Subtracting Negative Integers - Number Line

Date: 10/15/96 at 19:57:18
From: Anonymous
Subject: integers

How do you subtract integers like

5-(-6) ?

Anne-Marie
New Paltz MS
New Paltz NY

Date: 12/12/96 at 22:11:30
From: Doctor Sydney
Subject: Re: integers

Dear Anne-Marie,

Let's use the number line for this.
                
-+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---
-4  -3  -2  -1   0   1   2   3   4   5   6   7   8   9   10


When we add two positive numbers, we can think of this on the number 
line as corresponding to starting at one of the numbers and moving to 
the right from that number a distance equal to the other number.  

Think about the sum 5 + 2.

From the number 5 we have to hop two places to the right.
We hop two places to the right because we are adding two more places.

For the answer to 5 + 2, we will land on the number 7.
                
-+---+---+---+---+---+---+---+---+---x->-+->-x---+---+---+---
+
-4  -3  -2  -1   0   1   2   3   4   5   6   7   8   9   10

So 5 + 2 = 7.


Now let's think about subtraction: 5 - 2 = ?
   
From the number 5 we need to hop two places to the left.
(This time we have to take away from a number, so we hop to the left.)
We will land on the number 3.

-+---+---+---+---+---+---+---x-<-+-<-x---+---+---+---+---+---
+
-4  -3  -2  -1   0   1   2   3   4   5   6   7   8   9   10

So 5 - 2 = 3.


So, in summary, when we add two numbers, we start at one of the 
numbers and hop to the right the value of the other number.  

When we subtract numbers, we start at the first number and move to the 
left the value of the second number. 

But what if we are subtracting a NEGATIVE number?  This is the 
question you were asking.  For instance, consider the following 
problem:

    5 - (-2) = ?

Well, we know from the work we did above that when we subtract a 
number from another number, we hop to the left the value of the second 
number.  So, to figure out what 5 - (-2) is, we would "hop to the left 
-2 hops."  But what does that mean?  

We know that hopping to the left 2 hops corresponds to moving to the 
left 2 hops, but the negative sign reverses everything, so moving to 
the left -2 hops really means that we move to the right +2 hops!

So, 5 - (-2) = 5 + 2.

Therefore, from number 5, we have to hop two places forward.
We will land on the number 7.
              
-+---+---+---+---+---+---+---+---+---x->-+->-x---+---+---+---
-4  -3  -2  -1   0   1   2   3   4   5   6   7   8   9   10


For your sum of 5 -( -6) I am sure you have the idea by now.
              
-+---+---+---+---+---+---+---+---+---x->-+->-+->-+->-+>--+->-x---+---
-4  -3  -2  -1   0   1   2   3   4   5   6   7   8   9   10  11

5 -(-6) = 5 + 6 
        = 11


Once you understand the basics and practice them often enough, these  
types of problems will be easy.

-Doctors Lim and Sydney, The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/