Whether you need help solving equations or determining the slope of a line, this guide gives you the tools you need to find your answers! Beginning with the basics, you will learn and practice all the skills needed to enhance your algebra expertise.This comprehensive guide covers all the key concepts, including:
Variables and expressions
Linear equations and inequalities
Monomials and polynomials
Exponents
Rational expressions
The Pythagorean theorem
Area and perimeter
Graphs and charts
Inside you'll find hundreds of examples to illustrate the basics and plenty of exercises to ensure mastery of these fundamentals. No matter if you're a student looking for a companion to your textbook, or a curious learner who's been away from the classroom too long, this will be your indispensable algebra primer.
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In math, as in any form of communication, there are rules that are agreed upon so that everyone can understand exactly what is being communicated. Oral and written languages use vocabulary, grammar, and sentence structure to communicate effectively. Mathematics uses its own form of these entities as well. Learning the language of mathematics is a key aspect of understanding the concepts being communicated. Numbers (analogous in some ways to the alphabet in which a language is written) and how these numbers are combined (much as letters are combined in the spelling of words) are the foundation for the study of algebra. This chapter will introduce the basic building blocks of algebra and give you a sturdy foundation for your future study.
Real Numbers and Subsets
Children learn about numbers quickly. Youngsters are taught to hold up their fingers when asked their age. They learn quickly to notice whether they received the same number of cookies as their siblings during snack time. Counting comes naturally. The natural numbers (or counting numbers) are 1, 2, 3, 4, …. These are the basic values from which we start.
Children learn about zero when they do not get what they asked for. Including zero with the set of natural numbers gives you the set of whole numbers. Students, especially those from colder climates, learn about the negative numbers when “it is too cold to play outside.”
When combined, the whole numbers and their negatives make up the set of integers. Children encounter fractions when they break their cookies apart or share their toys with a playmate. The formal name for fractions, rational numbers, comes from the idea that a fraction is the ratio of two integers. You can see that children have been exposed to a fair amount of mathematics before they have even entered their first classroom!
Having learned about square roots of the “nice” numbers (1, 4, 9, 16, 25, and so on), students are often confused when their teacher asks about the square root of 3. There is some relief in the class when they learn that numbers such as the square root of 3 are irrational. For the record, the numbers are irrational merely because they are not rational. That is, irrational numbers cannot be written as the ratio of two integers.
There is a hierarchy for these sets of numbers. All of the natural numbers are included in the set of whole numbers; all of the whole numbers are included in the set of integers; and all of the integers are included in the set of rational numbers. Of course, there are no numbers that are common to both the set of rational numbers and the set of irrational numbers. However, when these two sets are combined (this is called taking the union of the sets), a new set of numbers is formed, the set of real numbers. A simple explanation for the set of real numbers is that it represents all the numbers that can be placed on a number line. A graphical representation for the relationship among these sets of numbers follows.
Set of Real Numbers and Its Subsets
The number 8, for example, is a counting number (N), a whole number (W), an integer (Z), a rational number (Q), and a real number (R).
is a rational number (Q) and a real number (R), but it belongs to none of the other categories.
is an irrational number (I) and a real number (R).
Integer Arithmetic
A geometric approach to the arithmetic of integers will tie together what you have known for a long time, will reinforce or clarify what may have been a fuzzy rule, and will, we hope, make sense. Start with a simple example: 2 + 3 = 5. You can recall any number of different manipulatives (chips, coins, marbles, or blocks, for instance) that helped you with this problem when you were younger. The following figure illustrates another way in which this problem may have been shown to you.
Adding on a Number Line
Note that the starting point of the second n...
Table of contents
Cover
The everything Guide to Algebra
Welcome to the EVERYTHING series!
Title Page
Dedication
Copyright
Contents
Acknowledgments
Top 10 Reasons Why—And How—: You Should Study Algebra
Introduction
Chapter 1 The Basic Building Blocks
Chapter 2 Working the Process Backward:: Solving Linear Open Sentences
Chapter 3 Beyond the Line:: Polynomial Expressions
Chapter 4 Graphing Linear Relationships
Chapter 5 Functions
Chapter 6 Systems of Linear Equations
Chapter 7 Factoring Polynomials:: A Key to Success in Algebra
Chapter 8 Solving Quadratic Equations
Chapter 9 Quadratic Relationships
Chapter 10 Quadratic Systems
Chapter 11 Rational Expressions
Chapter 12 Absolute Value and Irrational Expressions
