Subtracting Numbers by Walking a Number Line

Date: 09/12/2001 at 20:24:10
From: Stephanie
Subject: I need help on subtracting rational numbers

I am confused about subtracting rational numbers. 

For example 

     -8.1- -6.9 = -1.2 
and 
     -9.9- -11.1 = 1.2 

All the numbers are negative, but the answer to the first problem is a 
negative and the second is not.  Why?

Thanks.

Date: 09/13/2001 at 12:52:34
From: Doctor Ian
Subject: Re: I need help on subtracting rational numbers

Hi Stephanie,

Sometimes your body can be smarter about math than your mind is.  
Suppose you draw a line on the floor, and label it like this:

  <--------------------------------------------------->
    -12  -10  -8  -6  -4  -2   0   2   4   6   8   10

To add a positive number, face right and walk forward:

                        -4 + 6 = 2
                       o...........>
  <--------------------------------------------------->
    -12  -10  -8  -6  -4  -2   0   2   4   6   8   10


To add a negative number, face left and walk forward:

                        2 + -6 = -4
                       <...........o
  <--------------------------------------------------->
    -12  -10  -8  -6  -4  -2   0   2   4   6   8   10

To subtract a positive number, face right and walk _backward_:

                         2 - 6 = -4
                       >...........o
  <--------------------------------------------------->
    -12  -10  -8  -6  -4  -2   0   2   4   6   8   10

Now, this is interesting... subtracting a positive number looks just 
like adding a negative number!  And in fact, that is exactly the case:

  2 - 6 = 2 + (-6)

There is one case left. To subtract a negative number, face left and 
walk backward:

                        -4 - -6 = 2
                       o...........<
  <--------------------------------------------------->
    -12  -10  -8  -6  -4  -2   0   2   4   6   8   10

Now, this is also interesting... subtracting a negative number looks 
just like adding a positive number!  And in fact, that is exactly the 
case:

  2 - -6 = 2 + 6

Let's summarize what we're doing. We want to model an expression like

   ____ +/- _____

Here is how to do it:

  1. Move directly to the first number. 

  2. Look at the second number. If it's positive, face right
     (in the positive direction). If it's negative, face left. 

  3. Look at the operator.  If it's '+', walk forwards.  If it's
     '-', walk backwards. 

Let's try it with 

  -8.1 - -6.9 

First, move to -8.1:

              o
  <--------------------------------------------------->
    -12  -10  -8  -6  -4  -2   0   2   4   6   8   10

The second number is negative, so we're going to face left:

             <o
  <--------------------------------------------------->
    -12  -10  -8  -6  -4  -2   0   2   4   6   8   10

We're subtracting, so we want to walk backward:

              o..............<
  <--------------------------------------------------->
    -12  -10  -8  -6  -4  -2   0   2   4   6   8   10

What about 

  -9.9 - -11.1 

Again, we start at the first number:

            o
  <--------------------------------------------------->
    -12  -10  -8  -6  -4  -2   0   2   4   6   8   10

Again, the second number is negative, so we face left:

           <o
  <--------------------------------------------------->
    -12  -10  -8  -6  -4  -2   0   2   4   6   8   10

And again, we're subtracting, so we walk backward:

            o....................<
  <--------------------------------------------------->
    -12  -10  -8  -6  -4  -2   0   2   4   6   8   10

So why does the first one end up negative, while the second one ends 
up positive? In both cases, we started with a negative number, and 
moved back towards 0. In the first case, the _magnitude_ of the second 
number wasn't enough to get us back past 0; in the second case, it 
was. 

Note that while you can tell the sign of a product just by looking at 
the signs of the factors, e.g., 

  a *  b =  ab
 -a *  b = -ab
  a * -b = -ab
 -a * -b =  ab

you can't do this with addition or subtraction, except when both signs 
are the same:

  a +  b = positive
 -a +  b = positive or negative
  a + -b = positive or negative
 -a + -b = negative

Again, this is the sort of thing your body might be able to figure out 
pretty easily. If you start on one side of zero, and you move away 
from zero, there's no way for you to end up on the other side of zero.  
(Try it and see.)  

But if you start on one side of zero, and you move back in the 
direction of zero, the side you end up on depends on how far you move.  
(Again, try it and see.)  

Of course, you don't have to actually draw a line on the floor every 
time you want to add or subtract negative numbers. (That would look 
pretty strange during a test, wouldn't it?)  But you can imagine doing 
this, and it should work almost as well. 

In fact, if you think of numbers as arrows, you can accomplish the 
same thing by drawing simple pictures. (Some people think better with 
their bodies; other people think better with their eyes.) 

A positive number corresponds to an arrow that points to the right; a 
negative number corresponds to an arrow that points to the left.  

     5       -2       7        -3
   ---->     <-     ------>   <--

To add two numbers, you put the tail of the second arrow at the head 
of the first, and see what arrow would take you from where you started 
to where you'd end up:

      5           
    ---->         
      <--
        -2
    -->           5 + -2 is the same as 3
     3

     -3
    <-- 
    ------>
      7
       --->       -3 + 7 is the same as 4
        4

It's sort of like: 'Take this trip; and then take this other trip; and 
see where you end up'. 

To subtract a number, add the negative of the number:
 
          -4
        <---
   ---->       
     5
   <----          
     -5
   <--------       -4 - 5 = -4 + -5 = -9
       -9

    3
   --->
       <------
         -6
       ------>
          6
   ---------->     3 - -6 = 3 + 6 = 9
        9

    -4
  <---
  <-----
    -6
  ----->
     6
      ->
      2             -4 - -6 = -4 + 6 = 2

Now we can see immediately why the two cases you cited are different:

      -8.1
   <-------
   <------
     -6.9
   ------>
      6.9
          < 
         -1.2          -8.1 - -6.9 = -8.1 + 6.9 = -1.2

      -9.9
   <---------
   <----------
      -11.1
   ---------->
       11.1
             >
             1.2        -9.9 - -11.1 = -9.9 + 11.1 = 1.2

What makes the difference is whether the second arrow is shorter or 
longer than the first one. 

Give these methods a try on some other problems, and let me know if it 
isn't helping, so we can try to come up with something else. 

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