Applying the Distributive Property to Division or Subtraction

Date: 12/29/2006 at 11:32:36
From: Elena
Subject: Distributive property applied to division or subtraction

The distributive property is easily shown with multiplication and
addition.  Can it be used with subtraction or division?

I think it works sometimes but not all the time.  So, does it only
apply sometimes?

Date: 12/29/2006 at 22:39:55
From: Doctor Peterson
Subject: Re: Distributive property applied to division or subtraction

Hi, Elena.

The distributive property does work for multiplication over subtraction:

  a(b - c) = ab - ac
  (a - b)c = ac - bc

But you can only distribute division over addition (or subtraction) in
one direction:

  a/(b + c) = a/b + a/c is false
  (a + b)/c = a/c + b/c is true

This is most easily understood if you do as mathematicians do, and
think of subtraction as adding the opposite, and of division as
multiplying by the reciprocal.  Then the subtraction cases above
really mean this:

  a(b - c) = a(b + -c) = ab + a(-c) = ab + -ac = ab - ac
  (a - b)c = (a + -b)c = ac + (-b)c = ac + -bc = ac - bc

So we are really just distributing multiplication over addition.

The second case for division works this way:

  (a + b)/c = (a + b) * (1/c) = a*(1/c) + b*(1/c) = a/c + b/c

So there we are just distributing multiplication BY the reciprocal.
But this doesn't work in the other case:

  a/(b + c) = a * 1/(b + c)

and there is no rule to simplify the reciprocal of a sum.

If you have any further questions, feel free to write back.


- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/