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Section 1.4 Properties of Real Numbers -Relationships that are ALWAYS true for real numbers are called properties. -Equivalent expressions are two algebraic expressions that have the same value for all values of the variable(s). Commutative Property Changing the order of the addends/factors does not change the sum/product ADDITION: ◦a+b=b+a Ex: 8+4=4+8 MULTIPLICATION: ◦a•b=b•a Ex: 2•3=3•2 Associative Property Changing the grouping of the addends/factors does not change the sum/product ADDITION: ◦ (a + b) + c = a + (b + c) Ex: (20 + 3) + 5 = 20 + (3 + 5) MULTIPLICATION ◦ (a • b) • c = a • (b • c) Ex: (2 • 3) • 4 = 2 • (3 • 4) Identity Property ADDITION: The sum of any real number and 0 is the original number ◦a+0=a Ex: 6+0=6 MULTIPLICATION: The product of any real number and 1 is the original number ◦a•1=a Ex: 7•1=7 Zero Property of Multiplication The product of a and 0 is 0 ◦a•0=0 Ex: 5•0=0 Multiplication Property of -1 The product of -1 and a is –a ◦ -1 • a = -a Ex: -1 • 7 = -7 Identifying Properties What property is illustrated by each statement? 1) 42 • 0 = 0 2) (y + 2.5) + 28 = y + (2.5 + 28) 3) 10x + 0 = 10x Example A movie ticket costs $7.75. A drink costs $2.40. Popcorn costs $1.25. What is the total cost for a ticket, a drink and a popcorn? Use mental math!! 7.75 + 2.40 + 1.25 = Equivalent Expressions 5(3n) = = (4 + 7b) + 8 6xy= y (5 • 3)n 15n = = = 6x • y (7b + 4) + 8 7b + (4 + 8) 7b + 12 = 1 •y 6x • 1 = 6x Examples - Simplify each expression. Justify each step. 2.1(4.5x) 6 + (4h + 3) 8m 12mn Deductive Reasoning The process of reasoning logically from given facts to a conclusion A counterexample is needed to show that a statement is false. ◦ You need only one counterexample to prove that a statement is false Is the statement true or false? If it is false, give a counterexample. 1) For all real numbers a and b, a • b = b +a 2) For all real numbers a, b, and c, (a + b) + c = b + (a + c)