associate with subtraction and division. Although there are exceptions, a rule must work all the time, not just in special cases.
Here’s how the associative property works:
✓ 4 + (5 + 8) = 4 + 13 = 17 and (4 + 5) + 8 = 9 + 8 = 17, so 4 + (5 + 8) = (4 + 5) + 8
✓ 3 – (2 × 5) = 3 × 10 = 30 and (3 × 2) × 5 = 6 × 5 = 30, so 3 × (2 × 5) = (3 × 2) × 5
✓
✓ 3.2 × (5 × 4.8) = 3.2 × 24 = 76.8 and (3.2 × 5) × 4.8 = 16 × 4.8 = 76.8
Remember: This rule is special to addition and multiplication. It doesn’t work for subtraction or division. You’re probably wondering why even use this rule? Because it can sometimes make the computation easier.
✓ Instead of doing 5 + (–5 + 17), change it to [5 + (–5)] + 17 = 0 + 17 = 17.
✓ Instead of
Examples
Q. –14 + (14 + 23) =
A. Re-associate the terms and then add the first two together: –14 + (14 + 23) = (–14 + 14) + 23 = 0 + 23 = 23.
Q. 4(5 · 6) =
A. You can either multiply the way the problem is written, 4(5 · 6) = 4(30) = 120, or you can re-associate and multiply the first two factors first: (4 · 5) 6 = (20)6 = 120.
Practice Questions
1. 16 + (–16 + 47) =
2. (5 – 13) + 13 =
3.
4.
Practice Answers
1. 47.
2. 5.
3. 70.
4. 110.
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 traveling 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 · 3 or 3 · 2 yields 6.
Tip: 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,
In general, subtraction and division are not commutative. The special cases occur when you choose the numbers carefully. For example, 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. By the way, 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.
Take a look at how the commutative property works:
✓ 4 + 5 = 9 and 5 + 4 = 9, so 4 + 5 = 5 + 4
✓ 3 · (–7) = –21 and (–7) · 3 = –21, so 3 · (–7) = (–7) · 3
✓ 6.3 + 5.7 = 12 and 5.7 + 6.3 = 12, so 6.3 + 5.7 = 5.7 + 6.3
✓
You can use this rule to your advantage when doing math computations. In the following two examples, the associative rule finishes off the problems after changing the order.
Examples
Q.
A. You don’t really want to multiply fractions unless necessary. Notice that the first and last factors are multiplicative inverses of one another:
Q. –3 + 16 + 303 =
A. The second and last terms are reversed, and then the first two terms are grouped.
– 3 + 16 + 303 = –3 + 303 + 16 = (–3 + 303) + 16 = 300 + 16 = 316.
Practice Questions
1. 8 + 5 + (–8) =
2. 5 · 47 · 2 =
3.
4. –23 + 47 + 23 – 47 + 8 =
Practice Answers
1. 5.
2. 470.
3. 78.
4. 8.
In mathematics, the inverse of a number is tied to a specific operation.
The additive inverse of the number 5 is –5; the additive inverse of the number
