Order of Operations vs. Distributive Property

Date: 02/04/2005 at 20:39:08
From: Jeremy
Subject: order of operations vs factoring

Doesn't the distributive law contradict the rules of order of 
operations?  We should do parentheses before multiplication or 
division, but the distributive law tells us it's ok to solve x(a + b) 
by a*x + b*x, in spite of the parentheses.  Is this just an 
exceptional case?

Date: 02/04/2005 at 22:50:28
From: Doctor Peterson
Subject: Re: order of operations vs factoring

Hi, Jeremy.

The roles of the order of operations and the distributive property are
complementary, rather than contradictory.  That is, each applies to a
different aspect of algebra.

The order of operations tells us what an expression MEANS: if we
follow the rules, we will correctly evaluate the expression.

Properties like the distributive property tell us how we can REWRITE
an expression without changing its value.  So if we are faced with an
expression like

  a(b + c)

which (taken at face value) MEANS that we add b and c, then multiply
the result by a, we know that we will get the same value if we
rearrange it and INSTEAD evaluate

  ab + ac

So properties allow us to safely manipulate an expression to make it
easier to evaluate, or to solve a problem, knowing that it still has
the same value.  That doesn't change its meaning, only how we actually
calculate it.

The following page discusses a similar issue:

  Order of Operations vs. Associative Property
    http://mathforum.org/library/drmath/view/61447.html 

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

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