5.03 Properties of number operations
Introduction
We have just looked at using the distributive property to find equivalent numerical expressions. Let's look at some other properties that can also produce equivalent expressions.
Other properties of number operations
The associative property allows us to group numbers in parentheses so that we can find the sum or product before adding or multipying that to a third number. In an expression, this looks like we are adding or moving parentheses to change which pair of numbers we add or multiply first.
Let's look at the expression 29+38+12:
| \displaystyle 29+38+12 | \displaystyle = | \displaystyle 29+38+12 | Set up the expression equal to itself so we can compare |
| \displaystyle (29+38)+12 | \displaystyle = | \displaystyle 29+(38+12) | On the left-hand side group the first two terms and on the right-hand side group the second two terms. |
| \displaystyle 67+12 | \displaystyle = | \displaystyle 29+50 | Evaluate addition inside parentheses on each side |
| \displaystyle 79 | \displaystyle = | \displaystyle 79 | Evaluate the addition |
On the left hand side we added 29 and 38 together first so we got 67+12 and on the right-hand side we added 38 to 12 to get 29+50. On each side our final result was 79. Using this property we are able to group numbers and find a more friendly number set that helps us solve more efficiently. Which side was more efficient for you?
In the same way, we can use the commutative property to rearrange the numbers in the expression and get the same answer, but perhaps we can get the answer more efficiently.
Let's look at the expression 5 \times 17 \times 2:
| \displaystyle 5 \times 17 \times 2 | \displaystyle = | \displaystyle 5 \times 17 \times 2 | Set up the expression equal to itself so we can compare |
| \displaystyle 5 \times 17 \times 2 | \displaystyle = | \displaystyle 5 \times 2 \times 17 | On the left-hand side keep the order the same and on the right-hand side rearrange the order to multiply 5 and 2 first. |
| \displaystyle 85 \times 2 | \displaystyle = | \displaystyle 10 \times 17 | Evaluate the first multiplication on each side |
| \displaystyle 170 | \displaystyle = | \displaystyle 170 | Evaluate the remaining multiplication on each side |
As you can see, it didn't matter which order that we did the multiplication, we ended up with the same answer. However, by reordering the we were able to have more friendly numbers to multiply.
Note that these two properties don't apply to subtraction or division because those two operations have a specific order.
Examples
Example 1
Use the associative property of operations to find two expressions equivalent to:
6\times 4 \times 5
Example 2
Fill in the boxes to produce equivalent expressions and complete the steps.
| \displaystyle 27+49+13 | \displaystyle = | \displaystyle 27+⬚+⬚ | Apply the commutative property |
| \displaystyle = | \displaystyle ⬚+⬚ | Evaluate addition | |
| \displaystyle = | \displaystyle 89 | Evaluate |
The associative property allows us to group numbers in the parentheses so that we can evaluate their sum or product before adding or multipying that to a third number.
For example: 29+38+12 = 29+(38+12)
The commutative property allows us to change the order of the numbers in the expression.
For example: 5 \times 17 \times 2 = 5 \times 2 \times 17
Both properties result in equivalent expressions for both addition and multiplication