1.06 The properties of operations with integers
When we manipulate expressions with numbers, we follow a set of properties that apply to all numbers. Knowing these properties helps us to be able to rewrite expressions in a variety of different ways, and might make evaluating them easier.
The box below summarizes some of the properties of real numbers and gives an example of each.
| Property | Symbols | Example |
|---|---|---|
| Commutative property of addition | $a+b=b+a$a+b=b+a | $3+6=6+3$3+6=6+3 |
| Commutative property of multiplication | $a\times b=b\times a$a×b=b×a | $6\times3=3\times6$6×3=3×6 |
| Associative property of addition | $a+(b+c)=(a+b)+c$a+(b+c)=(a+b)+c | $6+3+2=6+3+2$6+3+2=6+3+2 |
| Associative property of multiplication | $a(bc)=(ab)c$a(bc)=(ab)c | $6\times3\times2=6\times3\times2$6×3×2=6×3×2 |
| Distributive property |
$a(b+c)=ab+ac$a(b+c)=ab+ac or $a(b-c)=ab-ac$a(b−c)=ab−ac
|
$4\left(3+5\right)=4\times3+4\times5$4(3+5)=4×3+4×5 or $4\left(3-5\right)=4\times3-4\times5$4(3−5)=4×3−4×5 |
| Identity property of addition | $a+0=a$a+0=a | $3+0=3$3+0=3 |
| Identity property of multiplication | $a\times1=a$a×1=a | $3\times1=3$3×1=3 |
| Inverse property of addition | $a+\left(-a\right)=0$a+(−a)=0 | $3+\left(-3\right)=0$3+(−3)=0 |
| Inverse property of multiplication | $a\times\frac{1}{a}=1$a×1a=1 | $3\times\frac{1}{3}=1$3×13=1 |
The commutative property is the reason that we can add numbers in any order or multiply numbers in any order. While it applies to multiplication and addition, it does not apply to expressions that are written as subtraction or division.
If we rotate the array in the applet below, we can see that the rectangles are the same size when the length and width are switched. This demonstrates the commutative property for multiplication.
The associative property is the reason that we can group sums of numbers differently and the result remains the same. The same is true for products of numbers. However, it's not true for subtraction and division. That's why we say they are not associative.
The distributive property shows us how the product of a number and a sum or difference is applied to each term in the sum or difference. We can see it demonstrated in the applet below when we split the rectangle into two smaller rectangles, but the area remains the same.
We know that adding zero to a number gives us the same number. That's the identity property for addition. In a similar way, multiplying a number by one doesn't change the number. That's the identity property for multiplication.
We also know that opposites add to zero. That's the inverse property of addition. When multiplying fractions, reciprocals multiply to give us one. That's the inverse property of multiplication.
All of these properties can be applied to help us evaluate expressions more easily. Let's walk through a few examples.
Worked examples
Question 1
Evaluate $6+\left(-5\right)+5$6+(−5)+5.
Think: Since we can add numbers in any order, it might be easier to evaluate the sum of ($-5$−5) and ($5$5) first. They are opposites, so they combine to make a zero pair.
Do:
| $6+\left(-5\right)+5$6+(−5)+5 | $=$= | $-5+5+6$−5+5+6 |
The commutative property of addition |
| $=$= | $0+6$0+6 |
$-5+5=0$−5+5=0 by the inverse property of addition |
|
| $=$= | $6$6 |
$0+6=6$0+6=6 by the identity property of addition |
So $6+\left(-5\right)+5=6$6+(−5)+5=6
Reflect: Could we have applied different properties to get the same result? Yes! Here's another example.
| $6+\left(-5\right)+5$6+(−5)+5 | $=$= | $6+0$6+0 |
Since $-5+5=0$−5+5=0, we can substitute $0$0 where we had $-5+5$−5+5. |
| $=$= | $6$6 |
By the identity property |
Either way, we have that $6+\left(-5\right)+5=6$6+(−5)+5=6.
Question 2
Evaluate $-7\times105$−7×105 without a calculator.
Think: We can substitute the expression $100+5$100+5 for $105$105. Then we can apply the distributive property to find the product. Multiplying by $100$100 and $5$5 seems much easier!
Do:
| $-7\times105$−7×105 | $=$= | $-7\left(100+5\right)$−7(100+5) |
Since $100+5=105$100+5=105, we can substitute $100+5$100+5 where we had $105$105. |
| $=$= | $-7\times100+\left(-7\times5\right)$−7×100+(−7×5) |
Apply the distributive property |
|
| $=$= | $-700+\left(-35\right)$−700+(−35) |
|
|
| $=$= | $-735$−735 |
Simplify |
Practice questions
Question 3
Use the commutative property of addition to fill in the missing number.
$19+15=15$19+15=15$+$+$\editable{}$
Question 4
Consider $11\left(7-3\right)$11(7−3).
Using the distributive law, complete the gap so that $11\left(7-3\right)$11(7−3) is rewritten as the difference of two integers.
$11\left(7-3\right)=77-\editable{}$11(7−3)=77−
Question 5
Which property is demonstrated by the following statement?
$4\cdot\left(9\cdot5\right)=\left(4\cdot9\right)\cdot5$4·(9·5)=(4·9)·5
Commutative property of multiplication
AAssociative property of multiplication
BDistributive property
CAssociative property of addition
DCommutative property of addition
E