Manipulating Positive and Negative Numbers

Date: 8/9/96 at 20:37:45
From: terry holman
Subject: Manipulating Positive and Negative Numbers

I am having a little trouble understanding the rules for adding, 
subtracting, multiplying, and dividing positive and negative numbers. 
Could you please help me? 

Date: 8/30/96 at 17:44:17
From: Doctor James
Subject: Re: Manipulating Positive and Negative Numbers

Negative numbers are very interesting. There are a couple of tricks 
you can do to remember the rules a little better.

Addition and subtraction are easy when you remember this:

3 - 1 = 3 + -1

So adding a positive and negative number is just like subtracting two
positive numbers. When the negative number is first it gets a little
tricky, but you can figure it out:

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

If the negative number is bigger, the final answer will be negative.
For example,

2 + -5 = 2 - 5 = -3

If both numbers are negative, it's just like adding positive numbers,
except that the answer is negative.

-3 + -5 = -8   (compare to 3 + 5 = 8)

Subtracting a negative number is just like adding a positive number.

1 - -3 = 1 + 3

So what's -3 - -8? well,

-3 - -8 = -3 + 8          because subtracting a negative number is
                          just like adding a positive number

-3 + 8 = 8 + -3           because you can change the order of addition

8 + -3 = 8 - 3            because adding a negative number is just
                          subtracting a positive number

8 - 3 = 5                 Just regular subtraction!


Now, the first thing to remember for multiplication is that if you 
multiply a positive number by a negative number, you get a negative 
number. Knowing that, we can say that a number like -5 is really just 
-1 * 5, where * means times. Remember that multiplication is 
'commutative', which means that it doesn't matter what order you 
multiply things (as long as you are just multiplying!) So if we have 
3 * -4, we can say:

    3 * -4 = 3 * -1 * 4         because -4 = -1 * 4

3 * -1 * 4 = -1 * 3 * 4         because multiplication is commutative

-1 * 3 * 4 = -1 * 12            because 3 * 4 = 12

   -1 * 12 = -12                because -12 = -1 * 12

so we get:

3 * -4 = -12, the answer!

similarly, see if you can get the following:

  3 * 4 = 12.
 -3 * 4 = -12
-3 * -4 = 12                   remember that -1 * -1 = 1

Now, if you just replace 3 and 4 with whatever two numbers with which 
you are working, you can do any multiplication problem!

Division is similar. There are a couple things to remember, though:

1/(-1) = -1     You can see this if you multiply the top and bottom
                of the fraction by -1 (since you are multiplying both
                the top and the bottom by the same thing, its ok).
                This gives you -1/(-1 * -1) = -1/(1) = -1.

-1/(-1) = 1     Can you figure this one out? do the same thing as the 
                last one.

Division, like multiplication, is commutative. This means that you can
switch the order in division too. You have to be careful, tho! You 
must switch the division sign with the number following it. For 
example,

3 * 4/5 = 3/5 * 4, but
3 * 4/5 doesn't equal 3/4 * 5 !

Also, although division and multiplication 'commute' with each other, 
they don't with addition or subtraction, so be careful there too.

So, if you have 6/(-2), you can do the following:

6/(-2) = 6/(-1 * 2)                  because -2 = -1 * 2

6/(-1 * 2) = (-1 * 6)/(-1 * -1 * 2)  because you can multiply the
                                     top and bottom by the same thing.

(-1 * 6)/(-1 * -1 * 2) = (-1 * 6)/(1 * 2)   because -1 * -1 = 1

(-1 * 6)/(1 * 2) = (-1 * 6)/2   because 1 * 2 = 2

(-1 * 6)/2 = -1 * (6/2)         because division and multiplication 
                                are commutative

-1 * (6/2) = -1 * 3             because 6/2 = 3

-1 * 3 = -3                     because -3 = -1 * 3

Hope this helps. If it's still confusing, please write back.

-Dr. James,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/