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Rules for Subtracting Integers


Date: 10/31/2001 at 10:41:08
From: Julie
Subject: Rules for subtracting integers

For my assignment I need to write rules for subtrating integers. I 
don`t know where to get this information.


Date: 10/31/2001 at 16:58:52
From: Doctor Peterson
Subject: Re: Rules for subtracting integers

Hi, Julie.

I don't tend to use "rules"; it works better to base what I do on an 
understanding of how things work. Let's look at how things work, and 
then you can see if you can express that in a simple set of rules.

If we put two numbers on the number line, then when we subtract A 
from B, we get the "directed distance" from A to B:

                         B-A
                 +----------------->
    <------------o-----------------o------------------------->
                 A                 B

That is, B-A is the number you have to add to A in order to get B:

    A + (B-A) = B

If B > (is greater than) A, as shown, then B-A is a positive number 
(the arrow goes to the right). But if B < (is less than) A, then the 
difference will be negative:

                         B-A
                 <-----------------+
    <------------o-----------------o------------------------->
                 B                 A

In fact, B-A is exactly the opposite of A-B:

    B-A = -(A-B)

So you can subtract a larger number from a smaller one by subtracting 
the other way around and sticking a negative sign on the answer. 
That's a rule you can use.

Another fact you may want to use is that if you have a number in 
between, the difference is the sum of the two intermediate 
differences:

                         B-A           C-B
                 +----------------->--------->
    <------------o-----------------o---------o--------------->
                 A                 B         C
                 +--------------------------->
                              C-A

    C-A = (B-A) + (C-B)

Notice that I deliberately didn't indicate on the number line whether 
A and B are positive or negative; I didn't show where zero is. That's 
because this definition of subtraction doesn't care about that. But 
when you actually try to subtract two numbers, it will matter. And 0 
may play the role of A, B, or C in this picture. For example, if B is 
zero, since 0-A is -A, and C-0 is just C, this says that

    C-A = -A + C

Since A is a negative number, -A is positive, and this gives a way to 
subtract a negative number from a positive by just adding two positive 
numbers, something you already know how to do.

There are three places where I could have put the zero, and each 
corresponds to a different case you may want to consider when you 
write your rules. Try writing a rule to find B-A in each of these 
cases:

1. A and B are both positive:

                          A            B-A
                 +----------------->--------->
    <------------o-----------------o---------o--------------->
                 0                 A         B
                 +--------------------------->
                               B

2. A is negative and B is positive:

                          -A            C
                 +----------------->--------->
    <------------o-----------------o---------o--------------->
                 A                 0         B
                 +--------------------------->
                              B-A

3. A and B are both negative:

                         B-A           -B
                 +----------------->--------->
    <------------o-----------------o---------o--------------->
                 A                 B         0
                 +--------------------------->
                              -A

In each case, you will want to consider which number is larger (and in 
case 2, that means which of the two is positive); I've only shown B 
larger. This will give you six cases; you may find a way to simplify 
your rules by combining related cases. Have fun finding a way to say 
all this neatly! Really, it can all be done with those two little 
"rules" I gave above:

    B-A = -(A-B)

and

    B - A = B + -A

Just put these together a little differently in each case.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/   
    
Associated Topics:
Elementary Subtraction

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