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Basic Math and Pre-Algebra All-In-One for Dummies (+ Chapter Quizzes Online) Zegarelli
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Library of Congress Control Number: 2023943405
ISBN 978-1-394-20124-2 (pbk); ISBN 978-1-394-20125-9 (ebk); ISBN 978-1-394-20126-6 (ebk)
Contents at a Glance
Unit 2: Getting
Unit 3: The Essentials of Trigonometry
Unit 4:
Unit 5: Analytic Geometry
Unit 6: Systems, Sequences, and Series
Unit 7: Onward to Calculus
Making Matrices Work
Introduction
Here you are: ready to take on these challenging pre-calculus topics — possibly on your way to calculus! Believe it or not, it was calculus that was responsible for my switching majors and taking on the exciting world of mathematics!
Pre-calculus books and classes are wonderful ways of taking the mathematics you’ve studied in the past and bolstering the experience with new, exciting, and challenging material. Some of what is presented in pre-calculus is review, but studying it and adding on to the topics is what will make you even more of a success in your next endeavor.
Maybe some of the concepts you’ve already covered in pre-calculus have given you a hard time, or perhaps you just want more practice. Maybe you’re deciding whether you even want to take pre-calculus and then calculus at all. This book fits the bill to help you with your decision for all those reasons. And it’s here to encourage you on your pre-calculus adventure.
You’ll find this book has many examples, valuable practice problems, and complete explanations. In instances where you feel you may need a more thorough explanation, please refer to Pre-Calculus For Dummies or Pre-Calculus Workbook For Dummies by Mary Jane Sterling (Wiley). This book, however, is a great stand-alone resource if you need extra practice or want to just brush up in certain areas.
About This Book
Pre-calculus can be a starting point, a middle point, and even a launching point. When you realize that you already know a whole bunch from Algebra I and Algebra II, you’ll see that precalculus allows you to use that information in a new way. Before you get ready to start this new adventure, you need to know a few things about this book.
This book isn’t a novel. It’s not meant to be read in order from beginning to end. You can read any topic at any time, but it’s structured in such a way that it follows a “typical” curriculum. Not everyone agrees on exactly what makes pre-calculus pre-calculus. So this book works hard at meeting the requirements of all those curriculums; hopefully, this is a good representation of any pre-calculus course.
Here are two different suggestions for using this book:
» Look up what you need to know when you need to know it. The index and the table of contents direct you where to look.
» Start at the beginning and read straight through. This way, you may be reminded of an old topic that you had forgotten (anything to get those math wheels turning inside your head). Besides, practice makes perfect, and the problems in this book are a great representation of the problems found in pre-calculus textbooks.
For consistency and ease of navigation, this book uses the following conventions:
» Math terms are italicized when they’re introduced or defined in the text.
» Variables are italicized to set them apart from letters.
» The symbol used when writing imaginary numbers is a lowercase i.
Foolish Assumptions
I don’t assume that you love math the way I do, but I do assume that you picked this book up for a reason special to you. Maybe you want a preview of the course before you take it, or perhaps you need a refresher on the topics in the course, or maybe your kid is taking the course and you’re trying to help them to be more successful.
It has to be assumed that you’re willing to put in some time and effort here. Pre-calculus topics include lots of algebraic equations, geometric theorems and rules, and trigonometry. You will see how these topics are used and intertwined, but you may need to go deeper into one or more of the topics than what is presented here.
And it’s pretty clear that you are a dedicated and adventurous person, just by the fact that you’re picking up this book and getting serious about what it has to offer. If you’ve made it this far, you’ll go even farther!
Icons Used in This Book
Throughout this book you’ll see icons in the margins to draw your attention to something important that you need to know.
You see this icon when I present an example problem whose solution I walk you through step by step. You get a problem and a detailed answer.
Tips are great, especially if you wait tables for a living! These tips are designed to make your life easier, which are the best tips of all!
The material following this icon is wonderful mathematics; it’s closely related to the topic at hand, but it’s not absolutely necessary for your understanding of the material being presented. You can take it or leave it — you’ll be fine just taking note and leaving it behind as you proceed through the section.
This icon is used in one way: It asks you to remember old material from a previous math course.
Warnings are big red flags that draw your attention to common mistakes that may trip you up.
When you see this icon, it’s time to tackle some practice questions. Answers and explanations appear in a separate section near the end of the chapter.
Beyond the Book
No matter how well you understand the concepts of algebra, you’ll likely come across a few questions where you don’t have a clue. Be sure to check out the free Cheat Sheet for a handy guide that covers tips and tricks for answering pre-calculus questions. To get this Cheat Sheet, simply go to www.dummies.com and type Pre-Calculus All In One For Dummies in the Search box.
The online quiz that comes free with this book contains over 300 questions so you can really hone your pre-calculus skills! To gain access to the online practice, all you have to do is register. Just follow these simple steps:
1. Register your book or ebook at Dummies.com to get your PIN. Go to www.dummies.com/go/getaccess.
2. Select your product from the dropdown list on that page.
3. Follow the prompts to validate your product, and then check your email for a confirmation message that includes your PIN and instructions for logging in.
If you do not receive this email within two hours, please check your spam folder before contacting us through our Technical Support website at http://support.wiley.com or by phone at 877-762-2974.
Now you’re ready to go! You can come back to the practice material as often as you want — simply log on with the username and password you created during your initial login. No need to enter the access code a second time.
Your registration is good for one year from the day you activate your PIN.
Where to Go from Here
Pick a starting point in the book and go practice the problems there. If you’d like to review the basics first, start at Chapter 1. If you feel comfy enough with your algebra skills, you may want to skip that chapter and head over to Chapter 2. Most of the topics there are reviews of Algebra II material, but don’t skip over something because you think you have it under control. You’ll find in pre-calculus that the level of difficulty in some of these topics gets turned up a notch or two. Go ahead — dive in and enjoy the world of pre-calculus!
If you’re ready for another area of mathematics, look for a couple more of my titles: Trigonometry For Dummies and Linear Algebra For Dummies.
1 Getting Started with Pre-Calculus
IN THIS UNIT . . .
Sharpening algebraic skills.
Reviewing number systems and their uses.
Describing basic function types.
Performing operations on real numbers and functions.
IN THIS CHAPTER
» Refreshing your memory on numbers and variables
» Accepting the importance of graphing
» Preparing for pre-calculus by understanding the vocabulary
Chapter 1
Preparing for Pre-Calculus
Pre-calculus is the bridge (drawbridge, suspension bridge, covered bridge) between Algebra II and calculus. In its scope, you review concepts you’ve seen before in math, but then you quickly build on them. You see some brand-new ideas, and even those build on the material you’ve seen before; the main difference is that the problems now get even more interesting and challenging (for example, going from linear systems to nonlinear systems). You keep on building until the end of the bridge span, which doubles as the beginning of calculus. Have no fear! What you find here will help you cross the bridge (toll free).
Because you’ve probably already taken Algebra I, Algebra II, and geometry, it’s assumed throughout this book that you already know how to do certain things. Just to make sure, though, I address some particular items in this chapter in a little more detail before moving on to the material that is pre-calculus.
If there is any topic in this chapter that you’re not familiar with, don’t remember how to do, or don’t feel comfortable doing, I suggest that you pick up another For Dummies math book and start there. If you need to do this, don’t feel like a failure in math. Even pros have to look up things from time to time. Use these books like you use encyclopedias or the Internet — if you don’t know the material, just look it up and get going from there.
Recapping Pre-Calculus: An Overview
Don’t you just love movie previews and trailers? Some people show up early to movies just to see what’s coming out in the future. Well, consider this section a trailer that you see a couple months before the Pre-Calculus For Dummies movie comes out! The following list presents some items you’ve learned before in math, and some examples of where pre-calculus will take you next.
» Algebra I and II: Dealing with real numbers and solving equations and inequalities.
Pre-calculus: Expressing inequalities in a new way called interval notation.
You may have seen solutions to inequalities in set notation, such as xx | 4 . This is read in inequality notation as x 4 . In pre-calculus, you often express this solution as an interval: 4, . (For more, see Chapter 2.)
» Geometry: Solving right triangles, whose sides are all positive.
You’ve learned that a length can never be negative. Well, in pre-calculus you sometimes use negative numbers for the lengths of the sides of triangles. This is to show where these triangles lie in the coordinate plane (they can be in any of the four quadrants).
» Geometry/trigonometry: Using the Pythagorean Theorem to find the lengths of a triangle’s sides.
Pre-calculus: Organizing some frequently used angles and their trig function values into one nice, neat package known as the unit circle (see Unit 3).
In this book, you discover a handy shortcut to finding the sides of triangles — a shortcut that is even handier for finding the trig values for the angles in those triangles.
» Algebra I and II: Graphing equations on a coordinate plane.
Pre-calculus: Graphing in a brand-new way with the polar coordinate system (see Chapter 16).
Say goodbye to the good old days of graphing on the Cartesian coordinate plane. You have a new way to graph, and it involves goin’ round in circles. I’m not trying to make you dizzy; actually, polar coordinates can make you some pretty pictures.
» Algebra II: Dealing with imaginary numbers.
Pre-calculus: Adding, subtracting, multiplying, and dividing complex numbers gets boring when the complex numbers are in rectangular form abi . In pre-calculus, you become familiar with something new called polar form and use that to find solutions to equations you didn’t even know existed.
Checking in on Number Basics and Processes
When entering pre-calculus, you should be comfy with sets of numbers (natural, integer, rational, and so on). By this point in your math career, you should also know how to perform operations with numbers. You can find a quick review of these concepts in this section. Also, certain
properties hold true for all sets of numbers, and it’s helpful to know them by name. I review them in this section, too.
Understanding the multitude of number types: Terms to know
Mathematicians love to name things simply because they can; it makes them feel special. In this spirit, mathematicians attach names to many sets of numbers to set them apart and cement their places in math students’ heads for all time.
» The set of natural or counting numbers: {1, 2, 3 . . .}. Notice that the set of natural numbers doesn’t include 0.
» The set of whole numbers: {0, 1, 2, 3 . . .}. The set of whole numbers consists of all the natural numbers plus the number 0.
» The set of integers: {. . . –3, –2, –1, 0, 1, 2, 3 . . .}. The set of integers includes all positive and negative natural numbers and 0.
Dealing with integers is like dealing with money: Think of positives as having it and negatives as owing it. This becomes important when operating on numbers (see the next section).
» The set of rational numbers: The numbers that can be expressed as a fraction where the numerator and the denominator are both integers. The word rational comes from the idea of a ratio (fraction or division) of two integers.
Examples of rational numbers include (but in no way are limited to) 1 5 , 7 2 , and 0.23. A rational number is any number in the form p q where p and q are integers, but q is never 0. If you look at any rational number in decimal form, you notice that the decimal either stops or repeats.
» The set of irrational numbers: All numbers that can’t be expressed as fractions. Examples of irrational numbers include 2 , 4 3 , and .
The decimal value of an irrational number never ends and never repeats.
» The set of all real numbers: All the sets of numbers previously discussed. For an example of a real number, think of a number . . . any number. Whatever it is, it’s real. Any number from the previous bullets works as an example. The numbers that aren’t real numbers are imaginary.
Like telemarketers and pop-up ads on the web, real numbers are everywhere; you can’t get away from them — not even in pre-calculus. Why? Because they include all numbers except the following.
• A fraction with a zero as the denominator: Such numbers don’t exist and are called undefined.
• The square root of a negative number: These numbers are part of complex numbers; the negative root is the imaginary part (see Chapter 15). And this extends to any even root of a negative number.
• Infinity: Infinity is a concept, not an actual number. It describes a behavior.
» The set of imaginary numbers: square roots of negative numbers. Imaginary numbers have an imaginary unit, like i, 4i, and –2i. Imaginary numbers used to be considered to be made-up numbers, but mathematicians soon realized that these numbers pop up in the real world. They are still called imaginary because they’re square roots of negative numbers, but they are a part of the language of mathematics. The imaginary unit is defined as i 1 . (For more on these numbers, head to Chapter 15.)
» The set of complex numbers: the sum or difference of a real number and an imaginary number. Complex numbers include these examples: 32i, 22i , and 4 2 3 i .
However, they also cover all the previous lists, including the real numbers (3 is the same thing as 30i) and imaginary numbers (2i is the same thing as 02i ).
The set of complex numbers is the most complete set of numbers in the math vocabulary because it includes real numbers (any number you can possibly think of), imaginary numbers (i), and any combination of the two.
In Questions 1 to 4, determine the different ways you can refer to each number. Your choices are: natural, whole, integer, rational, irrational, and real. Select all that apply.
Looking at the fundamental operations you can perform on numbers
From positives and negatives to fractions, decimals, and square roots, you should know how to perform all the basic operations on all real numbers. These operations include adding, subtracting, multiplying, dividing, taking powers of, and taking roots of numbers. The order of operations is the way in which you perform these operations.
The mnemonic device used most frequently to remember the order is PEMDAS, which stands for
1. Parentheses (and other grouping devices such as brackets and division lines)
2. Exponents (and roots, which can be written as exponents)
3. Multiplication and Division (done in order from left to right)
4. Addition and Subtraction (done in order from left to right)
One type of operation that is often overlooked or forgotten about is absolute value. Absolute value gives you the distance from zero on the number line. Absolute value should be included with the parentheses step because you have to consider what’s inside the absolute-value bars first (because the bars are a grouping device). Don’t forget that absolute value is always positive or zero. Hey, even if you’re walking backward, you’re still walking!
Q. Simplify the expression using the order of operations: 34 3
. First, simplify the numerator. Do this by squaring the binomial, multiplying through by 3 , and then combining like terms.
Next, factor the 3 from the terms in the numerator and simplify the denominator.
Divide the numerator and denominator by 3. And, finally, rationalize the denominator.
In Questions 5 and 6, simplify each expression using the order of operations.
Knowing the properties of numbers: Truths to remember
Remembering the properties of numbers is important because you use them consistently in pre-calculus. You may not often use these properties by name in pre-calculus, but you do need to know when to use them. The following list presents properties of numbers.
» Reflexive property: aa. For example, 1010
» Symmetric property: If ab, then ba. For example, if 53 8 , then 8 53
» Transitive property: If ab and bc, then ac . For example, if 53 8 and 8 42, then 53 42.
» Commutative property of addition: a bb a. For example, 2 33 2.
» Commutative property of multiplication: a bb a . For example, 2 33 2.
» Associative property of addition: ab ca bc . For example, 23 42 34 .
» Associative property of multiplication: ab ca bc . For example, 23 42 34
» Additive identity: aa 0 . For example, 30 3
» Multiplicative identity: aa 1 . For example, 4 14
» Additive inverse property: aa 0. For example, 2 20
» Multiplicative inverse property: a a 1 1. For example, 2 1 2 1. (Remember: a 0 .)
» Distributive property: a bc ab ac . For example, 10 23 10 2103 203050.
» Multiplicative property of zero: a 00 . For example, 50 0.
» Zero-product property: If ab 0, then a 0 or b 0 . For example, if xx 20, then x 0 or x 20.
If you’re trying to perform an operation that isn’t on the previous list, then the operation probably isn’t correct. After all, algebra has been around since 1600 b.c., and if a property exists, someone has probably already discovered it. For example, it may look inviting to say that 10 23 10 23 23, but that’s incorrect. The correct process and answer is 10 23 10 2103 50 . Knowing what you can’t do is just as important as knowing what you can do.
Q. Use the associative and commutative properties to simplify the expression: 53 45 611 xy x
A. 11x + 4y + 9. Rewrite with the variable terms grouped followed by the numbers: 56 43 511 xx y . Now combine the like terms: 56 4 xx y 35 1111 49xy .
