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§ 1.10 Properties of the Real Number System

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Angel, Elementary Algebra, 7ed 2 Commutative Property Commutative Property of Addition If a and b represent any real numbers, then a + b = b + a 3 + 2 = 2 + 3 Commutative Property of Multiplication If a and b represent any real numbers, then a · b = b · a 4 · 5 = 5 · 4 Commutative (commute) changes the order. *Note that the commutative property does not hold for subtraction and division 5 = 5 20 = 20

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Angel, Elementary Algebra, 7ed 3 Associative Property Associative Property of Addition If a, b, and c represent three real numbers, then (a + b) + c = a + (b + c) (2 + 4) + 5 = 2 + (4 + 5) Commutative Property of Multiplication If a, b, and c represent three real numbers, then (a · b) · c = a ·(b · c) (3 · 5) · 2 = 3 · (5 · 2) Associative (associate) changes the grouping. *Note that the associative property does not hold for subtraction and division 6 + 5 = 2 + 9 11 = 11 15 · 2 = 3 · 10 30 = 30

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Angel, Elementary Algebra, 7ed 4 Distributive Property If a, b, and c represent three real numbers, then a(b + c) = ab + ac Distributive involves two operations (usually multiplication and division). 2(3 + 4) = 2(3) + 2(4) 2(7) = 6 + 8 14 = 14

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Angel, Elementary Algebra, 7ed 5 Identity Properties If a represents any real number, then a + 0 = a and 0 + a = a a · 1 = a and 1 · a = a Identity Property of Addition Identity Property of Multiplication 4 + 0 = 4 0 + 4 = 4 13 · 1 = 13 1 · 13 = 13

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Angel, Elementary Algebra, 7ed 6 Inverse Properties If a represents any real number, then a + (-a)= 0 and (-a) + a = 0 Inverse Property of Addition Inverse Property of Multiplication a · = 1 and · a = a (a 0) 7 + (-7) = 0 (-7) + 7 = 0 12 · = 1 · 12 = 1

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