Multiplying Negatives

Date: 13 Jan 1995 19:15:37 -0500
From: Nathan Burg
Subject: (none)

Dear Dr.Math

        In pondering a question about math. I've gone back to my elementary 
school lessons of multiplication and negative numbers.  When we 
multiply 2 * 3 it could also be written as 2 + 2 + 2 or as 3 + 3 and 
they all equal 6.  When we multiply -2 * 3 we could also write it as 
-2 + -2 + -2 and they both equal -6 but how would we re-write -2 * -3?

                                             Nathan Burg
                                             nburg@walrus.mvhs.edu

Date: 14 Jan 1995 10:36:06 -0500
From: Dr. Ken
Subject: Re: Nathan Burg's addition question

Hello there!

As I've said to a few of your schoolmates, we think it's great that you're
thinking about the foundations of mathematics.  I'll see what I can do with
this one.

You're right that this is how multiplication is defined.  As far as -2 * 3
goes, sure, you could write it as -2 + -2 + -2, but you could also write it
as (-)3 + (-)3.  It's just like before, when there were three -2's, but now
there are negative two 3's.  

So how could we write -2 * -3?  Well, we can either think of it as negative
two -3's, in which case we'd get (-)-3 + (-)-3, or we can think of it as
negative three -2's, in which case we get (-)-2 + (-)-2 + (-)-2.

It's similar to positive number multiplication.  In that last figure, if we
count up the negative twos, we'll find that there are negative three of them.

I hope this helps your intuition a little bit.

-Ken "Dr." Math