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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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