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# The distributive property

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Author: Christopher Danielson
##### Description:

To support students trying to understand the distributive property for its mathematical importance, and to challenge the reader to think more deeply about the symbols we push around in algebra classes.

A very brief overview of the standard distributive property of multiplication over addition is offered, and then extended to examine relationships among other operations.

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Tutorial

## Introduction

The distributive property of multiplication over addition is important in elementary mathematics through higher, abstract college-level mathematics and beyond. This packet briefly examines this basic property and then gives an overview of the importance of the property through looking at distribution with other operations.

## The distributive property in arithmetic

The distributive property of multiplication over addition is this:

A*(B+C)=A*B+A*C

and

(B+C)*A=B*A+B*C

The first one is the left distributive property because the multiplication is on the left of the parentheses. The second one is the right distributive property because the multiplication is on the right of the parentheses.

All multiplication algorithms depend on the distributive property. When we multiply 45*9 by multiplying 5*9 and 40*9, we are using the right distributive property: (40+5)*9=40*9+5*9.

Anytime we finish a multiplication problem by adding, we are using the distributive property.

If we think of multiplication as finding the area of a rectangle, then the distributive property can be nicely illustrated with a diagram:

The sum of the areas of the two smaller rectangles is the same as the area of the large rectangle. So 40*9+5*9=45*9.

Finally, FOIL, the standard way in American algebra classromms of remembering how to multiply an algebraic expression such as (x+3)(x+2), is an application of the distributive property.

When we talk about "the distributive property" we usually mean "the distributive property of multiplication over addition". Saying "the distributive property" is shorthand-an abbreviation.

We could just as well ask about other distributive properties. For instance, does division distribute over addition? Let's look at what that would mean:

A/(B+C)=A/B+A/C

and

(B+C)/A=B/A+C/A

We really ought to check the truth of these equations.

Let's try the first one-the left distributive property of division over addition. Consider a=1, b=2 and c=3. Then a/(b+c)=1/5 but a/b+a/c=1/2+1/3=5/6. But 1/5 does not equal 5/6. So the left distributive property of division over addition does not exist.

Now the second one-the right distributive property of division over addition. Using the sample values from before, (2+3)/1=5/1=5 and 2/1+3/1=2+3=5. So we get the same result from computing both ways.

But it also works for a=5, b=6, c=10. Then (b+c)/a=(6+10)/5=16/5 and b/a+c/a=6/5+10/5=16/5.

This is not enough to prove that there is a right distributive property of division over addition. But it seems to suggest that it's true. And indeed, it is.

So division distributes over addition from the right but not from the left.

NOTE: Some of the text above has been corrected from a previous version which misstated the right-distributive property of division over addition. Thanks to student Yuriko for finding the error.

We don't typically think of exponentiation as an operation, but it is. In order to see this, it can be helpful to write ab as a^b. Now we can treat the "^" symbol just like we do the "+" symbol or any other operation.

And then we can ask whether there is a distributive property of exponentiation over addition, or of exponentiation over multiplication, etc.

If you consider the following four properties, one of them is true. The rest are false. Can you figure out which one is true?

1. The left-distributive property of exponentiation over addition. (i.e. that a^(b+c)=a^b+a^c or ab+c=ab+ac.)
2. The right-distributive property of exponentiation over addition.
3. The left-distributive property of exponentiation over multiplication.
4. The right-distributive property of exponentiation over multiplication.

In courses such as Modern Algebra, mathematicians consider the distributive property in the abstract-i.e. it is no longer grounded in statements about familiar operations on numbers, but in statements about new operations on new mathematical objects.

If you were able to find the correct property from the four above, you have taken a step towards higher mathematics. Congratulations!

(and one more thing...the property that is true in the list is actually a standard property of exponents-you probably already know it; you just have never noticed that it is an example of a distributive property.)

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