Example
Show that numbers may be added in any order without affecting the sum.
(−2)+7=5
Answer:
7+(−2)=5
Similarly, the
Example
Show that numbers may be multiplied in any order without affecting the product.
(−11)⋅(−4)=44
Answer:
(−4)⋅(−11)=44
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Caution! It is important to note that neither subtraction nor division is commutative. For example,
17−5 is not the same as
5−17. Similarly,
20÷5=5÷20.
Example
Show that you can regroup numbers that are multiplied together and not affect the product.
(3⋅4)⋅5=60
Answer:
3⋅(4⋅5)=60
Example
Show that regrouping addition does not affect the sum.
[15+(−9)]+23=29
Answer:
15+[(−9)+23]=29
Are subtraction and division associative? Review these examples.
Example
Use the associative property to explore whether subtraction and division are associative.
1)
8−(3−15)=?(8−3)−15
2)
64÷(8÷4)=?(64÷8)÷4
Answer:
1) 8−(3−15)=?(8−3)−15
8−(−12)=5−15
20=−10
2) 64÷(8÷4)=?(64÷8)÷4
64÷2=?8÷4
32=2
As we can see, neither subtraction nor division is associative.
Example
Use the distributive property to show that
4⋅[12+(−7)]=20
Answer:
Note that 4 is outside the grouping symbols, so we distribute the 4 by multiplying it by 12, multiplying it by –7, and adding the products.
To be more precise when describing this property, we say that multiplication distributes over addition.
The reverse is not true as we can see in this example.
Example
Rewrite the last example by changing the sign of each term and adding the results.
Answer:
12−(5+3)=12+(−5−3)=12+(−8)=4
This seems like a lot of trouble for a simple sum, but it illustrates a powerful result that will be useful once we introduce algebraic terms.
A General Note: Properties of Real Numbers
The following properties hold for real numbers
a,
b, and
c.
|
Addition |
Multiplication |
Commutative Property |
a+b=b+a |
a⋅b=b⋅a |
Associative Property |
a+(b+c)=(a+b)+c |
a(bc)=(ab)c |
Distributive Property |
a⋅(b+c)=a⋅b+a⋅c |
|
Identity Property |
There exists a unique real number called the additive identity, 0, such that, for any real number a
|
There exists a unique real number called the multiplicative identity, 1, such that, for any real number a
|
Inverse Property |
Every real number a has an additive inverse, or opposite, denoted –a, such that
a+(−a)=0 |
Every nonzero real number a has a multiplicative inverse, or reciprocal, denoted a1, such that
a⋅(a1)=1 |
Example
Use the properties of real numbers to rewrite and simplify each expression. State which properties apply.
- 3(6+4)
- (5+8)+(−8)
- 6−(15+9)
- 74⋅(32⋅47)
- 100⋅[0.75+(−2.38)]
Answer:
- 3⋅(6+4)=3⋅6+3⋅4Distributive property=18+12Simplify=30Simplify
- (5+8)+(−8)=5+[8+(−8)]Associative property of addition=5+0Inverse property of addition=5Identity property of addition
- 6−(15+9)=6+[(−15)+(−9)]=6+(−24)=−18Distributive propertySimplifySimplify
- 74⋅(32⋅47)=74⋅(47⋅32)=(74⋅47)⋅32=1⋅32=32Commutative property of multiplicationAssociative property of multiplicationInverse property of multiplicationIdentity property of multiplication
- 100⋅[0.75+(−2.38)]=100⋅0.75+100⋅(−2.38)=75+(−238)=−163Distributive propertySimplifySimplify
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