MCGRAW-HILL HIGHER EDUCATION AND BLACKBOARD HAVE TEAMED UP
1.7 Properties of the Real Numbers
Everyone knows that the price of a hamburger plus the price of a Coke is the same as the price of a Coke plus the price of a hamburger. But do you know that this example illustrates the commutative property of addition? The properties of the real numbers are commonly used by anyone who performs the operations of arithmetic. In algebra we must have a thorough understanding of these properties.
U 1 V The Commutative Properties
We get the same result whether we evaluate 35 or 53. This example illustrates the commutative property of addition. The fact that 4 6 and 6 4 are equal illustrates the commutative property of multiplication.
Commutative Property of Addition For any real numbers a and b,
abba.
Commutative Property of Multiplication For any real numbers a and b,
abba.
E X A M P L E 1 The commutative property of addition
Use the commutative property of addition to rewrite each expression.
a) 2(10) b) 8x2 c) 2y 4x
Solution
a) 2(10) 102 b) 8x2x28
c) 2y4x2y(4x) 4x2y
Now do Exercises 1–6
E X A M P L E 2 The commutative property of multiplication
Use the commutative property of multiplication to rewrite each expression.
a) n3 b) (x2)3 c) 5 yx
Solution
a) n3 3n3n b) (x2)33(x2) c) 5yx5xy
Now do Exercises 7–12
1-59 1.7 Properties of the Real Numbers 59
Addition and multiplication are commutative operations, but what about subtrac- tion and division? Since 532 and 35 2, subtraction is not commutative.
To see that division is not commutative, try dividing $8 among 4 people and $4 among 8 people.
U 2 V The Associative Properties
Consider the computation of 236. Using the order of operations, we add 2 and 3 to get 5 and then add 5 and 6 to get 11. If we add 3 and 6 first to get 9 and then add 2 and 9, we also get 11. So,
(23)62(36).
We get the same result for either order of addition. This property is called the associative property of addition. The commutative and associative properties of addition are the reason that a hamburger, a Coke, and French fries cost the same as French fries, a hamburger, and a Coke.
We also have an associative property of multiplication. Consider the following two ways to find the product of 2, 3, and 4:
(2 3)46 424 2(3 4)2 1224 We get the same result for either arrangement.
Associative Property of Addition For any real numbers a, b, and c,
(ab)ca(bc).
Associative Property of Multiplication For any real numbers a, b, and c,
(ab)ca(bc).
UHelpful Hint V
In arithmetic we would probably write (23)712 without thinking about the associative prop- erty. In algebra, we need the associa- tive property to understand that
(x3)7x(37) x10.
E X A M P L E 3 Using the properties of multiplication
Use the commutative and associative properties of multiplication and exponential notation to rewrite each product.
a) (3x)(x) b) (xy)(5yx)
Solution
a) (3x)(x)3(x x)3x2
b) The commutative and associative properties of multiplication allow us to rearrange the multiplication in any order. We generally write numbers before variables, and we usually write variables in alphabetical order:
(xy)(5yx)5xxyy5x2y2
Now do Exercises 13–18
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Consider the expression
39758413.
According to the accepted order of operations, we could evaluate this by computing from left to right. However, using the definition of subtraction, we can rewrite this expression as addition:
3(9)7(5)(8)4(13)
The commutative and associative properties of addition allow us to add these numbers in any order we choose. It is usually faster to add the positive numbers, add the nega- tive numbers, and then combine those two totals:
374(9)(5)(8)(13)14(35) 21
Note that by performing the operations in this manner, we must subtract only once. There is no need to rewrite this expression as we have done here. We can sum the positive num- bers and the negative numbers from the original expression and then combine their totals.
It is certainly not essential that we evaluate the expressions of Example 4 as shown. We get the same answer by adding and subtracting from left to right. However, in algebra, just getting the answer is not always the most important point. Learning new methods often increases understanding.
Even though addition is associative, subtraction is not an associative operation.
For example, (84)31 and 8(43)7. So, (84)38(43).
We can also use a numerical example to show that division is not associative. For instance, (16 4) 22 and 16 (4 2)8. So,
(16 4) 216 (4 2).
U 3 V The Distributive Property
If four men and five women pay $3 each for a movie, there are two ways to find the total amount spent:
3(45)3 927 3 43 5121527
E X A M P L E 4 Using the properties of addition Evaluate.
a) 3795 b) 4596248
Solution
a) First add the positive numbers and the negative numbers:
379512(12) 0
b) 4 59624814(24) 10
Now do Exercises 19–26
1-61 1.7 Properties of the Real Numbers 61
Since we get $27 either way, we can write
3(45)3 43 5.
We say that the multiplication by 3 is distributed over the addition. This example illustrates the distributive property.
Consider the following expressions involving multiplication and subtraction:
5(64)5 210 5 65 4302010 Since both expressions have the same value, we can write
5(64)5 65 4.
Multiplication by 5 is distributed over each number in the parentheses. This example illustrates that multiplication distributes over subtraction.
We can use the distributive property to remove parentheses. If we start with 4(x3) and write
4(x3)4x4 34x12,
we are using it to multiply 4 and x3 or to remove the parentheses. We wrote the product 4(x3) as the sum 4x12.
Distributive Property
For any real numbers a, b, and c,
a(bc)abac and a(bc)abac.
UHelpful Hint V
To visualize the distributive property, we can determine the number of cir- cles shown here in two ways:
º º º º º º º º º º º º º º º º º º º º º º º º º º º
There are 39 or 27 circles, or there are 34 circles in the first group and 35 circles in the second group for a total of 27 circles.
When we write a number or an expression as a product, we are factoring. If we start with 3x15 and write
3x153x3 53(x5),
we are using the distributive property to factor 3x15. We factored out the common factor 3.
E X A M P L E 5 Writing a product as a sum or difference Use the distributive property to remove the parentheses.
a) a(3b) b) 3(x2)
Solution
a) a(3b)a3ab Distributive property 3aab a33a
b) 3(x2) 3x(3)(2) Distributive property 3x(6) (3)(2) 6
3x6 Simplify.
Now do Exercises 27–38
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U 4 V The Identity Properties
The numbers 0 and 1 have special properties. Multiplication of a number by 1 does not change the number, and addition of 0 to a number does not change the number. That is why 1 is called the multiplicative identity and 0 is called the additive identity.
U 5 V The Inverse Properties
The idea of additive inverses was introduced in Section 1.3. Every real number a has an additive inverse or opposite,a, such that a(a)0. Every nonzero real number a also has a multiplicative inverse or reciprocal, written 1
a, such that a 1a1. Note that the sum of additive inverses is the additive identity and that the product of multiplicative inverses is the multiplicative identity.
Additive Inverse Property
For any real number a, there is a unique number a such that a(a)0.
Multiplicative Inverse Property
For any nonzero real number a, there is a unique number1
asuch that
a 1
a 1.
Additive Identity Property For any real number a,
a00aa.
Multiplicative Identity Property For any real number a,
a11 a a.
E X A M P L E 6 Writing a sum or difference as a product Use the distributive property to factor each expression.
a) 7x21 b) 5a5
Solution
a) 7x217x7 3 Write 21 as 7 3.
7(x3) Distributive property b) 5a55a5 1 Write 5 as 5 1.
5(a1) Factor out the common factor 5.
Now do Exercises 39–50
We are already familiar with multiplicative inverses for rational numbers. For example, the multiplicative inverse of2
3is 3
2because 2
3 3 2 6
6 1.
1-63 1.7 Properties of the Real Numbers 63
U 6 V Identifying the Properties
Zero has a property that no other number has. Multiplication involving zero always results in zero.
Multiplication Property of Zero For any real number a,
0 a0 and a 00.
E X A M P L E 7 Multiplicative inverses
Find the multiplicative inverse of each number.
a) 5 b) 0.3 c) 3
4 d) 1.7
Solution
a) The multiplicative inverse of 5 is 1 5because 5 1
5 1.
b) To find the reciprocal of 0.3, we first write 0.3 as a ratio of integers:
0.3 1 3
0 The multiplicative inverse of 0.3 is 1
3
0because 1
3 0 1
3 0 1.
c) The reciprocal of 34is 43because
34431.
d) First convert 1.7 to a ratio of integers:
1.711 7
0 1 1 7 0 The multiplicative inverse is 1
1 0 7.
Now do Exercises 51–62 UCalculator Close-Up V
You can find multiplicative inverses with a calculator as shown here.
When the divisor is a fraction, it must be in parentheses.
E X A M P L E 8 Identifying the properties
Name the property that justifies each equation.
a) 5 77 5 b) 4 1
4 1
c) 1 864864 d) 6(5x)(65)x
e) 3x5x(35)x f) 6(x5)6(5x)
g) x2y2(x2y2) h) 3250325
i) 330 j) 455 00
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Warm-Ups ▼
Fill in the blank.
1. According to the property of addition, abba for any real numbers a and b.
2. According to the property,
a(bc)abac for any real numbers a, b, and c.
3. According to the property of addition, a(bc)(ab)c for any real numbers a, b, and c.
4. is the process of writing a number or expression as a product.
5. The number 0 is the identity.
6. The number 1 is the identity.
True or false?
7. 99(3678)(9936)78 8. 24 (4 2)(24 4) 2 9. 9(43)(94) 3 10. 156 387387156 11. 156 387387 156
12. 5x55(x1) for any real number x.
13. The multiplicative inverse of 0.02 is 50.
14. The additive inverse of 0 is 0.
15. The number 2 is a solution to 3x5x9.
Solution
a) Commutative property of multiplication b) Multiplicative inverse property c) Multiplicative identity property d) Associative property of addition e) Distributive property f ) Commutative property of addition g) Distributive property h) Additive identity property i) Additive inverse property j) Multiplication property of 0
Now do Exercises 63–82
U1V The Commutative Properties
Use the commutative property of addition to rewrite each expression. See Example 1.
1. 9 r 2. t 6 3. 3(2x)
4. P(1 rt) 5. 45x 6. b2a
Use the commutative property of multiplication to rewrite each expression. See Example 2.
7. x 6 8. y (9) 9. (x4)(2)
10. a(bc) 11. 4 y 8 12. z 9 2
Exercises
UStudy Tips V
• Don’t stay up all night cramming for a test. Prepare for a test well in advance and get a good night’s sleep before a test.
• Do your homework on a regular basis so that there is no need to cram.
1.7
1-65 1.7 Properties of the Real Numbers 65
U2V The Associative Properties
Use the commutative and associative properties of multiplica- tion and exponential notation to rewrite each product. See Example 3.
13. (4w)(w) 14. (y)(2y) 15. 3a(ba) 16. (x x)(7x) 17. (x)(9x)(xz) 18. y( y 5)(wy) Evaluate by finding first the sum of the positive numbers and then the sum of the negative numbers. See Example 4.
19. 84310 20. 351210 21. 8 10787 22. 6 1179132 23. 411781520 24. 8139157225 25. 3.22.42.85.81.6 26. 5.45.16.62.39.1
U3V The Distributive Property
Use the distributive property to remove the parentheses.
See Example 5.
27. 3(x5) 28. 4(b1)
29. a(2t) 30. b(aw)
31. 3(w6) 32. 3(m5)
33. 4(5y) 34. 3(6p)
35. 1(a7) 36. 1(c8)
37. 1(t4) 38. 1(x7)
Use the distributive property to factor each expression.
See Example 6.
39. 2m12 40. 3y6
41. 4x4 42. 6y6
43. 4y16 44. 5x15
45. 4a8 46. 7a35
47. xxy 48. aab
49. 6a2b 50. 8a2c
U5V The Inverse Properties
Find the multiplicative inverse (reciprocal) of each number.
See Example 7.
51. 1
2 52. 1
3 53. 5
54. 6 55. 7 56. 8
57. 1 58. 1 59. 0.25
60. 0.75 61. 2.5 62. 3.5
U6V Identifying the Properties
Name the property that justifies each equation. See Example 8.
63. 3 xx 3 64. x55x 65. 2(x3)2x6 66. a(bc)(ab)c 67. 3(xy)(3x)y 68. 3(x1)3x3 69. 4(4)0 70. 1.3991.3 71. x2 55x2 72. 0 0 73. 1 3y3y 74. (0.1)(10)1 75. 2a5a(25)a 76. 303
77. 770 78. 1 bb 79. (2346)00 80. 4x44(x1) 81. ayyy(a1) 82. abbcb(ac)
Complete each equation, using the property named.
83. ay____, commutative property of addition 84. 6x6____, distributive property
85. 5(aw)____, associative property of multiplication 86. x3____, commutative property of addition 87. 1
2x 1
2 ____, distributive property 88. 3(x7)____, distributive property 89. 6x15____, distributive property
90. (x6)1____, associative property of addition 91. 4(0.25)____, multiplicative inverse property 92. 1(5y)____, distributive property 93. 096(____), multiplication property of zero 94. 3 (____)3, multiplicative identity property 95. 0.33(____)1, multiplicative inverse property 96. 8(1)____, multiplicative identity property dug84356_ch01b.qxd 9/14/10 9:23 AM Page 65
Getting More Involved 97. Writing
The perimeter of a rectangle is the sum of twice the length and twice the width. Write in words another way to find the perimeter that illustrates the distributive property.
98. Discussion
Eldrid bought a loaf of bread for $2.50 and a gallon of milk for $4.31. Using a tax rate of 5%, he correctly figured that the tax on the bread would be 13 cents and the tax on the milk would be 22 cents, for a total of $7.16. However, at the cash register he was correctly charged $7.15. How
could this happen? Which property of the real numbers is in question in this case?
99. Exploration
Determine whether each of the following pairs of tasks are “commutative.” That is, does the order in which they are performed produce the same result?
a) Put on your coat; put on your hat.
b) Put on your shirt; put on your coat.
Find another pair of “commutative” tasks and another pair of “noncommutative” tasks.
In This Section U1VUsing the Properties in
Computation
U2VCombining Like Terms
U3VProducts and Quotients
U4VRemoving Parentheses
U5VApplications