Commutative, Associative, and Distributive Properties

In working on last week’s post, I realized that I haven’t yet covered the general idea of properties, such as commutative, associative, and distributive. Here I’ll collect some introductory answers on that topic. Next week, we’ll dig a little deeper.

What are these properties?

We’ll start with this 1996 question:

Properties of Algebra

What are the properties used for algebra? What about the "distributive property"?

Doctor Lani answered:

Dear Justin,

When we're working with real numbers (integers, fractions, decimals, roots, and numbers like Pi), we like to say that certain things will always be true.

When we find these things, we give them the name "property."  The example you give, the distributive property, is something that works for *any* set of real numbers.  Because of this, properties are often stated with variables, which say that you can put any number in; I'll use a, b, and c in this answer.

All sorts of things have properties (chemical properties, physical properties, and so on), and so do numbers. The sort of properties we’re talking about here are actually properties of the operations we perform on numbers, such as addition or multiplication.

Now, for some examples:

Some properties true for multiplication and addition:
  
THE COMMUTATIVE PROPERTY: a + b = b + a
                          a * b = b * a
Basically, this says that you can switch the numbers that you are adding or multiplying and your answer stays the same: 
   2 + 3 = 3 + 2
   57 * 22.3 = 22.3 * 57

In effect, this property says that as long as you are doing only addition or only multiplication, it doesn’t matter if you change the order. Knowing that order doesn’t matter gives you a lot of freedom, as we’ll be seeing!

THE ASSOCIATIVE PROPERTY: (a + b) + c = a + (b + c)
                          (a * b) * c = a * (b * c)
Since parentheses mean "do this first," this property just tells you that you can add or multiply in any order without changing your answer:
   (2 + 3) + 6          2 + (3 + 6)
     \ /                      \ /
    = 5    + 6        = 2 +    9    
    = 11              = 11

Here we didn’t change the order of the numbers, but of the operations. Again, this is true of either addition or multiplication (alone).

Sometimes students mix up these names.  I just think of the words: "commute" means going from one place to another and "associate" means hanging out with. Numbers commute when they switch places and they associate in parentheses.  

Also, students sometimes think that properties are silly or obvious.  Well, test these properties with subtraction and division!

We’ll be doing that next week, seeing that addition and multiplication are very special. We’ll also see why these properties deserve names in the first place. At this point, all we’re doing is calling attention to some facts that are “obvious” (once you have experience with numbers), but not silly!

In other uses, “commute” means to “change” in various ways (such as a death sentence into a life sentence, or one form of money into another), but the idea of driving back and forth is the most familiar usage. We say addition or multiplication is commutative, because changing order has no effect.

The idea of an “association” as a group is more familiar. We say that addition is associative, because changing how we associate addends has no effect.

Now, finally, I'll explain distribution.  The official name is:

DISTRIBUTIVE PROPERTY OF MULTIPLICATION OVER ADDITION:

  a * (b + c) = a*b + a*c

This one is best explained by a couple of examples.

(1) Have you ever had to multiply something like 4 * 52 in your head? If so, you might have used the distributive property!  Like this:

  4 * 52 = 4 * (50 + 2) = 4 * 50  +  4 * 2 = 200 + 8 = 208

It's easier to multiply 4 by 50 and 4 by 2 and then just add the answers.  If you know that trick, you know the distributive property!

This is a property of how two operations work together, rather than of one operation alone.

(2) Why does it work?  Well, let's look at another example:

(a)  3*(2 + 5) = (2 + 5) + (2 + 5) + (2 + 5)
This is true because multiplying by 3 means adding the number three times.
   
(b)  (2 + 5) + (2 + 5) + (2 + 5) = 2 + 5 + 2 + 5 + 2 + 5
This works because the associative property tells me it doesn't matter which order I add in, so I don't need the parentheses.

(c)  2 + 5 + 2 + 5 + 2 + 5 = 2 + 2 + 2 + 5 + 5 + 5
This is true because the commutative property tells me I can move stuff around without changing the answer.

(d)  2 + 2 + 2 + 5 + 5 + 5 = 3*2 + 3*5
This is true because adding a number three times is the same as multiplying by 3.

If you look at the first line of (a) and the last line of (d) you'll see that 3*(2 + 5) = 3*2 + 3*5.  This is called a proof.  It should convince you that the distributive property works!

Of course, this is not really a proof that it works for all numbers, but it shows why it works, in a way that will work just the same whatever (whole) numbers you use. So you can tell that it is always true … for whole numbers. For others, you’d need a bigger proof.

Understanding the properties through pictures

This 2001 question led to visual “proofs” of these properties:

Properties, Defined and Illustrated

I need a page of information on the commutative, associative, and F.O.I.L. properties, to be used as a study sheet for a quiz.

Doctor Ian answered, starting with a comment on the very idea of memorizing such information:

If you understand the distributive property, you can ignore FOIL:

   When FOIL Fails
   http://mathforum.org/dr.math/problems/ryan.03.22.01.html   

For that matter, if you _understand_ the commutative and associative properties, then you don't need a study sheet to help you 'remember' them. (To put that another way, if you have to 'remember' these properties, then you probably don't _understand_ them.)

The page he’s referring to points out that FOIL is a way to multiply two binomials, like \((2x-3)(2x-3)\), but doesn’t apply to bigger factors, like \((4x^2-12x+9)(2x-3)\); it’s a specialized form of the distributive property, which applies more broadly. So we don’t really list FOIL as a separate property.

So, how can we visualize each of the three main properties, so as to make them understandable, believable, and memorable?

Take a look at the following diagrams, and construct enough other examples to be sure that you would be able to explain to someone else why the properties hold for _any_ numbers:
      
Commutative property of addition

       3      4
     ----- -------
     * * * * * * *               3 + 4 = 4 + 3
     ------- -----
        4      3

When I count objects, it doesn’t matter whether I count green, then red, or red, then green.

Commutative property of multiplication
    
     * * *       * * * *
     * * *    =  * * * *         3 * 4 = 4 * 3
     * * *       * * * *
     * * *

When I count an array of objects, it doesn’t matter whether I see them as three columns, or three rows.

Associative property of addition

         (2+3)+4
     -----------------
        2+3
     ---------
     * * * * * * * * *           (2 + 3) + 4 = 2 + (3 + 4)
         -------------
             3+4        
     -----------------
          2+(3+4)

It doesn’t matter whether I add the first two numbers first, or add the second two numbers first.

Associative property of multiplication

        +---+---+---+---+
       /   /   /   /   /|
      +---+---+---+---+ +
     /   /   /   /   /|/| 3
    +---+---+---+---+ + +        (2 * 3) * 4 = 2 * (3 * 4)
    |   |   |   |   |/|/|
    +---+---+---+---+ + +
    |   |   |   |   |/|/
    +---+---+---+---+ + 2
    |   |   |   |   |/
    +---+---+---+---+
          4

When I multiply three numbers, the picture becomes three dimensional! But this time, I can either multiply the height times depth first, or the depth times width; I get the same number of blocks.

Distributive property of multiplication over addition:

     2    3
    --- -----
    @ @ * * * |
    @ @ * * * | 4         (2 * 4) + (3 * 4) = (2 + 3) * 4
    @ @ * * * | 
    @ @ * * * |         
    ---------
      2 + 3

I can multiply each color first, then add them; or I can add each row first and then multiply. The two rectangles have the same total as the combined rectangle.

Details

For further explanations of the basic properties, consider this 1997 question:

Properties of Real Numbers

I have a problem that is all about properties of real numbers. I don't get it at all!  Help explain what they are!  Particularly, what are associative properties, commutative properties, and the zero product property?

Doctor Reno answered:

Hi Sarah!

These properties are simply fancy names for mathematical ideas that you probably already know very well.  Let's look at them one at a time to learn more.

ASSOCIATIVE PROPERTY

There are two associative properties, actually, but don't let that upset you!  One is the associative property of addition, and the other is the associative property of multiplication.  These are sometimes also called "grouping properties," and you will see why soon.

Let's say that you want to add 3 numbers: 25, 75, and 30.  We can add these up two different ways.  We can add the first two: 25 + 75 = 100. Then we can add that sum (100) to 30 and have a new sum of 130 (100 + 30 = 130).  But we could also add them in this way: 75 + 30 = 105, and then 105 + 25 = 130.  The answer is the same.

This says in words what the pictures said.

All that the associative property of addition says is that we will always get the same answer no matter how we "group" numbers when we add them!  That's it!  But we knew that, didn't we? It seems obvious. Sometimes mathematics is like that...it gives big, fancy names to things that seem easy and that we already know.  In the fancy language of mathematics, the associative property of addition says that if a, b, and c are any whole numbers, then

  (a + b) + c = a + (b + c).  

We have shown that to be true in my example above.

Next week, we’ll see why big, fancy names were given to these simple ideas! It’s not just to scare people, but to give us a way to talk about things that may not always be true in other circumstances. It makes our knowledge more explicit.

But note that this applies not only to whole numbers as Doctor Reno says, but to any real numbers, which the question referred to – even, in fact, to complex numbers, which most students have never seen when they first learn this.

The associative property of multiplication is written in the same fancy mathematical language:

  For any whole numbers a, b, and c, 
    (a x b) x c = a x (b x c). 

Once again, we all know and understand this property. If we want to multiply three numbers together....let's say 3 x 5 x 8.....we can do this two ways:

   3 x 5 = 15
  15 x 8 = 120...which is to say that 3 x 5 x 8 = 120.

OR...

   5 x 8 = 40
  40 x 3 = 120...which is to say that 3 x 5 x 8 = 120!  The same answer!

And that's all there is to the associative properties!  We use these properties regularly to solve equations and arithmetic problems, but they are so familiar to us that we may not even realize that we use them!

Technically, the last multiplication should have been written as \(3\times40=120\), because the 3 came first in the original expression. But that doesn’t really matter, because of the next property:

COMMUTATIVE PROPERTY

The commutative properties are more fancy names for stuff you already know about math!  And once again, there are two of them..one for addition and one for multiplication. 

These are the easier properties to remember.  They simply say that if 2 + 5 = 7, then 5 + 2 = 7; and if 2 x 5 = 10, then 5 x 2 = 10. That's it! 

The fancy words go like this: 

  If a and b are any whole numbers, then 

    a + b = b + a.

For multiplication, it says: 

  For any whole numbers a and b,

    a x b = b x a.

That seems pretty obvious, doesn't it?

Yet, as we’ll see next week, it could have been otherwise. In fact, there are some “multiplications” that are not commutative.

Three additional properties: Zero, Identity, Inverse

ZERO MULTIPLICATION PROPERTY OF WHOLE NUMBERS

This property is the easiest one of all! All it means is that whenever you multiply a number by zero, you get zero for an answer! That's all! 

Of course, we have to say it differently in math...

  For any whole number a,

    a x 0 = 0 = 0 x a.

That's all there is, Sarah! Relax and enjoy your math, and remember that we have to write these simple ideas in this fancy language in order to be exact and precise.

We are focusing here just on the three big properties, but there are several others. This is one of them; others are:

• the identity property: that each operation has a special number that does nothing (0 is the additive identity, 1 is the multiplicative identity), $$a+0=a\\a\cdot1=a$$
• the inverse property: that each number has a special number that undoes it, producing the identity (do-nothing) number, namely the negative (additive inverse) and the reciprocal (multiplicative inverse), $$a+(-a)=0\\a\cdot\frac{1}{a}=1$$

Using the properties in arithmetic

These properties are often taught in an algebra class, which is where they are perhaps most useful. But they can also be useful when you are just doing arithmetic. Consider this 2003 question:

Using Properties to Make Addition Easier

My daughter's homework is asking her to "add mentally, using the commutative, associative, and identity properties to help you add."

Here's one example:

     6    __
     3    __
     0    __
     2    __
     4    __
   + 5  + __
   ---  ----                        

She doesn't know what they are asking her to do.

Doctor Ian answered:

Hi Linda, 

Let me illustrate how I would use these properties to add these numbers.  

  6 + 3 + 0 + 2 + 4 + 5

The commutative property of addition says that when we add numbers, it doesn't matter what order we add them in.  That means we can move the numbers around so that convenient pairs are next to each other.  I'd want to move the 4 next to the 6, and the 2 next to the 3:

  6 + 4 + 3 + 2 + 0 + 5

Why would I want to do this?  Because 6 and 4 add up to 10, which is easy to deal with:

  10 + 3 + 2 + 0 + 5

When numbers work well together, we want to sit them together! This property says they are allowed to change seats.

Also, 3 and 2 add up to 5, which is easy to deal with, especially since I have another 5 sitting around:

  10 + 5 + 0 + 5

And now I can take advantage of the fact that if I add 0 (the 'additive identity') to something, it makes no difference:

  10 + 5 + 5

Since the 0 does nothing when added, we can just drop it. (Or you could just say we’re doing the addition.)

We haven't mentioned the associative property yet.  That says that when we're adding, we can group the additions any way we want.  So we can do this:

  10 + 5 + 5 = (10 + 5) + 5

             = 15 + 5

             = 20

or this:

  10 + 5 + 5 = 10 + (5 + 5)

             = 10 + 10

             = 20

Either way, we'll get the same answer, but the second one seems easier.

The “official” way (by the order of operations) is to add from left to right ; but this property says we can do the additions any way we want to, whatever seems easiest, and we’ll still get the right result. Of course, if it weren’t all additions, we couldn’t do that.

Compare doing it this way to doing the additions one at a time, in the order they happen to show up:

  6 + 3 + 0 + 2 + 4 + 5
 
      9 + 0 + 2 + 4 + 5

          9 + 2 + 4 + 5

             11 + 4 + 5
 
                 15 + 5

                     20

That wasn't too bad, but in the general case you have to do carries, and it can really turn into kind of a mess.  If you rearrange and group the additions, though, you can make things easier.

None of these tricks are required; you’ll get the same result regardless; but sometimes it’s worth the effort.

Now, I did this by moving things around, but if you're trying to do it in your head, you have to keep track of which things you've paired up.  So I would probably be scanning around with my eyes, matching things like this:

           ___________
       ___|___        |
      |       |       |
  6 + 3 + 0 + 2 + 4 + 5
  |_______________|

Then mentally, I would just have to add 10, 5, and 5, which is a lot easier than adding the numbers one at a time. 

Does this make sense?

When things get complicated, though, it’s better to write down at least some of your work, to keep track of everything.

Identifying the property

We’ll close with one last question, from 2005:

Which Property?

I am working on a problem that is asking "name the property shown by each statement".  The problem is "55+6 is equal to 6+55".  Is that a commutative property?

I get confused when I see this kind of problem because I get the properties mixed up.

That isn’t uncommon!  I answered:

Hi, Falon.

You're right about that example.  What you need to do is to look at the expressions on the two sides and ask WHAT CHANGED?

If the change is a change in the ORDER of two numbers (say, two numbers that are being added together), then the commutative property is being used.  For example:

  3(4+5) = 3(5+4)     (only change is 4+5 to 5+4)
    ---      ---

If the change is a change ONLY in the location of PARENTHESES around a pair of the SAME operation, the associative property is being used. For example:

  3+(4+5) = (3+4)+5   (only change is in where the parentheses are)
    -   -   -   -

If the change is from something times a sum to the sum of two products (changing the order in which BOTH addition and multiplication are done), it is the distributive property:

  3(4+5) = 3*4 + 3*5  (rather than first adding 4 and 5, we first
    ---    ---   ---   multiply each of them by 3)

There are other properties, but these are the most easily confused.

Notice that a property can be used within an expression. In my example of the commutative property, the \(4+5\) was replaced with \(5+4\), leaving everything else the same. Another example would be $${\color{DarkGreen}{3(4)}}+{\color{Red}5}={\color{Red}5}+{\color{DarkGreen}{3(4)}}.$$ There, the numbers being added are \(3(4)\) and \(5\), and we are swapping each entire unit. It’s the order of operations that tells us they are units, because the addition is done last.