What Sign Do You Use?

Date: 10/21/2001 at 18:42:24
From: butterfly
Subject: If two negatives equal a positive

If two negatives equal a positive, then how do you do it, and what 
sign do you take?

Like -4-(-7): would it be -4 + -7= 11?

Thank you,  
Butterfly

Date: 10/22/2001 at 11:33:43
From: Doctor Ian
Subject: Re: If two negatives equal a positive

Hi Butterfly,

When you _multiply_ two negative numbers, you get a positive number:

  Negative x Negative = Positive - Dr. Math FAQ
   http://mathforum.org/dr.math/faq/faq.negxneg.html   

But when you _add_ two negative numbers, you get another negative 
number. 

This illustrates one of the problems with trying to rely on 'rules' 
like 'two negatives equal a positive' without really understanding 
what the rules mean. The irony in the situation is that once you 
understand what the rules mean, there is no need to remember them any 
more!

If you owe one friend 4 dollars (i.e., you have '-4 dollars'), and you 
owe another friend 7 dollars (i.e., you have another '-7 dollars'), 
how much money do you have altogether? It's not a positive number, is 
it? If it were, we could all get rich by spending money instead of by 
earning it.

Now, suppose you owe a friend 7 dollars (i.e., you have -7 dollars), 
and I pay him 4 of those dollars for you (i.e., I've 'taken' -4 
dollars away from you). How much money do you have now?  

  -7 - (-4)  = ?

Here is another way to look at it.  I can rewrite a subtraction this 
way:
          
  7 - 4 = 1 + 1 + 1 + 1 + 1 + 1 + 1 - 1 - 1 - 1 - 1
         
Do you see why? But each of the subtractions cancels out one of the
additions, right? 

         = 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 + 1 + 1 + 1
           \___/   \___/   \___/   \___/
             0       0       0       0

         = 1 + 1 + 1

         = 3

Well, it works the same way with negative numbers:

-7 - (-4) 

   = -1 + -1 + -1 + -1 + -1 + -1 + -1 - (-1) - (-1) - (-1) -  (-1)

   = -1 - (-1) + -1 - (-1) + -1 - (-1) + -1 - (-1) + -1 + -1 + -1
                 \_______/   \_______/   \_______/   \_______/
                     0           0           0           0

   = -1 + -1 + -1

   = -3

Now, this would be really tedious for something like 

  -298 - (-197) = ?

And that's why it's so tempting to make up rules and shortcuts. But
applying the wrong rule or shortcut just gets you to the wrong answer 
more quickly.

Does this help? Write back if you'd like to talk about this some more, 
or if you have any other questions. 

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