Signs: Rules, Number Line

Date: 07/13/2001 at 11:20:15
From: Anita Hong
Subject: Algebra, integers

Hello, 

I am going to be in grade 7.

(-) - (-) = ?  
The integer will be +, right?

Then...

What is (-) - (+) = ? or (+) + (-) = ?

And I still don't get the integers addition questions; for example: 

    1        1
-  --- - (- ---) = ?
    4        2

Thank you.

Date: 07/13/2001 at 12:44:14
From: Doctor Peterson
Subject: Re: Algebra, integers

Hi, Anita.

It's easy to MULTIPLY signed numbers; you just look at the signs, and 
if they are the same (+ times + or - times -), the product is 
positive, while if they are different it is negative.

But addition and subtraction don't work that way. You have to know the 
actual values in order to know the sign. In particular, the difference 
between two negative numbers can be either positive or negative, or 
even zero if they are the same.

Let's start with addition. If you add two positive numbers, the sum 
will be positive: the first number puts you at some point to the right 
of zero on the number line, and adding a positive number moves you 
farther to the right.

      -5  -4  -3  -2  -1   0   1   2   3   4   5
    <--+---+---+---+---+---+---+---+---+---+---+-->
                           |------>|---------->|
                               2   +     3     =5

Since you are always going to the right, the sum has to be positive.

If you add two negative numbers, you are always moving to the left, 
since adding a negative number takes you to the left:

      -5  -4  -3  -2  -1   0   1   2   3   4   5
    <--+---+---+---+---+---+---+---+---+---+---+-->
       |<----------|<------|
             -3    +   -2         -2 + -3 = -5

But if the signs are different, you are going both left and right, and 
the sign of the answer depends on which is "bigger." It's sort of a 
tug-of-war; the stronger number wins. Let's add 2 + -3:

      -5  -4  -3  -2  -1   0   1   2   3   4   5
    <--+---+---+---+---+---+---+---+---+---+---+-->
                           |------>|    2
                       |<----------| + -3
                                     ----
                                       -1

Here I first went 2 units to the right, and then I went 3 units to the 
left. It took 2 units to get me to 0, and that left one more unit to 
the left, taking me to -1. Because 3 is bigger than 2, -3 "won" over 
+2, and made the answer negative. The size of the sum is the 
difference between the sizes of the two numbers.

The opposite thing happens if the first number is greater. Let's do 
-3 + 2:

      -5  -4  -3  -2  -1   0   1   2   3   4   5
    <--+---+---+---+---+---+---+---+---+---+---+-->
               |<----------|   -3
               |------>|     +  2
                             ----
                               -1

Here, I went 3 units to the left, and adding 2 didn't take me far 
enough to get back to zero, so the answer is negative. Its size again 
is the difference between the sizes of the numbers.

Now, notice that what we've been doing is the same as subtraction. 
Moving some amount to the left can be seen either as adding a negative 
number, or subtracting a positive number. And moving to the right can 
be thought of either as adding a positive number, or (and this is the 
one that might seem odd) as subtracting a negative number. That's 
because subtraction asks the question, what do we have to add to the 
second number to get the first? For example, when we say

    3 - 2 = 1

that's because

    3 = 2 + 1

That is, 1 is the amount we have to add to 2 to get 3.

So what is 2 - -3? Well,

    2 - -3 = 5

because

    2 = -3 + 5

So as I said, subtracting a negative moves us to the right. We can 
find the answer by changing the subtraction to an addition:

    2 - -3 = 2 + 3 = 5

On the other hand,

    -2 - -3 = -2 + 3 = 3 + -2 = 3 - 2 = 1

since after we change the addition to subtraction, the 3 takes us 
farther to the right than the -2 took us to the left. Notice that I 
did this without having to draw a number line; I just changed the 
problem by changing the order so the 3 comes before the -2, and then 
turning it back into a subtraction.

You should now be able to do any of your problems, even the one with 
fractions. But you should be aware that those are not integers. 
Because you have been introduced to signs as part of learning about 
integers, you may think that "integer" means "signed number"; but 
actually it means "whole numbers with signs." A signed fraction is a 
rational number.

If you're still not confident, please write back and show me your 
answers to several problems, so I can see whether you need any more 
help.

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

Date: 07/14/2001 at 09:25:07
From: Anita Hong
Subject: Re: Algebra, integers

Thank you very much for your help. But what if the number is more than 
the 10's column? You can't draw an arrow and know the answer, so then 
how are you going to know the answer without drawing an arrow? Is 
there another way to solve?

Date: 07/14/2001 at 22:42:57
From: Doctor Peterson
Subject: Re: Algebra, integers

Hi again, Anita.

Good question!

You don't have to actually draw the number line and the arrows; the 
purpose of explaining it that way is to help you understand how things 
work. What you can do instead is to imagine what the number line would 
look like, so you can see how to handle the numbers.

For example, let's add -189 + 312. We can just imagine something like 
this:

    <-------+------------+-------------------+------->
          -189           0                  312

All I really care about is that -189 is less than zero, that 312 is 
more than zero, and that 312 is more than 189. So when I start at -189 
and add 312, I am adding more than I need to get to zero. I will end 
up on the positive side, and the answer will be what is left of 312 
after using up 189, that is, 312 - 189.

Once you have a feel for how this works, there are more "formal" 
methods you can use, which require the sort of thinking you learn in 
algebra, just manipulating symbols following rules. Here is a summary 
of the rules you can apply:

1. The negative of a negative is the positive:

    -(-a) = a

2. Adding the negative is the same as subtracting the positive:

    a + -b = a - b

3. You can change the order of the terms in an addition without 
   changing the value:

    a + b = b + a

4. If you change the order of a subtraction, you negate the value of 
   the answer:

    a - b = -(b - a)

5. You can "pull the negative signs out of a sum or difference":

    -a + -b = -(a + b)
    -a - -b = -(a - b)

Using these rules, you can rearrange any addition or subtraction to 
turn it into a combination of negations and operations on positive 
numbers. Let's take my example:

    -189 + 312 = 312 + -189 = 312 - 189

Here I first saw that I was adding a negative, so I changed the order 
of the addition to put that second; then I could change addition of a 
negative to a subtraction. Now it's a problem you've been able to do 
for years!

What if it were a subtraction? Then

    -189 - 312 = -(189 + 312)

and we just have to do the addition.

This is probably what really goes on in my mind when I do this kind of 
problem. But I'm not really thinking of a list of rules (which is why 
I didn't want to put this approach first). I just have a sense of what 
works, developed from experience. I see subtractions as additions; I 
know that if the signs are opposite I will want to subtract something. 
I believe that visualizing these problems on the number line is a good 
way to develop this sense; just take it one step at a time, from 
actually drawing the line, to imagining how it would look if you did, 
to just thinking about which number is greater, and finally using the 
rules step by step.

I hope that helps to clarify a little.

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