1. Section 5.5 Real Numbers and Their Properties

2. What You Will Learn Properties of Real Numbers Closure Commutative
Associative
Distributive

3. Real Numbers
The set of real numbers is formed by the union of the rational and irrational numbers.
The symbol for the set of real numbers is .

4. Relationships Among Sets
Irrational numbers
Rational numbers
Integers
Whole numbers
Natural numbers
Real numbers

5. Relationships Among Sets
Natural
Numbers
Integers
Zero
Rational
Numbers
Negative
Integers
Fractions
Real
Numbers
Terminating or repeating decimal numbers
Irrational
Numbers

6. Properties of the Real Number System
Closure
If an operation is performed on any two elements of a set and the result is an element of the set, we say that the set is closed under that given operation.

7. Properties of the Real Number System
Commutative Property
Addition
a + b = b + a
Multiplication
a • b = b • a
for any real numbers a and b.

8. Properties of the Real Number System
Associative Property
Addition
(a + b) + c = a + (b + c)
Multiplication
(a • b) • c = a • (b • c)
for any real numbers a, b and c.

9. Properties of the Real Number System
Distributive Property of Multiplication over Addition
a • (b + c) = a • b + a • c
for any real numbers a, b and c.

10. Example 2: Identifying Properties of Real Numbers
Name the property illustrated.
a) = 5 + 2
Commutative property of addition
b) (x + 3) + 5 = x + (3 + 5)
Associative property of addition
c) 4 • (3 • y) = (4 • 3) • y
Associative property of multiplication

11. Example 2: Identifying Properties of Real Numbers
Name the property illustrated.
d) 9(w + 3) = 9 • w + 9 • 3
Distributive property of multiplication over addition
e) 5 + (z + 3) = 5 + (3 + z)
Commutative property of addition

12. Example 2: Identifying Properties of Real Numbers
Name the property illustrated.
f) (2p) • 5 = 5 • (2p)
Commutative property of multiplication

13. Example 3: Simplify by Using the Distributive Property

14. Example 4: Distributive Property
Use the distributive property to multiply 2(x + 5). Then simplify the result.