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Math 90 1.7 "Properties of Real Numbers" Objectives: * Closure, commutative, associative, distributive properties. * Identity elements. * Inverses for addition and multiplication. The Closure Properties The closure properties guarantee that the sum, di¤erence, product, or quotient (except for division by zero) of any two real numbers is also a real number. Closure Properties: If a and b are real numbers, then 1: a + b is a real number. 2: a Identity Elements: 2: ab is a real number a 3: is a real number (b 6= 0) b b is a real number. 0 is the identity element for addition. 1 is the identity element for multiplication Example 1: (Using the closure properties) Assume that x = 12 and y = x a) 2 y 2. Show that each expression represents a real number by …nding the real-number answer. 2x b) c) y 2 3y The Commutative Properties The commutative properties guarantee that addition or multiplication of two real numbers can be done in either order. Commutative Properties: If a and b are real numbers, then 1: a + b = b + a commutative property of addition 2: ab = ba commutative property of multiplication Example 2: (Verifying the commutative properties) Let x = 2, y = 3; and z = 1. Show that the two expressions have the same value. a) (x + y) + z; y + (x + z) b) (xy) z; x (yz) Page: 1 c) x2 yz 2 ; x2 y z 2 Notes by Bibiana Lopez Beginning and Intermediate Algebra by Gustafson and Frisk 1.7 The Distributive Property The distributive property shows how to multiply the sum of two numbers by a third number. Distributive Property: If a; b and c are real numbers, then a (b + c) = ab + ac : Example 3: (Using the distributive properties) Use the distributive property to write each expression without parentheses. Simplify each result, if possible. a) 2 (z 3) b) a (x + y) c) 4 x2 + x The Additive and Multiplicative Inverses Additive and Multiplicative Inverses: Because a + ( a) = 0 ; the numbers a and Because a 1 a = 1 (a 6= 0) a are called negative or additive inverses: the numbers a and ; 1 are called reciprocals or multiplicative inverses. a Example 4: (Using the identity properties and additive/multiplicative inverses) Give the additive and the multiplicative inverse of each number, if possible. a) 1 3 b) 0 c) Page: 2 2 Notes by Bibiana Lopez