Properties of Real Numbers

Sets In mathematics, a set is a collection of things Sets can be studies as a topic all on its own (known as set theory), but we only need to know a few basic ideas We often use braces { } to indicate a set Sets can have a finite number of objects, an infinite number of objects, or no objects This last set is called the empty set 2

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Sometimes we are concerned, not with an entire set, but some small part of the set This small part is also a set Because it is contained in a bigger set, we call it a subset of the larger set Every element in a subset is also in the larger set, but not every element in the larger set is in the subset 4

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The Real Numbers In algebra we will be dealing with the elements of a set called the set of real numbers This set has a number of subset Today you will learn the names of these subsets and how to know if a given number belongs in the subset or not 10

The Real Numbers In algebra you will be working with real numbers Unless otherwise indicated, any variable that you see or work with will represent some real number The real numbers form a set (a collection) and there are several subsets (a collection within a collection) of the real numbers that will be important for you to know about and to identify

The Real Numbers The real numbers contain both the rational numbers and the irrational numbers Every real number is either rational or irrational; it cannot be both! You must be able to tell whether a number is rational or irrational The next slide shows one way to think about how these sets are related

The Real Numbers

The Rational Numbers The rational numbers include three other subsets These are The integers The whole numbers The natural or counting numbers The next slide shows how these sets are related

The Rational Numbers

Natural Numbers

Whole Numbers

The Integers These integers extend the whole numbers to the left of zero on the number line

The Integers

The Rational Numbers

The Irrational Numbers

Example 1: Classify Real Numbers

Real Numbers Rational Numbers Integers Irrational Numbers Whole Numbers

Exercise 1.1 Handout

The Field of Real Numbers In the previous lesson you learned how the real numbers can be classified Now we must learn how to work with the real numbers In mathematics, what I will be describing is called a field A field consists of a set, two operations (i.e., two different ways to “act on” the things in the set), and the set of properties that you are about to study

The Field of Real Numbers In algebra II our field will consist of the set of real numbers, the operations of addition and multiplication, and these rules (one of each for addition and multiplication) Closure Commutativity Associativity Identity Element Inverse Element Distributive Property

The Field of Real Numbers

Closure Property

Commutative Property

Associative Property

Identity Property

Inverse Property

Distributive Property

Examples

Definitions of Subtraction and Division

Examples

Justification Definition of division Distributive Property Commutative Property for Multiplication Associative Property for Multiplication Multiplication Commutative Property for Addition

Examples Justification Definition of subtraction Commutative Property for Addition Associative Property for Addition Inverse Property of Addition Identity Property of Addition

Examples Justification Definition of Division Commutative Property for Multiplication Associative Property for Multiplication Inverse Property for Multiplication Identity Property for Multiplication

Examples Justification Definition of Division Distributive Property Commutative Property for Multiplication Associative Property for Multiplication Inverse Property for Multiplication Identity Property for Multiplication Addition

Examples Justification Commutative Property of Addition Associative Property of Addition Distributive Property (Factoring) Addition Commutative Property for Addition

Exercise 1.2: Number Properties Handout