Subtracting Negative Numbers

Date: 03/30/97 at 08:48:24
From: Kelly Buchan
Subject: Negative numbers

I am a teacher, and still have trouble explaining subtracting negative
numbers from negative numbers to my students.  I have tried many 
different ways.  I think the underlying problem is that I truly don't 
understand it myself.

If you owe me 5 dollars (-5) and then you owe me 5 more dollars (-5) 
that would be an addition of negative numbers right?

     (-5) + (-5) = (-10)

If you owe me 5 dollars (-5) and I suggest that you can forget the 
debt, that means you can subtract the (-5) right?

     (-5) -(-5) = 0

Do I have it yet?

Kelly

Date: 03/31/97 at 17:30:46
From: Doctor Keith
Subject: Re: Negative numbers

Hi,

Negative numbers get everyone. Your example seems good but allow me
to give a few examples and some comments to help clarify.

1) Accounting (similar to your example)
   Say I borrow $5 from you and buy lunch since I forgot my wallet. 
   I owe you $5 so on my balance sheet I have -5.  Later I get my 
   wallet and pay you the $5 back so I can subtract my debit (a net 
   credit, subtracting a negative is a positive).  So we have 
   -5 -(-5) = -5 +5 = 0, which is what I now owe.

2) Driving
   You are driving with cruise control set at 65mph (in a 65 zone, 
   of course), which we will call your reference speed.  You see a 
   sign stating that you are entering a 55 zone so you slow down 
   10 mph ( -10).  After a few miles a new sign informs you that 
   you are entering a 65 zone again so you resume your original 
   speed, thus removing (subtracting) the -10mph modification.  
   We thus have -10 - (-10) = 0, or no speed modification
   (thus you are moving at the reference speed of 65 again).

3) Books
   You borrow 3 books from a library.  You thus owe three books (-3).  
   You read one and discover it does not cover what you want, so you 
   return (subtract) it (a borrowed book is a minus, thus a -1) and 
   thus you have subtracted one book you owe, and now owe only two.  
   And we have:  -3 -(-1) = -3 + 1 = -2 

4) Behavior
   Johny swears and fights a lot (two negatives).  He feels he wants 
   to get better so he decides to stop (thus removing or subtracting) 
   fighting (a negative).  Thus he now has -2 - (-1) = -2 + 1 = -1, 
   or 1 negative behavior he does a lot.

Remember that subtracting a negative is adding a positive. Seeing this 
is a matter of perspective on the problem, as you can see from the 
above examples.

Also keep in mind that the negative numbers are the natural extension
of the positive numbers - that is, the basic mathematical operations 
(addition, subtraction, etc.) work the same on negative numbers. E.g.:

   -5 -(-3) = -(5-3) = -(2) = -2
           or
   -5 -(-3) = -5 + 3 = -2

So you can handle them using all the fun properties of algebra, like
the distributive property, which is modeled above.

As you can see, the math itself is straightforward. The real challenge
is explaining what it means in real life.  That has always been a
challenge to mathematicians and teachers, but what I have written here 
should help you out.  If anything is unclear, or you would like me to 
add more detail to any part of this, let me know.

-Doctor Keith,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/