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### Multiplying Negative by Negative

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Date: Sun, 6 Nov 1994 13:09:25 -0500
From: sspencer@k12.ucs.umass.edu (Sally Spencer (The CARE Center))
Subject: Multiplication of Positive and Negative Numbers

Dear Dr. Math,

I'm trying to make sense of these rules so that they'll be easier to
memorize:

Pos x Pos = Pos, makes sense.  I've been doing it since 3rd grade.  And I
can even think of a situation.  I get six birthday cards with \$5 in each.

Pos x Neg = Neg, I can think of a situation for this, too.  I get four bills
for \$20 each so I'd owe money.

But, Neg x Neg = Pos just doesn't make sense.  Does it ever happen in
real life?

My teacher said that you could say it would be the opposite of Pos x Neg
but that seems like cheating.  It's not realistic.

Thank you,
Sally
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Date: Sun, 6 Nov 1994 13:32:02 +0900
X-Sender: mpatter1@cc.swarthmore.edu

Hi Sally!

Thanks for writing to us.  This is a difficult question.  I wish that I had
a good explanation of it.  Someone else asked us this recently, so I am
going to give you the response that Dr. Demetri wrote.  The specific
example was -6*-6. I am afraid this might seem like "cheating", too.

find the result to this (which is exactly what you have above, ie -36) and
then consider what -[6(-6)] is. This is nothing other than the negative of
6(-6) or, if you prefer, its opposite (opposite numbers are two numbers
whose sum is 0- you may know this, but I said it just in case you don't
know). Clearly, the opposite of -36 is +36, because -36 +36 = 0. Therefore,
-6*(-6) = +36

I hope this helps.  Please feel free to write back if you have any questions.

-Margaret, Math Doctor on call

From: Ken Williams <ken@sccs.swarthmore.edu>
Date: Sun, 6 Nov 1994 13:32:21 -0500 (EST)

Sally!

Thanks for the question!  It's great that you want to actually make sense of
the situation instead of just doing it by rote.

I'll try to give you an example for the Neg x Neg case, based on your
receiving bills thing.

Let's say you got five bills in the mail for seven dollars each.  Then
you're right, you'd have 5 x -7 more dollars, i.e. -35 more dollars, i.e.
35 fewer dollars.

But what if you had _sent_out_ five bills instead of getting them?  Then, in
a sense, you've gotten negative five bills, so you have -5 x -7 = 35 more
dollars than you used to have.

Unfortnately, I can't think of another example right now to really drive the
point home.  But I'll keep thinking, and I'll try to get back to you soon!
Also, one of the other Math Doctors here might jump in and reply if they
think of anything clever.

Thanks!

-Ken

From: steve@mathforum.org
Date: Mon, 7 Nov 1994 14:40:57 -0500

Hi Sally, I think you've gotten some good answers and here's another
variation, again using your own example with bills.

Neg x Neg: Imagine that you buy five gift certificates worth \$5 each and
you pay for them using your credit card.  As you point out below, you now
owe money, so that's -\$25.  The bill comes from the credit card company,
but I TAKE IT AWAY from you and insist on paying it.  You now have \$25 of
gift certificates without having paid anything.  Taking away a debt is
analogous to negating a negative.  Take away five debts of \$5 (-5*-5)
equals a gain of \$25.

-- steve
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