CHAPTER 2 Operations with Rational Numbers and Introduction to Real Numbers
2.5 Properties of Real Numbers
Learning Objectives
By the end of this section, you will be able to:
- Use the commutative and associative properties
- Use the identity and inverse properties of addition and multiplication
- Use the properties of zero
- Simplify expressions using the distributive property
Use the Commutative and Associative Properties
Think about adding two numbers, say 5 and 3. The order we add them doesn’t affect the result, does it?
The results are the same.
As we can see, the order in which we add does not matter!
What about multiplying
Again, the results are the same!
The order in which we multiply does not matter!
These examples illustrate the commutative property. When adding or multiplying, changing the order gives the same result.
Commutative Property
When adding or multiplying, changing the order gives the same result.
The commutative property has to do with order. If you change the order of the numbers when adding or multiplying, the result is the same.
What about subtraction? Does order matter when we subtract numbers? Does give the same result as
The results are not the same.
Since changing the order of the subtraction did not give the same result, we know that subtraction is not commutative.
Let’s see what happens when we divide two numbers. Is division commutative?
The results are not the same.
Since changing the order of the division did not give the same result, division is not commutative. The commutative properties only apply to addition and multiplication!
- Addition and multiplication are commutative.
- Subtraction and Division are not commutative.
If you were asked to simplify this expression, how would you do it and what would your answer be?
Some people would think and then . Others might start with and then .
Either way gives the same result. Remember, we use parentheses as grouping symbols to indicate which operation should be done first.
| Add . Add. |
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| Add . Add. |
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When adding three numbers, changing the grouping of the numbers gives the same result.
This is true for multiplication, too.
| Multiply. Multiply. |
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| Multiply. . Multiply. |
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When multiplying three numbers, changing the grouping of the numbers gives the same result.
You probably know this, but the terminology may be new to you. These examples illustrate the associative property.
Associative Property
When adding or multiplying, changing the grouping gives the same result.
Let’s think again about multiplying . We got the same result both ways, but which way was easier? Multiplying and first, as shown above on the right side, eliminates the fraction in the first step. Using the associative property can make the math easier!
The associative property has to do with grouping. If we change how the numbers are grouped, the result will be the same. Notice it is the same three numbers in the same order—the only difference is the grouping.
We saw that subtraction and division were not commutative. They are not associative either.
When simplifying an expression, it is always a good idea to plan what the steps will be. In order to combine like terms in the next example, we will use the commutative property of addition to write the like terms together.
EXAMPLE 1
Simplify: .
| Use the commutative property of addition to re-order so that like terms are together. | |
| Add like terms. |
TRY IT 1.1
Simplify: .
Show answer
TRY IT 1.2
Simplify: .
Show answer
When we have to simplify algebraic expressions, we can often make the work easier by applying the commutative or associative property first, instead of automatically following the order of operations. When adding or subtracting fractions, combine those with a common denominator first.
EXAMPLE 2
Simplify: .
| Notice that the last 2 terms have a common denominator, so change the grouping. | |
| Add in parentheses first. | |
| Simplify the fraction. | |
| Add. | |
| Convert to an improper fraction. |
TRY IT 2.1
Simplify: .
Show answer
TRY IT 2.2
Simplify: .
Show answer
EXAMPLE 3
Use the associative property to simplify .
| Change the grouping. | |
| Multiply in the parentheses. |
Notice that we can multiply but we could not multiply 3x without having a value for x.
TRY 3.1
Use the associative property to simplify 8(4x).
Show answer
32x
TRY IT 3.2
Use the associative property to simplify .
Show answer
Use the Identity and Inverse Properties of Addition and Multiplication
What happens when we add 0 to any number? Adding 0 doesn’t change the value. For this reason, we call 0 the additive identity.
For example,
These examples illustrate the Identity Property of Addition that states that for any real number , and .
What happens when we multiply any number by one? Multiplying by 1 doesn’t change the value. So we call 1 the multiplicative identity.
For example,
These examples illustrate the Identity Property of Multiplication that states that for any real number , and .
We summarize the Identity Properties below.
Identity Property
Notice that in each case, the missing number was the opposite of the number!
We call . the additive inverse of a. The opposite of a number is its additive inverse. A number and its opposite add to zero, which is the additive identity. This leads to the Inverse Property of Addition that states for any real number . Remember, a number and its opposite add to zero.
What number multiplied by gives the multiplicative identity, 1? In other words, times what results in 1?
What number multiplied by 2 gives the multiplicative identity, 1? In other words 2 times what results in 1?
Notice that in each case, the missing number was the reciprocal of the number!
We call the multiplicative inverse of a. The reciprocal of number is its multiplicative inverse. A number and its reciprocal multiply to one, which is the multiplicative identity. This leads to the Inverse Property of Multiplication that states that for any real number .
We’ll formally state the inverse properties here:
Inverse Property
| of addition | For any real number , is the additive inverse of . A number and its opposite add to zero. |
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| of multiplication | For any real number , is the multiplicative inverse of . A number and its reciprocal multiply to one. |
EXAMPLE 4
Find the additive inverse of a) b) c) d) .
To find the additive inverse, we find the opposite.
- The additive inverse of is the opposite of . The additive inverse of is .
- The additive inverse of 0.6 is the opposite of 0.6. The additive inverse of 0.6 is .
- The additive inverse of is the opposite of . We write the opposite of as , and then simplify it to 8. Therefore, the additive inverse of is 8.
- The additive inverse of is the opposite of . We write this as , and then simplify to . Thus, the additive inverse of is .
TRY IT 4.1
Find the additive inverse of: a) b) c) d) .
Show answer
a) b) c) d)
Exercises
Find the additive inverse of: a) b) c) d) .
Show answer
a) b) c) d)
EXAMPLE 5
Find the multiplicative inverse of a) b) c) .
To find the multiplicative inverse, we find the reciprocal.
- The multiplicative inverse of 9 is the reciprocal of 9, which is . Therefore, the multiplicative inverse of 9 is .
- The multiplicative inverse of is the reciprocal of , which is . Thus, the multiplicative inverse of is .
- To find the multiplicative inverse of 0.9, we first convert 0.9 to a fraction, . Then we find the reciprocal of the fraction. The reciprocal of is . So the multiplicative inverse of 0.9 is .
TRY IT 5.1
Find the multiplicative inverse of a) b) c)
Show answer
a) b) c)
TRY IT 5.2
Find the multiplicative inverse of a) b) c) .
Show answer
a) b) c)
Use the Properties of Zero
The identity property of addition says that when we add 0 to any number, the result is that same number. What happens when we multiply a number by 0? Multiplying by 0 makes the product equal zero.
Multiplication by Zero
For any real number a.
The product of any real number and 0 is 0.
What about division involving zero? What is Think about a real example: If there are no cookies in the cookie jar and 3 people are to share them, how many cookies does each person get? There are no cookies to share, so each person gets 0 cookies. So,
We can check division with the related multiplication fact.
So we know because .
Division of Zero
For any real number a, except , and .
Zero divided by any real number except zero is zero.
Now think about dividing by zero. What is the result of dividing 4 by 0? Think about the related multiplication fact: means . Is there a number that multiplied by 0 gives 4? Since any real number multiplied by 0 gives 0, there is no real number that can be multiplied by 0 to obtain 4
We conclude that there is no answer to and so we say that division by 0 is undefined.
Division by Zero
For any real number a, except 0, and are undefined.
Division by zero is undefined.
We summarize the properties of zero below.
Properties of Zero
Multiplication by Zero: For any real number a,
| The product of any number and 0 is 0. |
Division of Zero, Division by Zero: For any real number
| Zero divided by any real number except itself is zero. | |
| Division by zero is undefined. |
EXAMPLE 6
Simplify: a) b) c) .
| a) The product of any real number and 0 is 0. |
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| b) The product of any real number and 0 is 0. |
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| c) Division by 0 is undefined. |
TRY IT 6.1
Simplify: a) b) c) .
Show answer
a) 0 b) 0 c) undefined
TRY IT 6.2
Simplify: a) b) c) .
Show answer
a) 0 b) 0 c) undefined
We will now practice using the properties of identities, inverses, and zero to simplify expressions.
EXAMPLE 7
Simplify: a) , where b) , where .
| a) Zero divided by any real number except itself is 0. |
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| b) Division by 0 is undefined. |
TRY IT 7.1
Simplify: a) , where b) , where .
Show answer
a) 0 b) undefined
TRY IT 7.2
Simplify: a) b) .
Show answer
a) 0 b) undefined
EXAMPLE 8
Simplify: .
| Notice that the first and third terms are opposites; use the commutative property of addition to re-order the terms. |
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| Add left to right. | |
| Add. |
TRY IT 8.1
Simplify: .
Show answer
TRY IT 8.2
Simplify: .
Show answer
Now we will see how recognizing reciprocals is helpful. Before multiplying left to right, look for reciprocals—their product is 1
EXAMPLE 9
Simplify: .
| Notice that the first and third terms are reciprocals, so use the commutative property of multiplication to re-order the factors. |
|
| Multiply left to right. | |
| Multiply. |
TRY IT 9.1
Simplify: .
Show answer
TRY IT 9.2
Simplify: .
Show answer
EXAMPLE 10
Simplify: .
| There is nothing to do in the parentheses, so multiply the two fractions first—notice, they are reciprocals. |
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| Simplify by recognizing the multiplicative identity. |
TRY IT 10.1
Simplify: .
Show answer
TRY IT 10.2
Simplify: .
Show answer
Simplify Expressions Using the Distributive Property
Suppose that three friends are going to the movies. They each need $9.25—that’s 9 dollars and 1 quarter—to pay for their tickets. How much money do they need all together?
You can think about the dollars separately from the quarters. They need 3 times $9 so $27, and 3 times 1 quarter, so 75 cents. In total, they need $27.75. If you think about doing the math in this way, you are using the distributive property.
Distributive Property
Back to our friends at the movies, we could find the total amount of money they need like this:
| 3(9.25) |
| 3(9 + 0.25) |
| 3(9) + 3(0.25) |
| 27 + 0.75 |
| 27.75 |
In algebra, we use the distributive property to remove parentheses as we simplify expressions.
For example, if we are asked to simplify the expression , the order of operations says to work in the parentheses first. But we cannot add x and 4, since they are not like terms. So we use the distributive property, as shown in (Example 11).
EXAMPLE 11
Simplify: .
| Distribute. | |
| Multiply. |
TRY IT 11.1
Simplify: .
Show answer
TRY IT 11.2
Simplify: .
Show answer
Some students find it helpful to draw in arrows to remind them how to use the distributive property. Then the first step in (Example 11) would look like this:
EXAMPLE 12
Simplify: .
| Distribute. | |
| Multiply. |
TRY IT 12.1
Simplify: .
Show answer
TRY IT 12.2
Simplify: .
Show answer
Using the distributive property as shown in (Example 13) will be very useful when we solve money applications in later chapters.
EXAMPLE 13
Simplify: .
| Distribute. | |
| Multiply. |
TRY IT 13.1
Simplify: .
Show answer
TRY IT 13.2
Simplify: .
Show answer
When we distribute a negative number, we need to be extra careful to get the signs correct!
EXAMPLE 14
Simplify: .
| Distribute. | |
| Multiply. |
TRY IT 14.1
Simplify: .
Show answer
TRY IT 14.2
Simplify: .
Show answer
EXAMPLE 15
Simplify: .
| Distribute. | |
| Multiply. | |
| Simplify. |
Notice that you could also write the result as . Do you know why?
TRY IT 15.1
Simplify: .
Show answer
TRY IT 15.2
Simplify: .
Show answer
(Example 16) will show how to use the distributive property to find the opposite of an expression.
EXAMPLE 16
Simplify: .
| Multiplying by −1 results in the opposite. | |
| Distribute. | |
| Simplify. | |
TRY IT 16.1
Simplify: .
Show answer
TRY IT 16.2
Simplify: .
Show answer
There will be times when we’ll need to use the distributive property as part of the order of operations. Start by looking at the parentheses. If the expression inside the parentheses cannot be simplified, the next step would be multiply using the distributive property, which removes the parentheses. The next two examples will illustrate this.
EXAMPLE 17
Simplify: .
Be sure to follow the order of operations. Multiplication comes before subtraction, so we will distribute the 2 first and then subtract.
| Distribute. | |
| Multiply. | |
| Combine like terms. |
TRY IT 17.1
Simplify: .
Show answer
TRY IT 17.2
Simplify: .
Show answer
EXAMPLE 18
Simplify: .
| Distribute. | |
| Combine like terms. |
TRY IT 18.1
Simplify: .
Show answer
TRY IT 18.2
Simplify: .
Show answer
All the properties of real numbers we have used in this chapter are summarized in the table below.
| Commutative Property | of addition If are real numbers, then | |
| of multiplication If are real numbers, then | ||
| Associative Property | of addition If are real numbers, then | |
| of multiplication If are real numbers, then | ||
| Distributive Property | If are real numbers, then | |
| Identity Property | of addition For any real number 0 is the additive identity |
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| of multiplication For any real number is the multiplicative identity |
||
| Inverse Property | of addition For any real number , is the additive inverse of |
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| of multiplication For any real number is the multiplicative inverse of . |
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| Properties of Zero | For any real number a,
For any real number For any real number |
is undefined |
Key Concepts
- Commutative Property of
- Addition: If are real numbers, then .
- Multiplication: If are real numbers, then . When adding or multiplying, changing the order gives the same result.
- Associative Property of
- Addition: If are real numbers, then .
- Multiplication: If are real numbers, then .
When adding or multiplying, changing the grouping gives the same result.
- Distributive Property: If are real numbers, then
- Identity Property
- of Addition: For any real number
0 is the additive identity - of Multiplication: For any real number
is the multiplicative identity
- of Addition: For any real number
- Inverse Property
- of Addition: For any real number . A number and its opposite add to zero. is the additive inverse of .
- of Multiplication: For any real number . A number and its reciprocal multiply to one. is the multiplicative inverse of .
- Properties of Zero
- For any real number ,
– The product of any real number and 0 is 0. - for – Zero divided by any real number except zero is zero.
- is undefined – Division by zero is undefined.
- For any real number ,
Glossary
- additive identity
- The additive identity is the number 0; adding 0 to any number does not change its value.
- additive inverse
- The opposite of a number is its additive inverse. A number and it additive inverse add to 0.
- multiplicative identity
- The multiplicative identity is the number 1; multiplying 1 by any number does not change the value of the number.
- multiplicative inverse
- The reciprocal of a number is its multiplicative inverse. A number and its multiplicative inverse multiply to one.
Practice Makes Perfect
Use the Commutative and Associative Properties
In the following exercises, use the associative property to simplify.
| 1. 3(4x) | 2. 4(7m) |
| 3. | 4. |
| 5. | 6. |
| 7. | 8. |
| 9. | 10. |
| 11. | 12. |
| 13. 17(0.25)(4) | 14. 36(0.2)(5) |
| 15. [2.48(12)](0.5) | 16. [9.731(4)](0.75) |
| 17. 7(4a) | 18. 9(8w) |
| 19. | 20. |
| 21. | 22. |
| 23. | 24. |
| 25. | 26. |
| 27. | 28. |
Use the Identity and Inverse Properties of Addition and Multiplication
In the following exercises, find the additive inverse of each number.
| 29. a) b) 4.3 c) d) |
30. a) b) 2.1 c) d) |
| 31. a) b) c) 23 d) |
32. a) b) c) 52 d) |
| 33. a) 6 b) c) 0.7 | 34. a) 12 b) c) 0.13 |
| 35. a) b) c) | 36. a) b) c) |
Use the Properties of Zero
In the following exercises, simplify.
| 37. | 38. |
| 39. | 40. |
| 41. | 42. |
| 43. | 44. |
Mixed Practice
In the following exercises, simplify.
| 45. | 46. |
| 47. | 48. |
| 49. , where | 50. , where |
| 51. , where | 52. , where |
| 53. , where | 54. , where |
| 55. where | 56. where |
| 57. | 58. |
Simplify Expressions Using the Distributive Property
In the following exercises, simplify using the distributive property.
| 59. | 60. |
| 61. | 62. |
| 63. | 64. |
| 65. | 66. |
| 67. | 68. |
| 69. | 70. |
| 71. | 72. |
| 73. | 74. |
| 75. | 76. |
| 77. | 78. |
| 79. | 80. |
| 81. | 82. |
| 83. | 84. |
| 85. | 86. |
| 87. | 88. |
| 89. | 90. |
| 91. | 92. |
| 93. | 94. |
Everyday Math
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95. Insurance copayment Carrie had to have 5 fillings done. Each filling cost $80. Her dental insurance required her to pay 20% of the cost as a copay. Calculate Carrie’s copay: a) First, by multiplying 0.20 by 80 to find her copay for each filling and then multiplying your answer by 5 to find her total copay for 5 fillings. b) Next, by multiplying [5(0.20)](80) c) Which of the properties of real numbers says that your answers to parts (a), where you multiplied 5[(0.20)(80)] and (b), where you multiplied [5(0.20)](80), should be equal? |
96. Cooking time Matt bought a 24-pound turkey for his family’s Thanksgiving dinner and wants to know what time to put the turkey in to the oven. He wants to allow 20 minutes per pound cooking time. Calculate the length of time needed to roast the turkey: a) First, by multiplying to find the total number of minutes and then multiplying the answer by to convert minutes into hours. b) Next, by multiplying . c) Which of the properties of real numbers says that your answers to parts (a), where you multiplied , and (b), where you multiplied , should be equal? |
|
97. Buying by the case. Trader Joe’s grocery stores sold a can of Coke Zero for $1.99. They sold a case of 12 cans for $23.88. To find the cost of 12 cans at $1.99, notice that 1.99 is . a) Multiply 12(1.99) by using the distributive property to multiply . b) Was it a bargain to buy Coke Zero by the case? |
98. Multi-pack purchase. Adele’s shampoo sells for $3.99 per bottle at the grocery store. At the warehouse store, the same shampoo is sold as a 3 pack for $10.49. To find the cost of 3 bottles at $3.99, notice that 3.99 is . a) Multiply 3(3.99) by using the distributive property to multiply . b) How much would Adele save by buying 3 bottles at the warehouse store instead of at the grocery store? |
Writing Exercises
| 99. In your own words, state the commutative property of addition. | 100. What is the difference between the additive inverse and the multiplicative inverse of a number? |
| 101. Simplify using the distributive property and explain each step. | 102. Explain how you can multiply 4($5.97) without paper or calculator by thinking of $5.97 as and then using the distributive property. |
Answers
| 1. 12x | 3. | 5. |
| 7. | 9. | 11. |
| 13. 17 | 15. 14.88 | 17. 28a |
| 19. | 21. 10p | 23. |
| 25. | 27. | 29. a) b) c) 8 d) |
| 31. a) b) 0.075 c) d) | 33. a) b) c) | 35. a) b) c) |
| 37. 0 | 39. 0 | 41. 0 |
| 43. 0 | 45. 44 | 47. d |
| 49. 0 | 51. 0 | 53. undefined |
| 55. undefined | 57. | 59. |
| 61. | 63. | 65. |
| 67. | 69. | 71. |
| 73. | 75. | 77. |
| 79. | 81. | 83. |
| 85. | 87. | 89. |
| 91. | 93. | 95. a) $80 b) $80 c) answers will vary |
| 97. a) $23.88 b) no, the price is the same | 99. Answers may vary | 101. Answers may vary |
Attributions
This chapter has been adapted from “Properties of Real Numbers” in Elementary Algebra (OpenStax) by Lynn Marecek and MaryAnne Anthony-Smith, which is under a CC BY 4.0 Licence. Adapted by Izabela Mazur. See the Copyright page for more information.