1. The Set of Real Numbers
Honors Math – Grade 8

2. KEY CONCEPT The Real Numbers
The Real Numbers consist of all rational and irrational numbers.
The Venn Diagram below shows the relationships among the sets of numbers.
Real Numbers
Rational Numbers
Integers
Whole Numbers
Natural Numbers
Irrational Numbers

3. Classify the following real numbers.
To decide to which set of numbers a real number belongs, you may need to rename the number in a different form.
This is a repeating decimal.
Rational number
This is a natural number.
Whole number
Integer
Rational number

4. A number line can help compare and order real numbers.
KEY CONCEPT The Completeness Property for Points
on the Number Line
Every real number corresponds to exactly one point on a number line and every point on the number line corresponds to exactly one real number.
The origin of a number line is zero.
Positive and negative numbers are often called signed numbers.
Points to the left of zero correspond to negative numbers.
Points to the right of zero correspond to positive numbers.
Zero is neither positive nor negative.
A number line can help compare and order real numbers.

5. Order the real numbers from least to greatest.
Graph the following real numbers.
The farther to the right a number is on the number line, the greater it is.
Order the real numbers from least to greatest.

6. KEY CONCEPT Additive Inverse Property
The absolute value of a real number, n, is the distance from 0 to n on a number line.
Since distance is always positive, the absolute value of a nonzero number is always positive.
Two real numbers are opposites (or additive inverses) if they are on opposite sides of 0 and they have the same absolute value.
KEY CONCEPT Additive Inverse Property
For any real number a,
a + (–a) = 0 and –a + a = 0
Read “the opposite of 5.6.”
Read “the opposite of -5.6.”

7. Find the value of each expression.
First find the opposite of each factor.
What is the opposite of -30?
What is the opposite of -12?
Then evaluate the expression.

8. Find the value of each expression.
First find the absolute value of each number.
What is the absolute value of ?
What is the absolute value of 2.9?
Then evaluate the expression.

9. Find the value of each expression.
Find the opposite of the absolute value of the product of 9 and 75.
First multiply.
Find the opposite.
Find the absolute value.
Find the opposite of the absolute value of the sum of 3.9 and 5.2.
First add.

10. The set of real numbers under addition.
When you apply an operation (for example addition) to any numbers in a set and the result is also a number in that set, the set is said to be closed under the operation. This is called the Closure Property.
Determine if each set of numbers is closed under the indicated operation.
The set of real numbers under addition.
This means to check whether the sum of any two real numbers will result in another real number.
The set of real numbers is closed under addition.
The set of integers under division.
This means to check whether the quotient of any two integers will result in another integer.
A counterexample shows that a set of numbers is not closed under an operation.
The set of integers is not closed under division.