, Subtraction, Multiplication, Division
Throughout this book, we use visual models to represent mathematical ideas. One important model is a number line, a line on which each point represents exactly one number. The numbers always increase from left to right. To show the scale, numbers are marked off at equal intervals. We draw an arrow at the end to indicate that the numbers extend infinitely in that direction.
Positive numbers, which we indicate with a + in front of the number, are numbers greater than 0. Negative numbers, which we indicate with a − in front of the number, are numbers less than 0. The word sign refers to the property of being positive or negative. The term signed numbers refers to numbers and their signs. Numbers that don't have a sign in front of them are understood to be positive.
On a horizontal number line (Figure 1.1), positive numbers lie to the right of 0, and negative numbers lie to the left of 0:
Did You Know?
The idea of positive numbers, negative numbers, and 0 may seem obvious to us now, but they actually developed around the world over thousands of years. By the 3rd century BCE, the Chinese were using counting rods of different colors to represent positive and negative numbers in their calculations. The 7th-century Indian mathematician Brahmagupta described rules in terms of “fortunes” (positive numbers) and “debts” (negative numbers). Ancient societies understood the concept of nothing (“we have no water”), but many cultures, such as the Egyptians, Romans, and Greeks, created complex mathematics without 0. The use of 0 didn't fully develop until the 5th century CE in India.
Figure 1.1 Number line
The absolute value of a number is its distance from 0 on a number line. Since the absolute value represents distance, it is always positive (unless we're talking about 0, which has an absolute value of 0). We use vertical bars to indicate absolute value. We read |+2| as “the absolute value of positive two.” For example, |+15| is equal to 15, |−15| is equal to 15, and |0| is equal to 0. Two numbers that are the same distance from 0 on the number line but have different signs, such as +2 and −2, are opposites. Zero is an exception—the opposite of 0 is itself.
In math, we have four basic operations (mathematical processes performed on quantities to get a result): addition, subtraction, multiplication, and division. When we combine quantities with operations, we make an expression, such as 5 + 3 and |+15| − 4.
Watch Out!
We use the + and − symbols to represent both addition and subtraction and the sign of a number.
When + and − represent the sign of a number (which only occurs
before
a number), we read + as “positive” and − as “negative.” We
never
put a space between the symbol and the number, so “negative 5” would be written −5, never − 5.
When + and − represent addition or subtraction (which only occurs
between
two numbers), we read + as “plus” and − as “minus,” and we put 1 space before and after the symbol. For example, 4 + 5, which is read as “4 plus 5,” means 5
is added to
4 to get a sum of 9.
The + and − symbols can represent both operations and signs in the same mathematical sentence. For example, +5 − −3 is read “positive 5 minus negative 3,” not “plus 5 minus minus 3.” Sometimes, we put parentheses around signed numbers to separate them from the addition or subtraction symbols, so we write +5 − −3 as (+5) − (−3). The parentheses are not pronounced.
You may recall working with number lines in elementary school. In this book, we also use squares to model signed numbers because they enable us to represent far more complicated ideas that we need to work with in algebra. To represent +1, we use a square whose area is +1. To represent −1, we use a square whose area is −1. (Don't worry about what a square with a negative area actually “means”—it's just a model!) A square with area +1 and a square with area −1 have a total area of 0. We call this pair a zero pair. We can group zero pairs into rectangles (think of them as “jumbo packs” of +1 or −1 squares) and use them to add signed numbers, as shown in Example 1.1:
Example 1.1 Evaluate −40 + 54.
Solution: When we evaluate an expression, we perform mathematical calculations to get a single number.
−40 + 54
= −40 + 40 + 14
Split +54 into +40 and +14.
= (−40 + 40) + 14
Group the −40 and +40 together to make 40 zero pairs.
= 14
The remainder is 14, the final answer.
In this example, we use the = symbol (which is called an equal sign and read “equals” or “is equal to”). The equal sign means that the expression on its left has the same value as the expression on its right. A mathematical statement containing an equal sign is called an equation. To make your work easier to read, do one part of the calculation at a time and write each step on a different line, starting each line with the equal sign.
Watch Out!
One common mistake when writing several equations on one line is to ignore the meaning of the equal sign. For example, when evaluating 2 + 3 + 4, some students write: 2 + 3 = 5 + 4 = 9. This “run-on” equation implies that 2 + 3, 5 + 4, and 9 are all equal, which isn't what we meant! Instead, write the following:
2 + 3 + 4
= 5 + 4
The sum of 2 and 3 is 5.
= 9
The sum of 5 and 4 is 9.
How to Add Signed Numbers
Determine the number with the larger absolute value.
Form zero pairs with the number with the smaller absolute value.
The remainder is the final answer, called the
sum
.
Addition and subtraction undo each other. For example, 5 + 3 − 3 equals 5. More formally, we say that addition and subtraction are inverse operations. This means that when we apply inverse operations on a number, the result is the original number. We can think of subtraction in terms of addition.
How to Subtract Signed Numbers
To subtract a
positive
number, add a negative number with the same absolute value, so 5 − 3 = 5 + (−3). The result, called the
difference
, is 2. (This models real-world behavior—adding debt lowers your net worth.)
To subtract a
negative
number, add a positive number with the same absolute value, so 5 − (−3) is the same as 5 + 3. The difference is 8. (This also models real-world behavior—removing debt raises your net worth.)
Example 1.2 illustrates how these rules work.
Example 1.2 Evaluate (−30) − (−46).
Solution:
−30 − (−46)
From −30, we remove 46 negative unit squares. Since we don't have any more negative unit squares, add 46 zero pairs (46 negative and 46 positive unit squares) and remove the 46 negative unit squares.
= −30 + 46
After removing the 46 negative unit squares, we have 30 negative and 46 positive unit squares.
= −30 + 30 + 16
To determine what we have left in step 2, we separate the +46 into 30 positive and 16 positive unit squares (since 46 − 30 equals 16).
= 16
The 30 negative and 30 positive unit squares make 30 zero pairs, which add up to 0, leaving 16 positive unit squares.
Technology Tip
Many calculators have different buttons for subtraction and negative numbers. Often, the subtraction button is located next to the buttons for addition, multiplication, and division. To change the sign of an entry, they have a button labeled +/- or (−), where the - symbol on the button is shorter than the − symbol. Some calculators will return an error if you try to use the subtraction button to change the sign of a number, so be careful! In contrast, most software applications and mathematical websites don't differentiate between the negative and subtraction symbols, so entering 5 − −3 will result in the correct answer of 8.
When we multiply numbers, we add groups of the same size.
How to Multiply Signed Numbers
Multiply the absolute values of the
factors
(the numbers being multiplied).
If we multiply two numbers with
different
signs, the result (called the
product
) is negative.
If we multiply two numbers with the
same
sign, the result is positive.
We write the multiplication of 3 times 2 using one of these methods:
with × between the numbers, as in 3 × 2
with · between the numbers, as in 3 · 2
with parentheses around one or both numbers, as in (3)(2), 3(2), or (3)2
We recommend not using the × symbol in algebra because it can easily be mistaken for the letter x, which has a special meaning that we discuss in Chapter 3.
Since the area of a rectangle is the product of its length and width, then we can use rectangles to represent multiplication. This idea dates back thousands of years to ancient Mesopotamia, Greece, and the Middle East. Unfortunately, we can't realistically show the difference between positive and negative dimensions with a rectangle, so we label the dimensions with the appropriate signed numbers and use the multiplication rules that we described above to find the correct sign of the product.
Example 1.3 Represent (−10)(−5) using a rectangle and evaluate the result.
Solution: We can represent this as a rectangle whose dimensions are −10 and −5:
NOTE: We can also think of this as removing 5 groups of −10, which results in a net increase of 50.
Here are some special cases of multiplication:
Any number multiplied by 0 equals 0.
For example, 4 groups of 0 is still 0.
A number multiplied by 1 equals itself.
We can explain this conceptually by noticing that 1 group of 4 is just that number, so 4(1) = 4.
A number multiplied by −1 equals its opposite.
For example, 4(−1) = −4 and −4(−1) = 4.
When we multiply a number by itself several times, we say that we raise it to a power. For example, we say that 2(2)(2)(2) equals 24, which we read as “two to the fourth power” or “two to the fourth.” In this case, 2 is called the base (the number being multiplied) and 4 is the power or exponent (the number of times the base is being multiplied). The exponent is written above and to the right of the base. The term power refers to both the number 16 (what 24 equals) as well as the exponent 4.
Here are some special cases for powers:
A number raised to the first power is equal to the number, so 2
1
= 2.
A number raised to the second power is
squared
, so 4
2
can be read as “four squared,” “four to the second power,” or “four to the second.” (We get this term from the formula for the area of a square, which is the length of its edge multiplied by itself.)
A number raised to the third power is
cubed
, so 4
3
can be read as “four cubed,” “four to the third power,” or “four to the third.” (We get this term from the formula for the volume of a cube, which is the length of its edge multiplied by itself three times.)
A positive number raised to a positive power is always positive. We can surround the base with parentheses, so (3)4, (+3)4, and 34 all represent the same quantity.
When we raise negative numbers to a power, we always surround the base with parentheses, so we write (−3)(−3)(−3)(−3) as (−3)4. If we raise a negative number to powers that are counting numbers, we see an interesting pattern in the signs:
(−3)
1
= −3
(−3)
2
= (−3)(−3) = +9
(−3)
3
= (−3)(−3)(−3) = −27
(−3)
4
= (−3)(−3)(−3)(−3) = +81
(−3)
5
= (−3)(−3)(−3)(−3)(−3) = −243
We summarize this pattern as follows:
A negative number raised to an odd power is negative.
A negative number raised to an even power is positive.
Reading and Writing Tip
We have no easy way to express in words the difference between numbers like −34 and (−3)4, since both can be pronounced as “negative 3 to the fourth power.” We find that people pronounce (−3)4 as “the quantity negative 3 to the fourth power,” “parentheses negative 3 to the fourth power,” or “negative 3 (pause) to the fourth power.” This is an example of a situation where mathematical symbols can communicate ideas more clearly and succinctly than words. Pay careful attention to how mathematical symbols are written. In the same way that a missing comma can completely change the meaning of a sentence, missing parentheses can give you a different answer!
When we divide numbers, we separate into groups of equal size. Multiplication and division are inverse operations.
How to Divide Signed Numbers
Divide the absolute values of the number that we divide (called the
dividend
) and the number that we divide by (called the
divisor
).
If we divide two numbers with different signs, the result (called the
quotient
) is negative.
If we divide two numbers with the same sign, the result is positive.
Division is often associated with fractions. A fraction is a quantity consisting of one number (called the numerator) divided by a nonzero number (called the denominator).
We write division using one of these methods:
Using the ÷ symbol (called the division symbol) between the two numbers, such as 8 ÷ 2
Using the / symbol between the two numbers, all written on the same line, such as 8/2
Writing one number on top of the other and separating the two with a
fraction bar
(sometimes called a
vinculum
), such as
All of these division examples are read as “8 divided by 2.” In this book, we prefer using the fraction bar to represent division. It provides the clearest separation of the quantities in division and minimizes the use of parentheses in more complicated mathematical statements.
When we use the ÷ or / symbol, the dividend appears before the symbol and the divisor appears after it. When we use a fraction bar, the dividend appears above it and the divisor appears below it. Figure 1.2 shows the terms associated with division:
Figure 1.2 Terms associated with division.
Some special cases of division deserve special attention:
Any number divided by 1 equals itself.
For example,
, which means 8 divided into 1 group, equals 8.
Any number divided by 0 is meaningless.
Another way of saying this is that a fraction can never have a denominator equal to 0. For example,
has no meaning since there is no number that when multiplied by 0 would give a product of 8 (this would have to be true since multiplication and division are inverse operations).
Any nonzero number divided by itself equals 1.
For example,
.
Zero divided by a nonzero number equals 0.
The fraction
equals 0.
The reciprocal of a number is 1 divided by that number.
The reciprocal of 8 is
.
The product of a number and its reciprocal is 1.
For example,
.
Example 1.4 Represent using a rectangle and evaluate the quotient.
Solution: Using a rectangle, we can think of this as dividing a rectangle that has an area of +6 and a side length of −3. Using the rules for dividing two numbers with different signs, we conclude that the quotient must be negative, so the answer is −2.
Table 1.1 summarizes the steps for operations with signed numbers:
Table 1.1 Operations with signed numbers.
Operation
Steps
Example
Addition
Determine the number with the larger absolute value.
Form zero pairs with the number with the smaller absolute value.
The remainder is the final answer.
−40 + 54 = −40 + 40 + 14 = 0 + 14 = 14
Subtraction
To subtract a
positive
number, add a negative number with the same absolute value.
To subtract a
negative
number, add a positive number with the same absolute value.
5 − 3 = 5 + (−3) = 2 5 − (−3) = 5 + 3 = 8
Multiplication
Multiply the absolute values of each number being multiplied.
If we multiply two numbers with
different
signs, the result is negative.
If we multiply two numbers with the
same
sign, the result is positive.
10(2) = 20 10(−2) = −20 −10(−2) = 20
Division
Divide the absolute values of each number being multiplied.
If we divide two numbers with
different
signs, the result is negative.
If we divide two numbers with the
same
sign, the result is positive.
One final note: although understanding the rules for operations with signed numbers is important, you can always use technology to help you with these calculations.
Exercises
Write the pronunciation of each expression.
−8 − (−12)
(+1) − (+3)
(+6) + (−4)
(+7)(+15)
(−2)(−32)
(−6)
4
Evaluate each expression:
|+7.5|
|−3|
|−889|
(+8) + (+5)
(+50) + (−10)
(−30) + (+20)
(−9) − (+5)
(−20) − (+30)
(−100) − (−40)
(+3)(−5)
(−3)(−7)
(−6)(+2)
(+2)
3
(+6)
2
(−7)
2
−(−1)
2
Questions to Think About
What is the difference between the words “plus” and “positive” as they are used in math?
What are two examples of real-life quantities that could be modeled by adding negative numbers?
What are two examples of real-life quantities that could be modeled by subtracting negative numbers?
Is (−1,234,567,890,000,000)
999
positive or negative? Explain.
1.2 Order of Operations
For many years, mathematicians didn't have a standardized set of rules for operations. When math education became more widespread in the 19th century, textbooks codified rules for what became known as the order of operations, the order in which mathematical operations should be performed. The growing popularity of computers in the last few decades has made the need for a standardized order of operations even more important.
In this book, we use the following convention (Figure 1.3) for order of operations:
GROUPING:
First, evaluate everything surrounded by parentheses, brackets, fraction bars, absolute value symbols, and other grouping symbols, working from the innermost symbols outwards. To group numbers inside parentheses, we use square brackets or another set of parentheses: 2 − (3 − [5 − 1]) or 2 − (3 − (5 − 1)).
Figure 1.3 Order of operations.
EXPONENTS:
Next, evaluate exponents.
MULTIPLICATION/DIVISION:
Next, when multiplication and division occur together, evaluate them
left to right
.
ADDITION/SUBTRACTION:
Finally, when addition and subtraction occur together, evaluate them
left to right
.
Watch Out!
Some textbooks use the mnemonic PEMDAS (which stands for Parentheses, Exponents, Multiplication, Division, Addition, Subtraction) to remember the order of operations. We recommend you avoid using PEMDAS since it implies that multiplication should be done before division and addition before subtraction. If you prefer using a mnemonic, we suggest PEMA (Parentheses, Exponents, Multiplication, Addition). Unfortunately, PEMA doesn't include division and subtraction, so you'll have to remember which operations are inverse operations (multiplication and division, addition and subtraction) and perform them left to right.
Some problems can be solved more easily using a calculator:
Example 1.5 Evaluate 2(−5)20.
Solution: The order of operations tells us that we need to evaluate the exponent first (in this case, the twentieth power) before the multiplication. The parentheses around −5 indicate that it, not −10 (which is the product of 2 and −5), is the base that is being raised to the twentieth power.
Multiplying −5 by itself 20 times is tedious, so we prefer using technology. To enter 2(−5)20 into a calculator, we use the exponent button (usually marked ^ or xy), typing something like:
With technology, we get an answer that looks like 1.9073486328125E14. This is your device's way of displaying 1.9073486328125 × 1014. This number is written in scientific notation, which consists of a number at least 1 and less than 10 that is multiplied by a power of 10. Translating this into the more familiar standard notation, this number is 190,734,863,281,250.
Just because you can solve a problem with technology doesn't mean that it's easy! Entering complicated expressions on the calculator can be quite challenging, as shown in Example 1.6:
Example 1.6 Evaluate .
Solution: The fraction bar acts as a grouping symbol, separating the numerator from the denominator, so we calculate each separately. In the denominator, we work with grouping symbols from the inside out.
=
Evaluate the power in the numerator (3 is squared).
=
Evaluate 4(9) in the numerator.
=
Evaluate addition and subtraction from left to right.
=
Add inside the parentheses in the denominator.
=
Multiply inside the absolute value symbols, which work as grouping symbols.
=
Calculate the absolute value of −18.
=
Subtract in the denominator.
= 3
Divide 45 by 15.
To enter this problem in the calculator, use the fraction tool if your device has one. This will put the numerator on top of the denominator, separated by a fraction bar. Doing so separates your numerator and denominator more clearly and reduces the likelihood of mistakes. Otherwise, you'd have to enter this problem on one line using many parentheses, which can be very confusing!
Technology Tip
When entering numbers into the calculator, keep the following in mind:
Use the +/− key, not the addition and subtraction keys, to make a number positive or negative.
Enter parentheses carefully. For example, (3)(2 + 1)
2
= 27, but (3(2 + 1))
2
= 81.
Use the exponent key (usually marked ^ or
x
y
) to enter powers.
To enter fractions, use the fraction tool if your calculator has one. (See
Example 1.6
.)
Exercises
Evaluate each expression.
6 − 3 + 2
1 − 7 + 8
12 ÷ 2(3)
8 ÷ 2(2 + 2)
7 + 5(8)
9 + (−3)
2
2(−4)
2
4(−2)
3
9 + |1 − 5|
2
Questions to Think About
Use the order of operations to explain why 3(4)
2
is 48 and not 144.
Use the order of operations to explain why 5(1)
2
is 5 and not 25.
Use the order of operations to explain why −4
2
is negative and not positive. (HINT: −4
2
can be rewritten as (−1)(4)
2
.)
1.3 Sets and Properties of Numbers
In math, we work with different groups, or sets, of numbers (Figure 1.4):
Counting numbers
are the numbers we use to count: 1, 2, 3, 4, and so on.
Whole numbers
are the counting numbers and zero: 0, 1, 2, 3, 4, and so on.
Figure 1.4 Sets of numbers.
Integers
are the whole numbers and their opposites: …, –3, –2, –1, 0, 1, 2, 3, …
Rational numbers
are numbers that can be expressed as an integer divided by a nonzero integer.
Example 1.7 Is every integer a whole number? Explain.
Solution: No. Whole numbers are the counting numbers and 0 (0, 1, 2, 3, …). Integers are the whole numbers and their opposites (…, –3, –2, –1, 0, 1, 2, 3, …), and negative integers are not whole numbers.
Example 1.8 Is every whole number a rational number? Explain.
Solution: The whole numbers are the counting numbers and 0: 0, 1, 2, 3, and so on. The rational numbers are numbers that can be represented as an integer divided by a nonzero integer. Every whole number can be represented as itself divided by 1. Thus, every whole number is a rational number.
Table 1.2 summarizes important properties of numbers, some of which we have already mentioned:
In this table, a, b, and c are variables—letters or other symbols that represent quantities that can change in value. We will discuss another important property that relates to addition and multiplication in Chapter 3.
Table 1.2 Properties of Numbers
Property
Description
Symbols
Example
Commutative property of addition
Numbers may be added in any order without changing the result.
a
+
b
=
b
+
a
3 + 4 = 4 + 3
Commutative property of multiplication
Numbers may be multiplied in any order without changing the result.
a
(
b
) =
b
(
a
)
4(3) = 3(4)
Associative property of addition
Numbers may be grouped in any way for addition without changing the result.
a
+ (
b
+
c
) = (
a
+
b
) +
c
3 + (4 + 5) = (3 + 4) + 5
Associative property of multiplication
Numbers may be grouped in any way for multiplication without changing the result.
a
(
bc
) = (
ab
)
c
3(4 · 5) = (3 · 4)5
Additive inverse property
A number added to its opposite equals 0.
a
+ (
−a
) = 0
4 + (−4) = 0
Additive identity property
A number added to 0 is unchanged.
a
+ 0 =
a
4 + 0 = 4
Multiplicative inverse property
A number multiplied by its reciprocal equals 1.
Multiplicative identity property
A number multiplied by 1 is unchanged.
a
(1) =
a
4(1) = 4
Exercises
Determine whether the following statements are true or false. Explain your answer.
Every integer is a counting number.
Every whole number is an integer.
Zero is a whole number but not a counting number.
Every counting number is positive.
Every whole number is positive.
Every integer is either positive or negative.
Every rational number is an integer.
The number 5 is a rational number.
Every whole number is rational.
Every rational number can be written as a fraction.
Every whole number can be written as a fraction.
Every quotient is a rational number.
Questions to Think About
Explain why subtracting two counting numbers does not always result in a counting number.
How are fractions different from rational numbers?
Is subtraction commutative? Explain your answer.
CHAPTER 1 TEST
Which number is a counting number?
0
−1
1
Which operation is the inverse of multiplication?
addition
subtraction
squaring
division
If a positive number is multiplied by a negative number, the result
is always positive.
is always negative.
can be positive or negative.
is always 0.
According to the order of operations, to calculate 1 + (3 − (6 − 4))
2
, which step must be done first?
6 − 4
1
2
1 + 3
3 − 6
Which number is equivalent to (−1)
15
?
+1
−1
+15
−15
What does the absolute value of a number represent?
the directed distance between any two points on a number line
the sum of two numbers on a number line
the positive distance between any two points on a number line
a number's distance from 0 on a number line
Which statement about
is correct?
It represents a rational number and a fraction, but not an integer.
It represents a rational number, a fraction, and an integer.
It represents an integer and a fraction but not a rational number.
It represents a fraction, but neither a rational number nor an integer.
If a negative number is subtracted from a positive number, which statement is correct?
The result is always positive.
The result is always negative.
The result can be positive or 0.
The result can be positive, negative, or 0.
Which of the following is equivalent to adding a negative number?
subtracting a positive number with the same absolute value
adding a positive number with the same absolute value
subtracting a negative number with the same absolute value
adding a positive number with a different absolute value
Is every integer a whole number? Explain.
Is 0 rational? Explain.
Write the pronunciation of 3
4
+ |−8| − (+5).
Use rectangles to calculate (−50) − (+40).
Calculate (12 − (3 + 1))
2
− 8.
Calculate
+ 1.
CHAPTER 1 SOLUTIONS
1.1.
“negative 8 minus negative 12”
“positive 1 minus positive 3”
“positive 6 plus negative 4”
“positive 7 times positive 15”
“negative 2 times negative 32”
“the quantity negative 6 to the fourth power”
7.5
3
889
13
40
−10
−14
−50
−60
−15
+21
−12
8
36
49
−1
−5
0
5
“Plus” is an operation that shows that two numbers are being added, while “positive” is a characteristic of one number.
Answers may vary. Examples include adding debt (which lowers the net worth) or adding ice cubes to a drink (which lowers its temperature).
Answers may vary. Examples include cancelling debt (which increases the net worth) or removing ice cubes from a drink (which raises its temperature).
Negative. A negative number raised to an odd power is negative.
1.2.
5
2
18
16
47
18
32
−32
25
6
9
100
The order of operations tells us to calculate powers before multiplication. We calculate 4
2
, which equals 16, before multiplying it by 3 to get 48. In contrast, (3(4))
2
= 12
2
= 144.
The order of operations tells us to calculate powers before multiplication. We calculate 1
2
, which equals 1, before multiplying it by 5 to get 5. In contrast, (5(1))
2
= 5
2
= 25.
The order of operations tells us to calculate powers before multiplication. We calculate 4
2
, which equals 16, before multiplying it by −1 to get −16.
1.3.
False. Some integers are negative or 0, and every counting number is positive.
True. Integers are whole numbers and their opposites.
True. Whole numbers are the counting numbers and 0.
True. The counting numbers are 1, 2, 3, 4, …, all of which are greater than 0.
False. Zero is a whole number but is neither positive nor negative.
False. Zero is an integer but is neither positive nor negative.
False. Some rational numbers, such as
, are not integers (
lies between the integers 0 and 1 on the number line).
True. The number 5 can be written as the quotient of 5 and 1, or
.
True. Every whole number can be written as the quotient of itself and 1, such as
.
True. A rational number consists of an integer divided by a nonzero integer, while a fraction consists of any quantity (not necessarily an integer) divided by another.
True. A whole number can be written as a fraction with a denominator of 1.
False. A noninteger fraction divided by an integer, such as
÷ 3, or
, can be written as a fraction but is not rational (since the numerator
is not an integer).
If the second number is larger than the first, then the result will be negative and will not be a counting number. For example, 2 − 3 is −1.
A rational number consists of an integer divided by a nonzero integer. A fraction consists of any quantity (not necessarily an integer) divided by a nonzero quantity.
Subtraction is not commutative. When the order in which numbers are subtracted is reversed, the sign of the difference is also reversed. For example, 5 − 4 = 1, but 4 − 5 = −1.
CHAPTER 1 TEST SOLUTIONS
(C)
(D)
(B)
(A)
(B)
(D)
(B)
(A)
(A)
No. Some integers are negative, and whole numbers (0, 1, 2, 3, …) cannot be negative.
Yes. Zero may be expressed as the quotient of 0 and a nonzero integer:
.
“Three to the fourth power plus the absolute value of negative 8 minus positive 5.”
−90
56
−2
