Lesson EXPLANATION of distributive property

Distributive property is a property of numbers that ties the operation of addition (or subtraction) and mutiplication together.

It says that for ANY numbers a, b and c, a*(b+c) = a*b+a*c

. For those used to multiplications without the multiplication sign, a(b+c) = ab+ac

.
The same property applies when there is subtraction instead of addition: a*(b-c) = a*b - a*c.

In the Solvers Section, there are two solvers that show CARTOONS of distributive property, for any numbers that you supply. Try them out, they are fun.
Here's a sample cartoon that shows how distributive property works:

 
cartoon( 
2(3+4), 
2*3+2*4, 
2*3+highlight(2*4), 
2*3+highlight(8), 
2*3+8, 
highlight(2*3)+8, 
highlight(6)+8, 
6+8, 
14 
)

<BR>
cartoon( <BR>
2(3+4),<BR>
2*3+2*4,<BR>
2*3+highlight(2*4),<BR>
2*3+highlight(8),<BR>
2*3+8,<BR>
highlight(2*3)+8,<BR>
highlight(6)+8,<BR>
6+8,<BR>
14<BR>
)<BR>

How it is used

The distributive property is used when something in paretheses is multiplied by something, or, in reverse, when you need to take some common multiplier OUT of the parentheses. Example:

Taking x INTO parentheses: 
cartoon( 
x(2y-3), 
x*2y-x*3, 
2xy-3x 
)

<BR>
cartoon(<BR>
x(2y-3),<BR>
x*2y-x*3,<BR>
2xy-3x<BR>
)<BR>

Taking a OUT of parentheses: 
cartoon( 
ax+ay, 
a(x+y) 
)

Why is the distributive property true?

Check out a separate lesson that I wrote that proves distributive property for integer numbers.

Addition vs. subtraction inside parentheses

As you know from studying addition of signed numbers (there is a module about this on this site), a subtraction is nothing more than addition of a number that's opposite to the one being subtracted. So, to work it out with an example: a*(b-c) = a*(b + (-c)). The right part of the formula is a regular subtraction. Here's the cartoon for it:

 
cartoon( 
a*(b-c), 
a*(b-c) = a*(b + (-c)), 
a*(b-c) = a*b + a*(-c), 
a*(b-c) = a*b - a*c 
)

<BR>
cartoon( <BR>
a*(b-c),<BR>
a*(b-c) = a*(b + (-c)),<BR>
a*(b-c) = a*b + a*(-c),<BR>
a*(b-c) = a*b - a*c<BR>
)<BR>