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/