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., sqrt of 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., sqrt of 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:

The topic: What is it and what will we learn?
The Real Number System is a classification or grouping of numbers based upon properties/definitions. You should be able to take any number and state what subsets the number is a part of. In addition, you should be comfortable showing relationships with Venn diagrams  a nesting Venn diagram for the entire Real Number System as well as intersecting Venn diagrams that compare/contrast two subsets.
Real Number SystemBasics of vocabulary associated with subsets.


Classifying Real NumbersInformation, explanations, and opportunities for you to try and then check to see how you've done.


Identifying Rational NumbersKhan Academy video clip that shows the difference between rational and irrational subsets.


Sets of Numbers in the Real Number SystemDefinitions and examples in diagram format with worksheet/answers to practice independently.

Real Number SystemWorksheet with answers for independent study and selfcheck.

Worksheets w/out Answer KeySome include information/notes.

Quia: Subsets of Real Number SystemOnline game for practice.

Sets of Numbers in the Real Number SystemFlow Chart (different than Venn diagram) and practice questions.

Math = Love: Real Number SystemPerhaps this is meant for teachers, but it has some great ideas for any student wanting to create something to study.

Common Number SetsThis site shares information that goes beyond what we need for class this year, but it has good information for those who wish to explore more information.
