Section 5.5 Real Numbers and Their Properties

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Presentation transcript:

Section 5.5 Real Numbers and Their Properties

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

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 .

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

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

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.

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.

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.

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.

Example 2: Identifying Properties of Real Numbers Name the property illustrated. a) 2 + 5 = 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

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

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

Example 3: Simplify by Using the Distributive Property

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