Order of Numbers in Subtraction and Division

Date: 11/09/1999 at 10:04:48
From: Julia
Subject: Adding or multiplying numbers

You can add or multiply numbers in any order, but you cannot switch 
order to divide or subtract. Why?

Date: 11/09/1999 at 12:43:51
From: Doctor Peterson
Subject: Re: Adding or multiplying numbers

Hi, Julia.

The difference is that in addition and multiplication, the two numbers 
you are working with do the same thing, while in subtraction and 
division they do opposite things.

In addition, each number tells you how far to move (to the right) on 
the number line:

                   4           2
           +-------------->o------>o
     --+---+---+---+---+---+---+---+---+---+--     4 + 2 = 6
           0                       6

               2           4
           +------>o-------------->o
     --+---+---+---+---+---+---+---+---+---+--     2 + 4 = 6
           0                       6

In multiplication, each number can be thought of as one side of a 
rectangle:

               4
       +---+---+---+---+
       |   |   |   |   |
     2 +---+---+---+---+     2 x 4 = 8
       |   |   |   |   |
       +---+---+---+---+

           2
       +---+---+
       |   |   |
       +---+---+
       |   |   |
     4 +---+---+             4 x 2 = 8
       |   |   |
       +---+---+
       |   |   |
       +---+---+

But when you subtract, you are using one number to move right from 
zero to the starting place, and the other to move left:

                       4
               +-------------->o
                       o<------o
     --+---+---+---+---+---+---+---+---+---+--     4 - 2 = 2
               0       2

                   2
               +------>o
       o<--------------o
     --+---+---+---+---+---+---+---+---+---+--     2 - 4 = -2
      -2       0

If you're familiar with negative numbers, subtracting is the same as 
adding the opposite of the subtrahend:

     4 - 2 = 4 + -2 = 2

     2 - 4 = 2 + -4 = -2

Only one of the numbers is negated, so the order matters. When you 
switch the two numbers, the result is negated.

Similarly, division is the same as multiplying by the reciprocal:

     4 / 2 = 4 x 1/2 = 2

     2 / 4 = 2 x 1/4 = 1/2

When you switch the numbers, the result is inverted.

Now, all of this shows that there is a way to apply the commutative 
property to subtraction and division: just rewrite each as addition or 
multiplication. For example, I can safely rearrange

     2 - 3 + 5
to
     2 + 5 - 3

by thinking of it as

     2 + -3 + 5 = 2 + 5 + -3

and dragging the minus sign along when I move the 3.

Similarly, I can do this with division:

     2 / 6 * 3 = 2 * 3 / 6

dragging the division with the 6, because it really means

     2 * 1/6 * 3 = 2 * 3 * 1/6

That's probably a lot more than you wanted to hear, but I think it 
should help.

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