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**Unformatted text preview: **CHAPTER 0: Preliminary Topics 0.1: Sets of Numbers 0.2: Logic 0.3: Rounding 0.4: Absolute Value and Distance 0.5: Exponents and Radicals: Laws and Forms 0.6: Polynomial, Rational, and Algebraic Expressions 0.7: Factoring Polynomials 0.8: Factoring Rational and Algebraic Expressions 0.9: Simplifying Algebraic Expressions 0.10: More Algebraic Manipulations 0.11: Solving Equations 0.12: Solving Inequalities 0.13: The Cartesian Plane and Circles 0.14: Lines 0.15: Plane and Solid Geometry 0.16: Variation • This chapter will introduce and review concepts, skills, techniques, and formulas needed in precalculus and calculus. (Section 0.1: Sets of Numbers) 0.1.1 SECTION 0.1: SETS OF NUMBERS LEARNING OBJECTIVES • Be able to identify different sets of numbers. • Know how to write sets of real numbers using set-difference, set-builder, graphical, and interval forms. PART A: DISCUSSION • Theorems and formulas often require constants (denoted by c , n , a 1 , a 2 , etc .) to be from a particular set of numbers , usually the set of real numbers (denoted by ¡ ). • Sets of real numbers can correspond to solutions of equations (see Section 0.11), solutions of inequalities (see Section 0.12), and domains and ranges of functions (see Section 1.1). There are several ways to describe these sets. PART B: SETS OF NUMBERS A set is a collection of objects, called the elements or members of the set. • Two sets are equal when they consist of the same elements. Typically, order is irrelevant, and elements are not repeated. ¡ denotes the empty set , or null set , the set consisting of no elements. Let A and B be sets. A is a subset of B , denoted by A ¡ B , when every element of A is also an element of B . If A ¡ B , but A ¡ B , then A is a proper subset of B , denoted by A ¡ B . This means that B contains all of the elements of A , as well as at least one other element not in A . (Section 0.1: Sets of Numbers) 0.1.2 Some important sets of numbers are: ¡ , or Z , the set of integers . • This set consists of 1, 2, 3, etc.; their opposites, ¡ 1 , ¡ 2 , ¡ 3 , etc.; and 0. • ¡ comes from “Zahlen,” the German word for “integer.” • ¡ is in blackboard bold typeface. It is more commonly used than Z . ¡ , or Q , the set of rational numbers . • This is the set of all numbers that can be written in the form: integer nonzero integer • It is the set of numbers that can be written as finite (or terminating ) decimals or repeating decimals . • Examples include: Fraction form Decimal expansion 1 2 0.5 1 3 0.3 , or 0.3333 … ¡ 823 9900 ¡ 0.831 , or ¡ 0.8313131 … 7 1 , or 7 7 • As demonstrated by the last example, every integer is a rational number . That is, ¡ ¡ ¢ ....

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