Subtracting a Negative Number

Date: 08/16/99 at 22:49:53
From: Elaina Boetto
Subject: Subtracting a negative from a negative

I recall that when you subtract a negative number, the two "negative 
signs" change the number following them to a positive. Therefore 
-7 - -5 = -2. Then I question how -7 + 5 = -2 as well. It doesn't make 
sense to me.

Thanks for your help!

Date: 08/17/99 at 18:49:09
From: Doctor Rick
Subject: Re: Subtracting a negative from a negative

Hi, Elaina.

A good way to picture addition and subtraction of positive and 
negative numbers is to imagine standing on a number line. 

To add, start on the first number, face right (toward higher numbers), 
and walk the number of "steps" named by the second number.

     5 + 3 = ?

Start on 5, face right, go 3 steps to 8.

                                              X=============>O
      +----+----+----+----+----+----+----+----+----+----+----+
     -3   -2   -1    0    1    2    3    4    5    6    7    8

To subtract, start on the first number, face left, and walk the number 
of steps named by the second number.

     5 - 3 = ?

Start on 5, face left, go 3 steps to 2.

                               O<=============X
      +----+----+----+----+----+----+----+----+----+----+----+
     -3   -2   -1    0    1    2    3    4    5    6    7    8

If the second number is negative, you will need to walk backward. 
Thus:

     5 + (-3) = ?

Start on 5, face right, go back 3 steps to 2.

                               O<=============X
      +----+----+----+----+----+----+----+----+----+----+----+
     -3   -2   -1    0    1    2    3    4    5    6    7    8

Notice that 5 + (-3) is the same as 5 - 3, except that you are "facing 
the other way."

Now for the hard one:

     5 - (-3) = ?

Start on 5, face left (to subtract), go backward (right) 3 steps to 8.

                                              X=============>O
      +----+----+----+----+----+----+----+----+----+----+----+
     -3   -2   -1    0    1    2    3    4    5    6    7    8

Thus 5 - (-3) is the same as 5 + 3.

What if the first number is negative? Everything works just the same.

     -5 - (-3) = ?
     -5 +   3  = ?

These are the same problem. Start on -5, go right 3 steps to -2.

       X=============>O
  +----+----+----+----+----+----+----+----+----+----+----+
 -6   -5   -4   -3   -2   -1    0    1    2    3    4    5

If this gives you trouble, you can trade in the negative numbers for 
positive numbers that are easier to work with. You have to change 
_all_ numbers, and change them back when you're done. In other words:

1. Replace positive numbers with negative numbers and negative numbers 
   with positive numbers.

2. Do the addition or subtraction.

3. Switch the sign of the result.

You can do this whenever you want to reduce the number of negative 
signs in a problem. Let's do your example:

     -7 - (-5) = ?

      7 -   5  = ?

I changed -7 to 7 and -5 to 5. The answer is, of course, 2. Now we do 
the third step and change the sign of this result from 2 to -2. This 
is the answer:

  -7 - (-5) = -2

I hope this helps.

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