|
Algebra operations follow certain rules, and those rules have certain properties. Here, you find out about two of those properties — the commutative property and the associative property.
Reordering operations: The commutative property
Before discussing the commutative property, take a look at the word commute. You probably commute to work or school and know that whether you're going from home to work or from work to home, the distance is the same: The distance doesn't change because you change directions (although getting home during rush hour may make that distance seem longer).
The same principle is true of some algebraic operations: It doesn't matter whether you add 1 + 2 or 2 + 1; the answer is still 3. Likewise, multiplying 2 x 3 or 3 x 2 yields 6.
 | The commutative property means that you can change the order of the numbers in an operation without affecting the result. Addition and multiplication are commutative. Subtraction and division are not. So, |
a + b = b + a
a x b = b x a
a – b ≠ b – a (except in a few special cases)
a ÷ b ≠ b ÷ a (except in a few special cases)
In general, subtraction and division are not commutative. The special cases occur when you choose the numbers carefully. For instance, if a and b are the same number, then the subtraction appears to be commutative because switching the order doesn't change the answer. In the case of division, if a and b are opposites, then you get -1 no matter which order you divide them in. This is why, in mathematics, big deals are made about proofs. A few special cases of something may work, but a real rule or theorem has to work all the time.
Look at the following examples:
- 4 + 5 = 9 and 5 + 4 = 9 so 4 + 5 = 5 + 4
- 3 x (-7) = -21 and (-7) x 3 = -21 so 3 x (-7) = (-7) x 3
- (-5) - (+2) = (-7) and (+2) - (-5) = +7 so (-5) – (+2) ≠ (+2) - (-5)
- (-6) ÷ (+1) = -6 and (+1) ÷ (-6) = –1/6 so (-6) ÷ (+1) ? (+1) ÷ (-6)
| Keep in mind that the commutative property holds true only for addition and multiplication. |
Associating expressions: The associative property
The commutative property has to do with the order of the numbers when you perform an operation. The associative property has to do with how the numbers are grouped when you perform an operation on more than two numbers.
Think about what the word associate means. When you associate with someone, you're close to the person, or you form a group with the person. Say that Anika, Becky, and Cora associate. Whether Anika drives over to pick up Becky and the two of them go to Cora's and pick her up, or Cora is at Becky's house and Anika picks up both of them at the same time, the same result occurs — the same people are in the car at the end.
| The associative property means that if the grouping of the operation changes, the result remains the same. Addition and multiplication are associative. Subtraction and division are not associative operations. So, |
a + (b + c) = (a + b) + c
a x (b x c) = (a x b) x c
a - (b - c) ≠ (a - b) - c (except in a few special cases)
a ÷ (b ÷ c) ≠ (a ÷ b) ÷ c (except in a few special cases)
You can always find a few cases where the property works even though it isn't supposed to. For example, in the subtraction problem 5 - (4 - 0) = (5 - 4) - 0 the property seems to work. Also, in the division problem 6 ÷ (3 ÷ 1) = (6 ÷ 3) ÷ 1, it seems to work. Although there are exceptions, a rule must work all the time.
Some real-number examples may make this clearer:
- 4 + (5 + 8) = 4 + 13 = 17 and (4 + 5) + 8 = 9 + 8 = 17
So 4 + (5 + 8) = (4 + 5) + 8
- 3 x (2 x 5) = 3 x 10 = 30 and (3 x 2) x 5 = 6 x 5 = 30
So 3 x (2 x 5) = (3 x 2) x 5
- 13 - (8 - 2) = 13 - 6 = 7 and (13 - 8) - 2 = 5 - 2 = 3
So 13 - (8 - 2) ? (13 - 8) - 2
- 48 ÷ (16 ÷ 2) = 48 ÷ 8 = 6 and (48 ÷ 16) ÷ 2 = 3 ÷ 2 = 3/2
So 48 ÷ (16 ÷ 2) ? (48 ÷ 16) ÷ 2
The commutative and associative properties come in handy when you work with algebraic expressions. You can change the order of some numbers or change the grouping to make the work less messy or more convenient. Just keep in mind that you can commute and associate addition and multiplication operations, but not subtraction or division.
| 
