Adding and Subtracting Negative Numbers

Date: 02/07/2002 at 20:52:15
From: Devin Myers
Subject: Negative problem

What is (-1) - 5 + -8 equal to?

Date: 02/07/2002 at 23:14:57
From: Doctor Peterson
Subject: Re: Negative problem

Hi, Devin.

You can picture this on a number line.

You are starting at -1, which is 1 unit to the left of zero. Now you 
subtract 5; that takes you 5 more units to the left (to -6). Finally, 
you add -8. This is the same as subtracting 8: it takes you 8 units to 
the left. That takes you to -14.

Notice that you have gone left three times, so you are 1+5+8 units to 
the left. This can be expressed by thinking of subtraction as addition 
of the negative, then thinking of a negative as multiplication by -1 
and using the distributive property:

    -1 - 5 + -8 = -1 + -5 + -8
                = (-1)(1) + (-1)(5) + (-1)(8)
                = (-1)(1 + 5 + 8)
                = -(1 + 5 + 8)
                = -14

This is the way I think about negative numbers, without ever having to 
write it out this way. When I see subtraction, I THINK addition of the 
negative; when I see negation, I THINK multiplication by -1.  With 
experience, I don't have to write anything down to answer your 
question. But you can take your time; think it through the way you 
probably had to think about every step when you first learned to walk, 
and soon you won't be falling down so often!

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