Associative, Commutative, and Distributive Properties
Purplemath
There are three basic properties of numbers, and your textbook will probably have just a little section that covers these properties, somewhere near the beginning of the book.
After you finish that section, you'll probably never see these properties again — until, maybe, the beginning of the next math course.
What are properties of numbers?
Number properties are descriptions of things that numbers do; they are names for how numbers behave.
Many things have properties. For instance, matter (any physical object) has the property of density, because an object has a certain amount of material (mass) that occupies a certain amount of volume. Dividing the mass by the volume tells you how dense the object is. All physical objects fill a certain volume with a certain amount of stuff, so "the property of density" is just a description of one thing that all matter does.
What are the three basic number properties?
The basic number properties are as follows:
- the Associate Property (of Addition or Multiplication, depending on the context)
- the Commutative Property (of Addition or Multiplication, depending on the context)
- the Distributive Property (of multiplication over addition)
(My impression is that covering these properties at this stage in your studies is a holdover from the "New Math" fad of the mid-1900s. While these number properties will start to become relevant in matrix algebra and calculus — and become amazingly important in advanced math, a couple years after calculus — they may seem fairly useless to you right now. That's okay; view this topic as easy points on the next test.)
Why do these number properties seem so pointless?
These number properties may seem obvious (so the "study" of them seems pointless) because every math system you've *ever* worked with has obeyed these properties. You have, for instance, never dealt with a system where a×b did not in fact equal b×a, or where (a×b)×c did not equal a×(b×c).
You have always been able to move terms around (as long as you kept track of their signs) and you have always been able to regroup terms. Which is why the associative and commutative properties probably seem painfully obvious to you, and thus hardly worth the time or effort.
Don't worry about their "relevance" for now; just make sure you can keep the properties straight so you can pass the next test. The lesson below explains how I keep track of the properties.
What is the Associative Property?
The word "associative" comes from "to associate" or "to group". The Associative Property is the rule that refers to grouping; the regrouping can be of added terms, or of multiplied factors. For adding numbers, the rule is:
a + (b + c) = (a + b) + c
Using numerical values, this looks:
2 + (3 + 4) = (2 + 3) + 4
For multiplying numbers, the rule is:
a(bc) = (ab)c
Using numerical values, this looks like:
2(3×4) = (2×3)4
Any time they refer to the Associative Property, they want you to re-associate, or regroup, things; any time a computation depends on things being regrouped, they want you to say that the computation uses the Associative Property.
- Rearrange, using the Associative Property: 2(3x)
They want me to regroup things, not simplify things. In other words, they do not want me to say "6x". They want to see me do the following regrouping:
(2×3)x
- Simplify 2(3x), and justify your steps.
In this case, they do want me to simplify, but I have to say why it's okay to do... just exactly what I've always done. Here's how this works:
2(3x) : original (given) statement
(2×3) x : by the Associative Property
6x : simplification of 2×3
- Why is it true that 2(3x) = (2×3)x?
Since all they did was regroup things, this is true by the Associative Property.
What is the Commutative Property?
The word "commutative" comes from "to commute" or "to move around", so the Commutative Property is the one that refers to moving stuff around. For adding numbers, the rule is:
a + b = b + a
Using numerical values, this looks like:
2 + 3 = 3 + 2
For multiplying numbers, the rule is:
ab = ba
Using numerical values, this looks like:
2×3 = 3×2
Any time they refer to the Commutative Property, they want you to commute, or move, stuff around; any time a computation depends on moving stuff around, they want you to say that the computation uses the Commutative Property.
- Use the Commutative Property to restate "3×4×x" in at least two ways.
They want me to move stuff around, not simplify. In other words, my answer should not be "12x"; the answer instead can be any two of the following:
4 × 3 × x
4 × x × 3
3 × x × 4
x × 3 × 4
x × 4 × 3
- Why is it true that 3 + (4 + x) = (4 + x) + 3?
Since all they did was move stuff around (they didn't regroup), this statement is true by the Commutative Property.
What is the Distributive Property?
The Distributive Property is easy to remember, if you keep in mind the mantra "multiplication distributes over addition". This phrase means that two (or more) things are being added together, and their sum is being multiplied by something. A real-life model might be a group of seven kids, three of whom are boys and four of whom are girls, and you're distributing two cookies to each kid. You could find the number of cookies handed out by multiplying the 2 by the total number of kids (that is, by 7), or you can multiply each of the gender-totals by 2. Either way, you'll have distributed fourteen cookies.
Formally, the Distributive Property is written as:
a(b + c) = ab + ac
Using numerical values, this looks like:
2(3 + 4) = 2×3 + 2×4
Any time they refer in a problem to using the Distributive Property, they want you to take something through the parentheses (or else to factor something out); any time a computation depends on multiplying through a parentheses (or else on factoring something out), they want you to say that the computation used the Distributive Property.
Unlike the Associative and Commutative Properties, there are not two versions (one for addition and another for multiplication) of the Distributive Property. Instead, both multiplication and addition occur within the one rule.
- Why is the following true? 2(x + y) = 2x + 2y
Since they distributed through the parentheses, this is true by the Distributive Property.
- Use the Distributive Property to rearrange: 4x − 8
The Distributive Property either takes something through a parentheses or else factors something out. Since there aren't any parentheses to go in-to, you must need to factor out-of. (Yes; when using the Distributive Property to go backwards, all you're doing is factoring.) Then the answer is:
By the Distributive Property, 4x − 8 = 4(x − 2).
"But wait!" I hear you cry; "the Distributive Property says multiplication distributes over addition, not over subtraction! What gives?" You make a good point. This is one of those times when it's best to be flexible. You can either view the contents of the parentheses as the subtraction of a positive number ("x − 2") or else as the addition of a negative number ("x + (−2)"). In the latter case, it's easy to see that the Distributive Property applies, because you're still adding; you're just adding a negative.
Once you've learned these properties, you will be asked to demonstrate your knowledge.
- Simplify 3a − 5b + 7a. Justify your steps.
I'm going to do the exact same algebra I've always done, but now I have to give the name of the property that says its okay for me to take each step. The answer looks like this:
3a − 5b + 7a : original (given) statement
3a + 7a − 5b : Commutative Property
(3a + 7a) − 5b : Associative Property
a(3+7) − 5b : Distributive Property
a(10) − 5b : simplification (3 + 7 = 10)
10a − 5b : Commutative Property
The only fiddly part was moving the "− 5b" from the middle of the expression (in the first line of my working above) to the end of the expression (in the second line). If you need help keeping your negatives straight, convert the "− 5b" to "+ (−5b)". Just don't lose that minus sign!
- Simplify 23 + 5x + 7y − x − y − 27. Justify your steps.
I'll do the exact same steps I've always done. The only difference now is that I'll be writing down the reasons for each step.
23 + 5x + 7y − x − y − 27 : original (given) statement
23 − 27 + 5x − x + 7y − y : Commutative Property
(23 − 27) + (5x − x) + (7y − y) : Associative Property
(−4) + (5x − x) + (7y − y) : simplification (23 − 27 = −4)
(−4) + x(5 − 1) + y(7 − 1) : Distributive Property
−4 + x(4) + y(6) : simplification (5 − 1 = 4, 7 − 1 = 6)
−4 + 4x + 6y : Commutative Property
- Simplify 3(x + 2) − 4x. Justify your steps.
3(x + 2) − 4x : original (given) statement
3x + 3×2 − 4x : Distributive Property
3x + 6 − 4x : simplification (3×2 = 6)
3x − 4x + 6 : Commutative Property
(3x − 4x) + 6 : Associative Property
x(3 − 4) + 6 : Distributive Property
x(−1) + 6 : simplification (3 − 4 = −1)
−x + 6 : Commutative Property
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Why is it true that 3(4 + x) = 3(x + 4)?
All they did was move stuff around inside the parentheses. Moving around means commuting, so the reason the two sides of the equation are equal is:
the Commutative Property
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Why is 3(4x) = (3×4)x?
All they did was regroup. Grouping means associating, so the reason that the two sides of the equation are equal is:
the Associative Property
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Why is 12 − 3x = 3(4 − x)?
They factored something out of the two terms, leaving a subtraction inside parentheses. This factoring-out is the reverse of distributing-through, so the reason that the two sides of the equation are equal is:
the Distributive Property
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