Multiplying Negative by Negative

Date: Sun, 6 Nov 1994 13:09:25 -0500
From: Sally Spencer
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

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.

     So, one way to think about this is to take 6*(-6) (that is 6 times -6),
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: Dr. Ken
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.

Unfortunately, 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: Anonymous
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

From: Tom Rocklin
Date: Tue, 7 May 1996 12:55:48 -0500
Subject: Multiplying negatives

My son found the debt metaphor pretty helpful.  I have since been told
about one I like even better.  Imagine a number line on which you walk.
Multiplying x*y is taking x steps, each of size y.  Negative steps require
you to face the negative end of the line before you start walking and
negative step sizes are backward (i.e., heel first) steps.  So, -x*-y means
to stand on zero, face in the negative direction, and then take x backward
steps, each of size y.

Date: 02/02/98 at 22:34:37
From: Jim Caprio
Subject: Negative times negative

My wife is going to love your site, as she is a high school math teacher (LaSalle Sr. High, Niagara Falls, NY), teaching algebra & geometry.

Here's one that's taught in 7th grade, but is still impossible to
explain effectively: if you know of an intuitive explanation for why the product of two negative numbers is positive, please share it with me (and I'll pass it on to my wife, because her SENIOR HIGH REGENTS
STUDENTS are constantly getting confused about it!).

I think their confusion is on what exactly does it "mean" to multiply
something a negative number of times. Perhaps expanding the product out in the traditional manner might illustrate the "problem":

2 x 3    =  2 +  2 + 2 = 6     Multiply 2 three times - no problem
2 x (-3) = -3 + -3 = -6        Multiply -3 two times - no problem
(-2) x (-3) = ?                Confusion! How do you expand this?

Thanks for the insight,

Jim Caprio

Date: 02/03/98 at 01:26:27
From: Doctor Pete
Subject: Re: Negative times negative

Hi,

Imagine the following situation:

You're standing at a street corner. At exactly noon, a car passes by you, going east, at a constant rate of 20 miles/hour. After one hour has elapsed, that is, at 1:00 p.m., it would be 20 miles east of where you're standing. And one hour ago, at 11:00 a.m., it was 20 miles west of where you're standing.

Now, what would it mean if the car was travelling east at -20 miles/hour? It'd be in reverse, going west, of course. So, if it were going at -20 miles/hour east, at 1:00 p.m. it would be 20 miles west of where you're standing. Finally, if it were going -20 miles/hour east, at 11:00 a.m., where would it be? 20 miles east, of course. What does this mean?

     (-20 miles/hour)(-1 hour) = 20 miles.

Hence here is a "concrete" explanation of why two negatives multiply to make a positive number.

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