Distributive Property

Welcome to Omni Calculator's distributive property article, where we'll learn one of the basic arithmetic tools: the distributive property of multiplication over addition. In fact, we can extend it to division (under some conditions) and subtraction, as long as we're careful. But don't worry: the distributive property definition is not the only thing you'll see here. Once we explain what it is, we'll introduce a few nice distributive property examples.

"So what is the distributive property in math exactly?", you might ask. Well, how fortunate that's precisely the title of the first section!

Roughly speaking, the distributive property lets us expand one big, complicated expression into several smaller, simpler ones. It must always involve two different operations (hence, the long name distributive property of multiplication over addition).

Let's take a look at a symbolic distributive property definition and analyze it:

x × (a + b + c + d + ...) = x × a + x × b + x × c + x × d + ...

So, what is the distributive property in math? As you can see, it's all about taking an expression with just one × sign and turning it into quite a few of them. That is precisely the essence of the distributive property of multiplication over addition: we distribute the multiplication sign × over all the terms separated by the addition sign +. As a result, we go from a lengthy expression into an even longer but much simpler one.

It's important to remember a few directions in which we can extend the above distributive property definition. As the nice guys that we are, we list them all neatly below.

1. Multiplication is commutative. Therefore, x × (a + b + c + d + ...) is the same as (a + b + c + d + ...) * x. What is more, the same property lets us change the order of multiplication on the right side of the distributive property definition: x × a + x × b + x × c + x × d + ... = a × x + b × x + c × x + d × x + ...
2. Subtraction is similar to addition. In other words, we can have subtraction instead of addition in the distributive property definition, or even have a mixture of both. However, make sure to take good care of the signs. For instance, x × (a - b - c + d + ...) = x × a - x × b - x × c + x × d + ...
3. Division is somewhat similar to multiplication. Like above, we can sometimes have division instead of multiplication. However, here we have to be extra careful since division is not commutative and so the distributive property of division only works one way: (a + b + c + d + ...) / x = a/x + b/x + c/x + d/x + .... In other words, we cannot use a similar formula for x / (a + b + c + d + ...). Nevertheless, point 2. still applies: we can have both pluses and minuses inside the bracket.
4. The distributive property appears in many areas of mathematics. It applies to other, more complicated operations done not only on numbers but on objects such as sequences (e.g., the one in the arithmetic sequence calculator) or functions. In some sense, it describes well-structured spaces, and weird things happen when it fails. Fortunately, we don't have to care too much about it: the distributive property of multiplication over addition is all we need for now (and most probably the rest of our lives)!

Phew, that seems like enough mathematical mumbo-jumbo. Let's move on from symbols to numbers and see how to do the distributive property in practice. We'll take on a few problems, but we'll make sure to go over them slowly and thoroughly. After all, we promised some nice distributive property examples and nice examples you shall get!

We will see here several distributive property examples in increasing order of difficulty.

• We begin with the simplest case.

  3 × (2 + 4 + 11 + 0) = 3 × 2 + 3 × 4 + 3 × 11 + 3 × 0 = 6 + 12 + 33 + 0 = 51

  We simply applied the distributive property definition from the above section and computed each piece.

• Let's mix things up a little. Now, we'll have division instead of multiplication (remember that the distributive property of division only works from one side!) and have some minuses in the brackets.

  (13 - 1 + 7 + 3 - 2) / 4 = 13/4 - 1/4 + 7/4 + 3/4 - 2/4 = 3.25 - 0.25 + 1.75 + 0.75 - 0.5 = 5

  Observe how we copied the pluses and minuses in the corresponding places.

• Now let's try multiplying by a negative number.

  (-2) × (3 + 1 - 9 - 5) = (-2) × 3 + (-2) × 1 + (-2) × (-9) + (-2) × (-5) = -6 - 2 + 18 + 10 = 20

  Note how we have -2 in every term, i.e., we copied the number with its sign. What is more, the -9 and -5 also appeared with their sign. Alternatively, we could have written:

  (-2) × (3 + 1 - 9 - 5) = (-2) × 3 + (-2) × 1 - (-2) × 9 - (-2) × 5 = 6 - 2 + 18 + 10 = 20

  Signs are extremely important in such calculations: take good care of them and don't miss any!

• Lastly, we give a more complicated example with nested brackets. Recall the order of operations in mathematics and see how to use the distributive property in such cases.

  (3 + 2 × (4 - 5)) × ((11 - 1)/5 + 2) = (3 + 2 × 4 - 2 × 5) × (11/5 - 1/5 + 2)

  = (3 + 8 - 10) × (2.2 - 0.2 + 2)

  = 3 * (2.2 - 0.2 + 2) + 8 × (2.2 - 0.2 + 2) - 10 × (2.2 - 0.2 + 2)

  = 3 × 2.2 - 3 × 0.2 + 3 × 2 + 8 × 2.2 - 8 × 0.2 + 8 × 2 - 10 × 2.2 - 10 × (-0.2) - 10 × 2

  = 6.6 - 0.6 + 6 + 17.6 - 1.6 + 16 - 22 + 2 - 20

  = 4

  The third line is how we use the distributive property of multiplication over addition when both factors are sums. Of course, we could have also added up one of the brackets and used the formula from the above section, e.g.,

  (3 + 8 - 10) × (2.2 - 0.2 + 2) = (3 + 8 - 10) × 4

Hmm, let's see if we covered everything.

• What is the distributive property in math? ✓
• Possible extensions ✓
• How to do distributive property in practice ✓
• Distributive property examples ✓

Oh, it seems like we have covered everything!

This article was written by Maciej Kowalski and reviewed by Steven Wooding.