Lesson EXPLANATION of distributive property

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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 <A HREF=Solvers.html>Solvers Section</A>, 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:

<CENTER>
{{{
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
)
}}}

</CENTER>

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
)
}}}

Taking <B>a</B> OUT of parentheses: {{{
cartoon( 
  ax+ay,
  a(x+y)
) }}}

<H4> Why is the distributive property true?</H4>

Check out a <A HREF=proof-of-distributive-property.lesson>separate lesson</A> that I wrote that proves distributive property for integer numbers.

<H4>Addition vs. subtraction inside parentheses</H4>

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:

<CENTER>{{{
cartoon( 
  a*(b-c),
  a*(b-c) = a*(b + (-c)),
  a*(b-c) = a*b + a*(-c),
  a*(b-c) = a*b - a*c
)
}}}
</CENTER>