SOL Objective(s):
8.2 The student will describe orally and in writing the relationships between the subsets of the real number system.
The Basics and Background Information
• The set of real numbers includes natural numbers, counting numbers, whole numbers, integers, rational and irrational numbers.
• The set of natural numbers is the set of counting numbers {1, 2, 3, 4, ...}.
• The set of whole numbers includes the set of all the natural numbers or counting numbers and zero {0, 1, 2, 3…}.
• The set of integers includes the set of whole numbers and their opposites {…2, 1, 0, 1, 2…}.
• The set of rational numbers includes the set of all numbers that can be expressed as fractions in the form a/b where a and b are integers and b does not equal zero (e.g., 25 , 1/4, 2.3, 75%, 4.59 ) .
• The set of irrational numbers is the set of all nonrepeating, nonterminating decimals. An irrational number cannot be written in fraction form {e.g.,π , 2 , 1.232332333…}.
• The set of natural numbers is the set of counting numbers {1, 2, 3, 4, ...}.
• The set of whole numbers includes the set of all the natural numbers or counting numbers and zero {0, 1, 2, 3…}.
• The set of integers includes the set of whole numbers and their opposites {…2, 1, 0, 1, 2…}.
• The set of rational numbers includes the set of all numbers that can be expressed as fractions in the form a/b where a and b are integers and b does not equal zero (e.g., 25 , 1/4, 2.3, 75%, 4.59 ) .
• The set of irrational numbers is the set of all nonrepeating, nonterminating decimals. An irrational number cannot be written in fraction form {e.g.,π , 2 , 1.232332333…}.
Essential Knowledge and Skills
The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to...
• Describe orally and in writing the relationships among the sets of natural or counting numbers, whole numbers, integers, rational numbers, irrational numbers, and real numbers.
• Illustrate the relationships among the subsets of the real number system by using graphic organizers such as Venn diagrams. Subsets include rational numbers, irrational numbers, integers, whole numbers, and natural or counting numbers.
• Identify the subsets of the real number system to which a given number belongs.
• Determine whether a given number is a member of a particular subset of the real number system, and explain why.
• Describe each subset of the set of real numbers and include examples and nonexamples.
• Recognize that the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.
• Describe orally and in writing the relationships among the sets of natural or counting numbers, whole numbers, integers, rational numbers, irrational numbers, and real numbers.
• Illustrate the relationships among the subsets of the real number system by using graphic organizers such as Venn diagrams. Subsets include rational numbers, irrational numbers, integers, whole numbers, and natural or counting numbers.
• Identify the subsets of the real number system to which a given number belongs.
• Determine whether a given number is a member of a particular subset of the real number system, and explain why.
• Describe each subset of the set of real numbers and include examples and nonexamples.
• Recognize that the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.
Vocabulary and Things to Know:

Be Able To:Define subsets of Real Number System
Classify numbers. Create and/or Use a Venn diagram to show relationships amongst number classifications. 
The topic: What is it and what will we learn?
Numbers can appear in more than one subset, e.g., 4 is an integer, a whole number, a counting/natural number and a rational number. The attributes of one subset can be contained in whole or in part in another subset.
Classification and the Real Number SystemTeacher Lesson, but great resource using the Venn diagram, classification, and the Real Number System as an example, if you'd like to look.

Logic Diagrams of John Venn 
John Venn's WritingsOnline texts

Introduction to Set Concepts & Venn DiagramsThe basics are given in an easy to understand and format. Common set concept vocabulary is defined and used. The first video is a bit long, but certainly doesn't have to be viewed in it's entirety.


Problem Solving with Venn DiagramsProblems modeled and practice problems to try on your own.

