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Section 1.1 Sets of Numbers and Interval Notation

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Review of Basic Algebraic Concepts

1.1 Sets of Numbers and Interval Notation 1.2 Operations on Real Numbers 1.3 Simplifying Expressions 1.4 Linear Equations in One Variable 1.5 Applications of Linear Equations in One Variable 1.6 Literal Equations and Applications to Geometry 1.7 Linear Inequalities in One Variable 1.8 Properties of Integer Exponents and Scientific Notation

In Chapter 1, we present operations on real numbers, solving equations,

and applications. This puzzle will help you familiarize yourself with some basic terms and geometry formulas. You will use these terms and formulas when working the exercises in Sections 1.5 ...view middle of the document...

In mathematics, a collection of elements is called a set, and the symbols { } are used to enclose the elements of the set. For example, the set {a, e, i, o, u} represents the vowels in the English alphabet. The set {1, 3, 5, 7} represents the first four positive odd numbers. Another method to express a set is to describe the elements of the set by using set-builder notation. Consider the set {a, e, i, o, u} in set-builder notation.

description of set

Set-builder notation: {x | x is a vowel in the English alphabet}

“the set of”

“all x ”

“such that”

“x is a vowel in the English alphabet”

Consider the set {1, 3, 5, 7} in set-builder notation.

description of set

Set-builder notation: {x | x is an odd number between 0 and 8}

“the set of ”

“all x ”

“such that”

“x is an odd number between 0 and 8”

Several sets of numbers are used extensively in algebra. The numbers you are familiar with in day-to-day calculations are elements of the set of real numbers. These numbers can be represented graphically on a horizontal number line with a point labeled as 0. Positive real numbers are graphed to the right of 0, and negative real numbers are graphed to the left. Each point on the number line corresponds to exactly one real number, and for this reason, the line is called the real number line (Figure 1-1).

5 4 3 2 1 0 1 2 3 4 Negative numbers Positive numbers Figure 1-1 5

Several sets of numbers are subsets (or part) of the set of real numbers. These are The The The The The set set set set set of of of of of natural numbers whole numbers integers rational numbers irrational numbers

Section 1.1 Sets of Numbers and Interval Notation

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Definition of the Natural Numbers, Whole Numbers, and Integers

The set of natural numbers is {1, 2, 3, . . . }. The set of whole numbers is {0, 1, 2, 3, . . . }. The set of integers is { . . . , 3, 2, 1, 0, 1, 2, 3, . . . }. The set of rational numbers consists of all the numbers that can be defined as a ratio of two integers.

The set of rational numbers is 5 q 0 p and q are integers and q does not equal zero}.

p

Definition of the Rational Numbers

Example 1

Identifying Rational Numbers

Show that each number is a rational number by finding two integers whose ratio equals...

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