Subtracting a Negative from a Positive

Date: 04/11/2006 at 12:14:02
From: Dianne
Subject: How do you explain subtracting a - number from a + number?

How do you explain subtracting a negative number from a positive?  For
example:

10 - (-6) = 16

I know the rule and just apply it, but I want to know why it works. 
Can you also give me a real life example of when this occurs?  I am a
teacher looking to explain this to my 5th grade students.

Date: 04/11/2006 at 12:54:04
From: Doctor Ian
Subject: Re: How do you explain subtracting a - number from a + number?

Hi Diane,

The way I think about it is this:

  1.  Any subtraction is the same as an addition of the
      additive inverse.  So (a - b) is the same as (a + -b).

  2.  The additive inverse of any number is the number 
      times -1. 

  3.  -1 times -1 is 1. 

Applying this to your example, 

  10 - (-6) = 10 + -(-6)

            = 10 + -1*(-1 * 6)

            = 10 + (-1* -1*)*6

            = 10 + 6     

The whole point of having negative numbers is so you don't have to
have separate rules for dealing with things like east and west, profit
and loss, and other pairs of opposites. 

When does subtraction occur in real life?  When some change occurs: 
John grows from 48 inches to 52 inches.  How much did John grow?

  52 inches - 48 inches = 4 inches

What if we have measurements that can have associated directions? 
John was at 5 degrees west longitude, and he drove to 7 degrees east
longitude.  How far did he travel?

If we use negative numbers for west, and positive for east, this is
just like the previous problem:

  7 degrees - -5 degrees = 12 degrees

Other measurements like this might include elevations (above or below
sea level), temperatures (above or below zero), electrical charges,
and so on. 
 
The nice thing about using the rules above is that they'll all
translate directly to problems in algebra, e.g., manipulating
polynomials.  

There's a second way you can approach it, too, which is to go back and
revisit the definition of subtraction.  That is, 

   a - b = c    and     a = b + c 

are just two different ways of saying the same thing.  So the answer to 

   a - b = ?

is 

   a = b + ?    or what would I have to add to b, to get a? 

From this point of view, starting from

  10 - -6 = ?

we can rewrite it as

       10 = -6 + ?

Now, what do you have to add to -6 to get 10?  

Does this help? 

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