1.06 The properties of operations with integers
Introduction
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.
Properties of operation with integers
The box below summarizes some of the properties of real numbers and gives an example of each.
| Property | Symbols | Example |
|---|---|---|
| \text{Commutative property of addition} | a+b=b+a | -3+6=6+\left(-3\right) |
| \text{Commutative property of multiplication} | a \times b=b \times a | 6 \times (-3)=(-3) \times 6 |
| \text{Associative property of addition} | a+\left(b+c\right) \\ =\left(a+b\right)+c | 6+\left(\left(-3\right)+2\right)=\\\left(6+\left(-3\right)\right)+2 |
| \text{Associative property of multiplication} | a\times(b\times c)=(a\times b)\times c | 6 \times \left(-3 \times 2\right)=\\ \left(6 \times (-3\right)) \times 2 |
| \text{Distributive property} | a\times\left(b+c\right)= \\ a\times b+a\times c \\ \text{or} \\ a\times\left(b-c\right)= \\a\times b-a\times c | 4\left(-3+5\right)= \\ 4 \times \left(-3\right)+4 \times 5 \\ \text{or} \\ 4 \left(\left(-3\right)-5\right)= \\ 4 \times\left(-3\right)-4 \times 5 |
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.
Exploration
The applet below uses two sliders that you can use to adjust the number of rows of columns of the array. A multiplication expression is shown that represents the array. Use the bottom slider to generate the rotated version of the left array.
What do you notice when you rotated the array?
If we rotate the array in the applet, 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 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.
All of these properties can be applied to help us evaluate expressions more easily.
Examples
Example 1
Find the value of: 6+\left(-5\right)+5
Example 2
Use the commutative property of addition to fill in the missing number:
19 + \left(-15\right) = \left(-15\right) + ⬚
Example 3
Which property is demonstrated by the following statement?
4 \times (9 \times (-5)) = (4 \times 9) \times (-5)
Example 4
Using the distributive property, rewrite -11\times\left(7-3\right) as the sum or difference of two integers.
| Property | Symbols |
|---|---|
| \text{Commutative property of addition} | a+b=b+a |
| \text{Commutative property of multiplication} | a \times b=b \times a |
| \text{Associative property of addition} | a+\left(b+c\right) =\left(a+b\right)+c |
| \text{Associative property of multiplication} | a\times(b\times c)=(a\times b)\times c |
| \text{Distributive property} | a\times\left(b+c\right)= a\times b+a\times c \\ \text{or} \\ a\times\left(b-c\right)=a\times b-a\times c |