Multiplying Positive and Negative Numbers

Date: 10/27/2003 at 18:15:17
From: kirsten
Subject: math

In a problem like 

 -3 x 7 x 4 = ?

what do I do with the negative numbers?

Date: 10/27/2003 at 22:26:37
From: Doctor Ian
Subject: Re: math

Hi Kirsten,

Here's the basic idea.  A negative number is the same as -1 times the
corresponding positive number.  For example, -3 is just -1*3, -18 is
just -1*18, and so on.  (I'm using '*' to represent multiplication.)

How does this help?  Well, suppose we have a bunch of positive and
negative numbers being multiplied, like 

  3 * -4 * 5 * -2 * -7 = ?

We can start by replacing the negatives with -1 times the
corresponding positives:

  3 * -1 * 4 * 5 * -1 * 2 * -1 * 7 = ?

How does this help us?  Well, since multiplication is commutative,
order doesn't matter; so we can move all the -1's out to one side:

  -1 * -1 * -1 * 3 * 4 * 5 * 2 * 7 = ?

The next thing to know is that -1 * -1 = 1.  So whenever we have a
pair of -1's multiplied together, we can replace them with a 1... and
since multiplying by 1 doesn't change anything, we can just eliminate
the 1:

  (-1 * -1) * -1 * 3 * 4 * 5 * 2 * 7 = ?

        (1) * -1 * 3 * 4 * 5 * 2 * 7 = ?

              -1 * 3 * 4 * 5 * 2 * 7 = ?

Now, we just have to multiply the other numbers together, and since we
still have a -1 hanging around, the sign of the whole thing is negative. 

Now, there is a shortcut we can use, but it's only worth doing if we
realize _why_ it works.  Instead of dragging the -1's out front to
kill each other off, we can just kill of pairs of signs in place.  For
example, given 

  3 * -4 * 5 * -2 * -7 = ?

we could kill off a pair of signs:

  3 * -4 * 5 * -2 * -7 = ?
      ^        ^
      |________|_________________ these signs cancel


  3 *  4 * 5 *  2 * -7 = ?

Now that there's only one sign left, we can bring it out front:

  -(3 * 4 * 5 * 2 * 7) = ?

Does this help?

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