Algebra Know-It-ALL

What if we reverse the order of this sum? We start with 2 and travel upward −3 units. We’re
talking about displacement here, not simple distance, so negatives can make sense! An upward
displacement of −3 units is the same as a downward displacement of 3 units. This process is
shown on the right in Fig. 4-2. When we add the integers −3 and 2 in either order as shown,
we end up at the same point, which corresponds to −1. We have now analyzed these two
facts:

− 3 + 2 =− 1

and

2 + (−3)=− 1

To subtract, move downward

If you think it’s ridiculous to imagine downward movement as negative upward movement,
you’re right, except for one little catch. You are going to come up with situations in mathemat-
ics where you’ll get a negative quantity for an answer to a problem, and it won’t seem to make
sense. Suppose you fly a rush-hour traffic observation helicopter for your local TV station.
You see that a rain shower has caused an existing jam in the westbound traffic on Boxelder Bug

0

1

2

3

1

2

• 3


• 
  


• 
  

Start here

Start here

Move upward

by 2 units

Finish here Finish here

Move upward

by-3 units

Figure 4-2 On the left, we add − 3 + 2. On the right,

we add 2 + (−3). When we move negatively

upward, we move downward by the equivalent

distance.

Moving Up and Down 53