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### Working with Integers

```
Date: 01/12/99 at 22:45:14
From: Britni E.
Subject: Integers (dividing and multiplication)

Hello,

I'm having trouble figuring out how to divide, multiply, add, and
subtract integers. Could you explain to me how to do integers? Maybe
include some examples? It would really help!
```

```
Date: 01/13/99 at 12:49:28
From: Doctor Peterson
Subject: Re: Integers (dividing and multiplication)

Hi, Britni. Integers are simply numbers with a sign attached. The sign
says which way you go on the number line: plus means to the right;
minus means to the left. I'll call the number part the "value" of the
integer. It would be better to call it the "absolute value," but I want
to save some typing. So for example the integer -3 consists of the sign
"-" and the value "3". If there's no sign, it means +. We only write +
when we want to emphasize that the number is a positive integer.

Let's start with multiplication and division, because that's the easy
part (surprise!). They both follow the same rule: you multiply the
values together, and choose a sign for the answer using this table:

| + | - |
--+---+---+
+ | + | - |
--+---+---+
- | - | + |
--+---+---+

That is, when you multiply or divide + by +, or - by - (the same sign),
the answer is +. If the signs are + and -, or - and + (different
signs), the answer is -. You can think of - as flipping the whole
number line over, so if you flip it twice in a row everything is back
where it started: - - = +.

Here are some examples:

-2 * 3 = -+ 2*3 = -6
4 * -9 = +- 4*9 = -36
-3 * -5 = -- 3*5 = +15

-6 / 2 = -+ 6/2 = -3
4 / -2 = +- 4/2 = -2
-9 / -3 = -- 9/3 = +3

For adding, I prefer not to use a rule, but just to think about what it
means. Think of a positive number as an arrow whose length is the value
of the number, pointing to the right, and a negative number as an arrow
pointing to the left. Adding them means starting at zero, following one
arrow, then starting the next arrow there and following it to its end.
For example, here's 2 + 3 = 5:

---+---+---+---+---+---+---+---+---+---+---+---
-5  -4  -3  -2  -1   0   1   2   3   4   5

+------->   3
2   +---------->
+------------------>
5

We went 2 to the right and then 3 more to the right, which is the same
as going 5 to the right.

If the arrows go in the same direction, it's easy: just add the values
and keep the sign. To add -2 + -3 you just add a negative sign and get
-5:

---+---+---+---+---+---+---+---+---+---+---+---
-5  -4  -3  -2  -1   0   1   2   3   4   5

-3   <-------+
<----------+   -2
<------------------+
-5

If the signs are opposite, like 2 + -3, one will be undoing part of
what the other does, so the result will be whatever is left over of the
larger one:

---+---+---+---+---+---+---+---+---+---+---+---
-5  -4  -3  -2  -1   0   1   2   3   4   5

+2
+------->
<----------+
<--+  -3
-1

Here I first went 2 to the right, then 3 to the left. 2 of the 3 were
used up in just getting back to the start of the 2. The remaining 1
continues in the negative direction, giving us -1 for the answer. To
do this, you just notice that the number with the larger value is
negative (-3). You subtract the smaller value, 2, from the larger value,
3, leaving 1, and use the negative sign. You'd do exactly the same thing
if the numbers were in the other order,-3 + 2, and the answer would be
the same.

You probably would have no trouble figuring out that if I told you I
took two steps forward and three steps back, I ended up going one step
backward. Just use the same sort of thinking to add integers, and
you'll be on your way.

Finally, you can change subtraction to addition by just combining the
minus sign with the negative sign. For example, to do -3 - -2, I would
combine the - - to make a +: -3 + 2. Then you can add that just as we
did above.

If you still need more help, try sending us one or two of the problems
you got wrong, showing what you did, so we can see what part gives you
trouble.

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

```
Associated Topics:
Elementary Division
Elementary Multiplication
Elementary Number Sense/About Numbers
Elementary Subtraction
Middle School Division
Middle School Negative Numbers
Middle School Number Sense/About Numbers

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