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### Multiplying Negative by Negative

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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
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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: 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

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/
```

```
Date: 09/11/2011 at 05:53:45
From: Mrs. Nehila
Subject: Negative x negative

Hi Dr. Math!

I am a new teacher who will be teaching 7th and 8th graders multiplication
with integers. I'm using your site to see what people commonly struggle
with, and to generally verify my lessons, and so found Jim's question,
above.

Most of my kids seem to have it drilled into their heads that when two
negative signs are together, they make a positive. So would it be correct
to think of Jim's example this way?

-(-3) + -(-3) = 6

In other words, by representing the -2 as an extra negative sign (which
would be the opposite of +), you are now accounting for the (-2).

I am going to try it out and hope it's as clear to my students as it seems
to me!

```

```
Date: 09/11/2011 at 11:12:15
From: Doctor Ian
Subject: Re: Negative x negative

Hi Mrs. Nehila,

I don't see how that really addresses the problem, since it's basically
circular. That is, saying ...

-(-a) = a      BECAUSE      -1*-a = a.

... doesn't really explain anything.

Personally, I think of this topic as an opportunity to teach a much deeper
'intuition' about mathematics as a whole, rather than focusing on this one
particular result.

The intuition I have in mind is that the nature of math is this: We choose
some definitions, and then we trace out the consequences of those
definitions, to see where they take us. And the reason we do that -- what
makes it fun, and interesting (at least for mathematicians, if not always
for students in math classes) -- is that they sometimes take us to
surprising conclusions!

This is one of those times. Having defined a negative number as the
additive inverse of a positive number ...

a + -a = 0

... we then see where that takes us. And one of the places it takes us is
here:

http://mathforum.org/library/drmath/view/55717.html

That is, what is happening here is that our CHOICE of a definition for
negative numbers FORCES a particular conclusion, which is that the product
of two negatives is positive.

And this is how math works. If you just think of it a bunch of rules that
have been known forever, and that you have to memorize, you miss all the
fun.

To put this another way, the deepest intuition you can have about why the
product of negatives is positive is that it's not something we DECIDED up
front, but rather something that we are REQUIRED to accept because of the
definitions we started with. If your students can really get a handle on
that, it will make everything about math easier to understand, for as long
as they continue to study math.

I like to explain this kind of thing with an analogy to other games, like
chess. It turns out that there are certain endgames in chess, where for
example, if one guy has just a king and a bishop, and the other guy has
just a king, the latter (apparently weaker!) player can play out
a stalemate.

(I don't know whether that's true; I don't really play chess. But what's
important is that situations LIKE this exist. For now, we can just pretend
that it's true.)

Now, we could say that this is a 'rule of chess.' But it's not really like
what we normally think of as 'the rules of chess' -- what pieces there
are, how they're arranged at the start of a game, how they can move, and
how you take turns. Those are STARTING rules. But the impossibility of
forcing checkmate with a king and a bishop against just a king is a
DERIVED rule.

No one set out with the intention of making it impossible to force
checkmate if you're down to a king and a bishop. It's just something that
follows from the initial definitions -- and the first time anyone figured
it out, and proved it to be true, I'm sure it was quite a surprise.

In the same way, the rule that a negative number is an additive inverse is
a STARTING rule. And the rule that the product of two negatives is
positive is a DERIVED rule. And the first time anyone figured it out, and
proved it to be true, I'm sure it was quite a surprise. :^D

But if we blur the distinction between those kinds of rules, then students
lose 'the element of surprise,' so to speak. Do you see what I mean?

Having said all that, one major science application where this rule shows
up is forces. For example, if I have two positive charges (these can be
electrical or magnetic), or two negative charges, those forces repel;
whereas a positive charge and a negative one attract.

If I have particles with charges of 1 and -1,

k * 1 * 1
F = ---------   = positive force (repulsion)
r^2

k * -1 * 1
F = ----------  = negative force (attraction)
r^2

k * 1 * -1
F = ----------  = negative force (attraction)
r^2

k * -1 * -1
F = ----------- = positive force (repulsion)
r^2

To make this more intutive, you can let your students play with some bar
magnets. They'll quickly notice, without your even telling them, that like
poles repel, while unlike poles attract. That manipulative experience puts
some real world context to the rule that the product of two negatives
behaves the same way as the product of two positives.

Would this help?

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

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Associated Topics:
Middle School Negative Numbers

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