5.5 Real Numbers and Their Properties 263 The Properties of Real Numbers Commutative property of addition + = + a b b a Commutative property of multiplication ⋅ = ⋅ a b b a Associative property of addition + + = + + a b c a b c ( ) ( ) Associative property of multiplication ⋅ ⋅ = ⋅ ⋅ a b c a b c ( ) ( ) Distributive property of multiplication ⋅ + = ⋅ + ⋅ a b c a b a c ( ) over addition Learning Catalytics Keyword Angel-SOM-5.5 (See Preface for additional details.) Exercises Warm Up Exercises In Exercises 1–8, fill in the blanks with an appropriate word, phrase, or symbol(s). 1. The union of the rational numbers and the irrational numbers is the set of ________ numbers. Real 2. The symbol used to represent the set of real numbers is ________. R 3. If an operation is performed on any two elements of a set and the result is always an element of the set, we say that the set is ________ under that given operation. Closed 4. The equation + = + 2 3 3 2 demonstrates the ________ property of addition. Commutative 5. The equation ⋅ = ⋅ 4 5 5 4 demonstrates the ________ property of multiplication. Commutative. 6. The equation + + = + + (2 3) 4 2 (3 4) demonstrates the ________ property of addition. Associative 7. The equation ⋅ ⋅ = ⋅ ⋅ (5 6) 7 5 (6 7) demonstrates the ________ property of multiplication. Associative 8. The equation ⋅ + = ⋅ + ⋅ 2 (3 4) 2 3 2 4 demonstrates the ________ property of multiplication over addition. Distributive Practice the Skills/Problem Solving In Exercises 9–12, determine whether the natural numbers are closed under the given operation. 9. Addition Yes 10. Multiplication Yes 11. Subtraction No 12. Division No In Exercises 13–16, determine whether the integers are closed under the given operation. 13. Multiplication Yes 14. Division No 15. Subtraction Yes 16. Addition Yes In Exercises 17–20, determine whether the rational numbers are closed under the given operation. 17. Addition Yes 18. Division No 19. Subtraction Yes 20. Multiplication Yes In Exercises 21–24, determine whether the irrational numbers are closed under the given operation. 21. Division No 22. Addition No 23. Multiplication No 24. Subtraction No In Exercises 25–28, determine whether the real numbers are closed under the given operation. 25. Addition Yes 26. Multiplication Yes 27. Division No 28. Subtraction Yes 29. Does + + = + + x x (5 ) 6 ( 5) 6 illustrate the commutative property or the associative property? Explain your answer. * 30. Does + + = + + x x ( 5) 6 (5 6) illustrate the commutative property or the associative property? Explain your answer. * 31. Does the commutative property hold for the rational numbers under the operation of division? Give an example to support your answer. No. ÷ ≠ ÷ 3 4 4 3. 32. Does the commutative property hold for the integers under the operation of subtraction? Give an example to support your answer. No. − ≠ − 4 3 3 4. 33. Give an example to show that the commutative property of multiplication may be true for the negative integers. ( 3)( 4) ( 4)( 3) 12 − − = − − = 34. Give an example to show that the commutative property of addition may be true for the negative integers. (3) ( 4) ( 4) (3) 7 − + − = − + − = − 35. Give an example to show that the associative property of addition may be true for the negative integers. (2) (3) ( 4) (2) ( 3) ( 4) 9 [ ] −+− +− =−+−+− ⎣⎡ ⎤⎦ = − SECTION 5.5 *See Instructor Answer Appendix
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