Adding and Subtracting Integers

Date: 04/17/2006 at 21:06:36
From: Constance
Subject: Adding and subtracting integers

I am having a hard time with integer operations.  I have the most
problems with subtraction when negatives are involved.

Date: 04/18/2006 at 09:52:21
From: Doctor Ian
Subject: Re: Adding and subtracting integers

Hi Constance,

Let's start with adding and subtracting integers.

The basic rule is that you get partial cancellation.  For example, 

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

         = 0 + -2

         = -2

In this case, the negative number had the larger magnitude, so we end
up with some negative "left over".   When the positive has the larger
magnitude, the result is positive:

  8 + -2 = (6 + 2) + -2

         = 6 + (2 + -2)

         = 6 + 0

         = 6

That's the basic idea.  Another useful thing to know is that you can
convert any subtraction into the addition of the corresponding
negative, i.e., 

  3 - 4 = 3 + -4

What happens when you subtract a negative?

  3 - -4 = ?

To understand this, it's good to know that a negative number is just
-1 times the corresponding positive number:

  -4 = -1*4

This means the negative of a negative is positive:

  -(-4) = -1*(-1*4)

        = (-1*-1)*4

        = 1*4

        = 4

So 

  3 - -4 = 3 + -(-4)

         = 3 + 4

Does this make sense? 

In practice, whenever I have both additions and subtractions, the very
first thing I do is convert everything to additions, using these rules
for changing signs.  Why do I do that?

Well, for one thing, then I just have to know how to add numbers, and
I don't need extra rules for dealing with subtractions.  That makes my
life simpler, since I don't like to have to remember more than I
absolutely have to!

Second, the nice thing about additions is that addition is commutative
and associative, which means I can reorder and group additions any way
I want:

    3 - 4 + 5 - 6 + 7 - 8

  = 3 + -4 + 5 + -6 + 7 + -8       Change to additions

  = 3 + 5 + 7 + -4 + -6 + -8       Addition is commutative

  = (3 + 5 + 7) + (-4 + -6 + -8)   Addition is associative

  = 15 + -18

  = -3

But subtraction is neither commutative nor associative.  So addition
is much, much easier to work with.

Is this clear so far?   

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

Date: 05/06/2006 at 10:57:11
From: constance
Subject: Thank you (Adding and subtracting integers)

Thank you.  It was a great help.