Subtracting Negative Numbers

Date: 12/06/2005 at 14:49:15
From: Rashi
Subject: How do you subtract negative numbers

Why do we have to change a subtraction problem to an addition problem
and change the second term to the opposite sign?  For example,

  -9 - (-3) = -9 + (3) = -6

My teacher made me memorize the rule but I want to fully understand
why it works.  I don't just want a rule, I want an example that I can
understand and use.

Date: 12/06/2005 at 20:15:53
From: Doctor Rick
Subject: Re: How do you subtract negative numbers

Hi, Rashi.

I like it when students want to UNDERSTAND math!  Memorizing rules is 
OK sometimes, but it's much better to understand the reasoning behind 
them, because you can do a lot more with reasoning than with rules. 

Let's start by understanding what subtraction means.  To do that, I'll 
start with addition.  Consider the addition 6 + 3 = 9 on a number 
line.  You start at the 6 and go 3 units to the right:

                                 6      +3     =9
                                 O=============>O
 --+----+----+----+----+----+----+----+----+----+----+----+--
   0    1    2    3    4    5    6    7    8    9   10   11

Now, what does the subtraction 9 - 3 = 6 mean?  It is the answer to 
the problem

  ___ + 3 = 9

In other words, the subtraction is "undoing" the addition above: we 
start at the 9 and back up three spaces.  (Read backwards: 9 - 3 = 6)

                                =6      -3      9
                                 O<=============O
 --+----+----+----+----+----+----+----+----+----+----+----+--
   0    1    2    3    4    5    6    7    8    9   10   11

Now what about negative numbers?  To add 9 + -3, I start at 9 on the 
number line and move LEFT 3 spaces:

                                =6    + -3      9
                                 O<=============O
 --+----+----+----+----+----+----+----+----+----+----+----+--
   0    1    2    3    4    5    6    7    8    9   10   11

That's 9 + -3 = 6.  Notice that it's the same as 9 - 3 = 6: adding a 
negative is the same as subtracting the opposite of the negative 
(which is positive).

Finally, we can subtract -3 from 6: the result is the answer to the 
problem

  ___ + -3 = 6

As before, I work backward: I start at the 6 and "undo" the addition 
of -3.  I reverse the arrow I drew above, going 3 spaces to the right 
instead of left.

                                 6    - -3     =9
                                 O=============>O
 --+----+----+----+----+----+----+----+----+----+----+----+--
   0    1    2    3    4    5    6    7    8    9   10   11

What do you notice this time?  I do exactly the same thing that I did 
for 6 + 3.

  6 - (-3) is the same as 6 + 3

Does this help?

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

Date: 12/07/2005 at 08:10:01
From: Rashi
Subject: Thank you (How do you subtract negative numbers)

Yes, that helped a lot.  Thank you.