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Introduction to Division with Negative Numbers

Date: 05/05/2004 at 21:01:04
From: Kallie
Subject: do not know

I don't know how to do division with negative numbers, like
+18/-9   or   -24/-8   or   +21/-3.  It is hard for me.



Date: 05/05/2004 at 22:50:24
From: Doctor Ian
Subject: Re: do not know

Hi Kallie,

Here's one way that I would handle these.  We can always rewrite
something like -9 as -1 * 9, so 

  18    1 * 18
  -- = -------
  -9   -1 *  9

How does this help? Well, let's take another step:

  18    1 * 18
  -- = -------
  -9   -1 *  9

        1   18
     = -- * --
       -1    9

We can do this no matter what combination of signs we have:

               __________ We know how to do this part.
              |

  -18   -1   18
  --- = -- * --
   -9   -1    9

   18    1   18
  --- = -- * --
   -9   -1    9

  -18   -1   18
  --- = -- * --
    9    1    9

   18    1   18
  --- = -- * --
    9    1    9
 
         |________________ We just have to figure out this part!

So we can really break these into two problems:  dividing the positive
numbers (which you've already learned how to do), and dealing with the
signs.  

Does this make sense so far?  If so, then we just have to learn how to
deal with the following:

  -1
  -- = ?    
  -1 

   1
  -- = ?    
  -1 

  -1
  -- = ?    
   1 

   1
  -- = ?    
   1 

So, how do we make sense of these?  Well, let's think back to when you
first learned about division.  Recall that all three of the following
statements, 

      12 = 4 * 3

  12 / 4 = 3

  12 / 3 = 4

are just different ways of expressing exactly the same fact.  So when
we see one, we can replace it with either of the others.  So let's try
that with our cases, above:

  -1
  -- = ?     -->    -1 = ? * -1    
  -1 

   1
  -- = ?     -->     1 = ? * -1
  -1 

  -1
  -- = ?     -->    -1 = ? * 1
   1 

   1
  -- = ?     -->     1 = ? * 1
   1 

Three of the cases are pretty straightforward:


  -1
  -- = ?     -->    -1 = ? * -1     --> ? must be 1, so -1/-1 = 1
  -1 

   1
  -- = ?     -->     1 = ? * 1      --> ? must be 1, so 1/1 = 1
   1 

  -1
  -- = ?     -->    -1 = ? * 1      --> ? must be -1, so -1/1 = -1
   1 

But what about the remaining case?  

   1
  -- = ?     -->     1 = ? * -1     
  -1 

Here, we use the rule that 

  -1 * -1 = 1

which is really the only new piece of information that you need to
have to solve these kinds of problems.  So our last case is

   1
  -- = ?     -->     1 = ? * -1      --> ? must be -1, so 1/-1 = -1   
  -1 

Now, I know that this looks like a lot to learn!  But as I've pointed
out, almost all of it follows from things you've already learned.  

Let me know if this helps. 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
Elementary Division
Elementary Fractions
Middle School Division
Middle School Fractions
Middle School Negative Numbers

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