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Section 5.5 Real Numbers and Their Properties Copyright 2013, 2010, 2007, Pearson, Education, Inc. What You Will Learn Properties of Real Numbers Closure Commutative Associative Distributive 5.5-2 Copyright 2013, 2010, 2007, Pearson, Education, Inc. 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 . 5.5-3 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Relationships Among Sets Real numbers Rational numbers Integers Whole numbers Natural numbers 5.5-4 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Irrational numbers Relationships Among Sets Natural Numbers Rational Numbers Real Numbers Integers Zero Fractions Negative Integers Terminating or repeating decimal numbers Irrational Numbers 5.5-5 Copyright 2013, 2010, 2007, Pearson, Education, Inc. 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. 5.5-6 Copyright 2013, 2010, 2007, Pearson, Education, Inc. 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. 5.5-7 Copyright 2013, 2010, 2007, Pearson, Education, Inc. 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. 5.5-8 Copyright 2013, 2010, 2007, Pearson, Education, Inc. 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. 5.5-9 Copyright 2013, 2010, 2007, Pearson, Education, Inc. 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 5.5-10 Copyright 2013, 2010, 2007, Pearson, Education, Inc. 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 5.5-11 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 2: Identifying Properties of Real Numbers Name the property illustrated. f) (2p) • 5 = 5 • (2p) Commutative property of multiplication 5.5-12 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 3: Simplify by Using the Distributive Property Simplify. 5 2 5 3 10 5 3 7 4 2 7 4 7 2 a) 5 2 3 b) 4 7 14 5.5-13 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 4: Distributive Property Use the distributive property to multiply 2(x + 5). Then simplify the result. 2x 5 2 x 2 5 2x 10 5.5-14 Copyright 2013, 2010, 2007, Pearson, Education, Inc.