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Subtracting Negative Numbers


Date: 05/11/98 at 01:22:06
From: William
Subject: Subtracting Negative And Positive Numbers, like 3-(-4)=?

How do you subtract a negative number from a positive number or a 
positive number from a negative number? I got stuck when I saw it. I 
can do this kind of problem:

   3 - (-4) = 3 + 4 = 7

That one was easy. The opposite makes me stuck.

Please help.


Date: 05/18/98 at 11:51:02
From: Doctor Jeremiah
Subject: Re: Subtracting Negative And Positive Numbers, like 3-(-4)=?

Hi William:

The easy way to remember how to do this is with a number line. You 
don't have to have a real one, you just have to imagine one in your 
mind. A number line looks like this:

 +---+---+---+---+---+---+---+---|---+---+---+---+---+---+---+---+---+
-8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7   8   9

When you need to answer a problem, go through these steps.

Step 1:

You already know how to change -(-4) into + 4. That is step one.

Step 2:

Now you have something that looks like:

   3 + 4, 3 - 4, -3 + 4 or -3 - 4

Here is where you use the number line. Take the first number and 
figure out where it goes on the number line.

Step 3:

Then move from that spot however far the second number says to go. The 
direction you move is left if the second number is being subtracted 
and right if the second number is being added.

Example 1:  -3 - (-4)

 Step 1: Change the -(-4) into a + 4:

        -3 - (-4) = -3 + 4

 Step 2: Start at -3

                     O
 +---+---+---+---+---+---+---+---+---|---+---+---+---+---+---+---+---+
-8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7   8   9

 Step 3: We are adding four because the expression is -3 + 4 so we 
         want to move to the right 4 numbers:

                     O--- --- --- -->O
 +---+---+---+---+---+---+---+---+---|---+---+---+---+---+---+---+---+
-8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7   8   9

So the answer is +1.


Example 2:  -3 - 4

 Step 1: There is no double negative so we don't do anything.

 Step 2: Start at -3

                     O
 +---+---+---+---+---+---+---+---+---|---+---+---+---+---+---+---+---+
-8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7   8   9

 Step 3: We are subtracting four because the expression is -3 - 4 so 
         we want to move to the left 4 numbers:

     O<-- --- --- ---O
 +---+---+---+---+---+---+---+---+---|---+---+---+---+---+---+---+---+
-8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7   8   9

So the answer is -7.

See how that works? I usually just imagine a number line in my head. 
It works great for small numbers. But what about large numbers?

When you do 34 - 86, you can't draw a number line big enough. You have 
to think: what if I start at 34 and subtract 86? I need to move left. 
If I move left 86 I will cross zero because 86 is bigger than 34. How 
far past zero will I go? Well, the first 34 numbers out of 86 are on 
the right side of the zero and the rest (86 - 34) are on the left side 
of the zero. So I am on the left side of zero by 86 - 34. Being on the 
left side in the end means that the answer is negative. So the answer 
to 34 - 86 is 52 on the left side of zero or -52.

When you do -34 - (-86), again, you can't draw a number line big 
enough, so change -(-86) to + 86. Then you have to think: what if I 
start at -34 and add 86? I start on the left side of the zero and I 
need to move right. If I move right 86 I will cross zero because 86 is 
bigger than 34. How far past zero will I go? Well, the first 34 
numbers out of 86 are on the left side of the zero and the rest 
(86 - 34) are on the right size of the zero, so I am on the right side 
of zero by 86 - 34. Being on the right side in the end means that the 
answer is positive. So the answer to -34 - (-86) is 52 on the right 
side of zero or 52.

Does that help explain it? If you need more help please write me back 
and I will give you more details.

-Doctor Jeremiah,  The Math Forum
Check out our web site! http://mathforum.org/dr.math/   
    
Associated Topics:
Elementary Subtraction
Middle School Negative Numbers

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