32 CHAPTER R Review of Basic Concepts Properties of Real Numbers Recall the following basic properties. Properties of Real Numbers Let a, b, and c represent real numbers. Property Description Closure Properties a +b is a real number. ab is a real number. The sum or product of two real numbers is a real number. Commutative Properties a +b =b +a ab =ba The sum or product of two real numbers is the same regardless of their order. Associative Properties 1 a +b2 +c =a + 1b +c2 1ab2c =a1bc2 The sum or product of three real numbers is the same no matter which two are added or multiplied first. Identity Properties There exists a unique real number 0 such that a +0 =a and 0 +a =a. There exists a unique real number 1 such that a # 1 =a and 1 # a =a. The sum of a real number and 0 is that real number, and the product of a real number and 1 is that real number. Inverse Properties There exists a unique real number -a such that a + 1 −a2 =0 and −a +a =0. If a≠0, there exists a unique real number 1 a such that a # 1 a =1 and 1 a # a =1. The sum of any real number and its negative is 0, and the product of any nonzero real number and its reciprocal is 1. Distributive Properties a1b +c2 =ab +ac a1b −c2 =ab −ac The product of a real number and the sum (or difference) of two real numbers equals the sum (or difference) of the products of the first number and each of the other numbers. Multiplication Property of Zero 0 # a =a # 0 =0 The product of a real number and 0 is 0. CAUTION With the commutative properties, the order changes, but with the associative properties, the grouping changes. Commutative Properties Associative Properties 1x + 42 + 9 = 14 + x2 + 9 1x + 42 + 9 = x + 14 + 92 7 # 15 # 22 = 15 # 22 # 7 7 # 15 # 22 = 17 # 52 # 2
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