Associative Property of Multiplication and Grouping Symbols

Date: 10/22/2006 at 15:50:37
From: Erika
Subject: Associative Property of Multiplication Grouping Symbols

Will the associative property of multiplication delete grouping 
symbols? 

Ex: (a x b) x c 
    (b x c) x a
     a x b x c

My algebra teacher says that each algebraic "tool" has certain things
it can do.  While you may be able to find a simple answer he always
wants to know which tools you used to teach us to synthesize.  I want
to make sure I know what each property can do and what it can't.

I would think that it could delete grouping symbols and not affect the
answer because 2 x (3 x 4) = 24, (2 x 3) x 4 = 24, 2 x 3 x 4 = 24.

Date: 10/22/2006 at 23:08:39
From: Doctor Peterson
Subject: Re: Associative Property of Multiplication Grouping Symbols

Hi, Erika.

Note that the associative property says

  (ab)c = a(bc)

which does not EXPLICITLY drop the parentheses, just moves them around.

But also note that when we write it without parentheses, as abc, the
order of operations tells us to work left to right, so

  abc = (ab)c

by definition.  We multiply ab first, then multiply that by c.

So to drop the parentheses in (ab)c, we can just use the order of
operations; those parentheses are not really needed!  But to drop the
parentheses in a(bc), we have to first apply the associative property
to change it to (ab)c, and THEN use the order of operations.

The examples you give don't all have the same order for a, b, and c,
so you would have to use the commutative property too for the middle
one to be equal to the others.  The first two have their parentheses
on the left, so they don't require associativity.

The net effect of associativity and the order of operations is that we
can ignore parentheses where only multiplication (or only addition) is
involved; we don't normally pay much attention to which we are
actually doing, but it's good to notice the details once in a while!

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


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