Why is the Product of Two Negative Numbers Positive?

Updated on January 19, 2015

Introduction

A negative number times a negative number equals a positive number. That idea that really bothered me when I was learning Algebra in high school--especially because I understood everything else. Many people have had similar experiences with algebraic sign conventions. And for a typical Algebra student, that triggers the same puzzled expression as that of Maxwell Smart, in the photo. However the NNP (negative-negative-positive) principle was easy for me to remember, because it caused so much consternation. I filed it away under Revisit.

Although this hub is primarily geared toward high school math teachers, anyone with an active sense of curiosity can learn something. Before we hunker down on NNP, let's do a warm-up exercise on a simpler case: negative-positive-negative. Why is a negative number times a positive number negative? Here's an example to illustrate the concept.

To estimate a person's net worldly fortune at any given time, you put all of the assets--including the equity in one's home--in one column. And you put all of the debts--with negative signs in front of them--in a second column. Then add up each column.

Net Worth?

The algebraic sum of the two subtotals is called Net Worth. To be honest, I'm not completely comfortable with that expression, because a single quantity cannot do justice to the essence of a person. For example, Net Worth does not include intangible qualities, like moral character. However Net Worth is the most common English expression used to describe the concept, and we're stuck with it.

Net Worth can be positive. Or it can be negative, in the case of a person who has fallen on hard times, because he was 'downsized' during a recession.

Imagine that a person's Net Worth one year ago was -400 dollars. During the past year, he accumulated another 400 dollars of debt, with no new assets. In other words, his (negative) Net Worth has doubled. What is the new Net Worth? Obviously it's -800 dollars. We can express this idea with multiplication as follows:

-400*2 = -800

Thus by analogy, a negative number times a positive number is always a negative number.

This analogy should be taught very early in basic algebra. However it would be best to wait for the NNP case, until after the lesson on the Distributive Law.

Teaching the Distributive Law

One is in a position to truly understand NNP only after mastering the Distributive Law. If you haven't thought about algebra for awhile, you may want to review the standard usage of parentheses in equations.

Using algebraic symbols, the Distributive Law states:

a(b + c) = ab + ac (Equation 1)

About nomenclature: The term ab is understood to mean a times b.

Why is the distributive Law true? There are three ways to reinforce the concept. The Great White Father approach is the most common. The Distributive Law is true, because I say that it's true. And I know that, because a long line of wise men before me have said so.

A slightly better approach: Assign your students a Distributive Law problem set for homework. Have them arbitrarily assign numerical values for a, b, and c. Using these three values, calculate the left-hand-side of Equation 1, and then do the same for the right-hand-side. Do the two sides match up numerically? Then have each student create nine more examples to test the distributive Law. (The next homework assignment would be to apply the Distributive Law.)

Failing to find a counterexample is not the same thing as a proof. But it can increase one's confidence in the veracity of the Distributive Law, and one's comfort level with algebra.

The third and best approach involves visual thinking. Here's a link to a website that gives an intuitive aha approach to the Distributive Law.
http://www.cut-the-knot.org/Curriculum/Arithmetic/DistributiveLaw.shtml

Taming the NNP Monster

Now that we have the Distributive Law under our belts, there's a straightforward way to understand why the product of two negative numbers is positive.

Now we'll use the Distributive Law in the following equality chain:

25 = 5*5 = (10 - 5)*(10 - 5) = (10 + (-5))*(10 + (-5))

= 10*10 + 10*(-5) + (-5)*10 + (-5)*(-5) = 100 - 50 - 50 + (-5)*(-5)

= (-5)*(-5)

If you accept the premise that multiplication is distributive over addition, which the above link shows you how to visualize, then it necessarily follows that a negative number times a negative number will always be a positive number.

Professional mathematicians may balk at this simple argument. With some justification, they could say that it doesn't measure up to the standards of a rigorous proof. Nevertheless most people who are grounded in the other basic algebra skills can follow it.

The Purple Elephant

One more thing. The above equality chain illustrates an important problem-solving technique. Dr. James Householder, a mathematics professor at my alma mater, called it the Principle of the Purple Elephant (PPE). If you have a white elephant, and the color bothers you, then spray-paint it purple! Here's how to apply it.

If you're struggling to grasp an abstract mathematical problem, substitute friendly numbers for the variables. (To me, 5's and 10s are friendly numbers.) It'll be less intimidating, and more clear. After you've solved the friendly-number version of the problem, analyze your logic. Then apply the same logic to the abstract version of the problem.

The PPE is a special case of the Simplification Technique.

Creative problem-solvers frequently use this approach, but few will admit it. They want everyone else to think that they're ultra-logical Vulcans, like Mr. Spock, in the original Star Trek series.

Copyright 2011 and 2013 by Larry Fields

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• AUTHOR

Larry Fields

5 years ago from Northern California

Hi Gershon. Good job.

5 years ago

oh forgive me, made a mistake in my example. Now corrected, thanks.

Example

-2 x -5 = (-1)(2) x (-1)(5)

= (-1)(-1) x (2)(5)

-2 x -5 = (-1) ^2 x 10

-2 x -5 = (1) ^2 x 10 [since a^2 = (-a)^2 or (1)^2 = (-1)^2]

-2 x -5 = 1 x 10

-2 x -5 = 10

5 years ago

Hello Larry,

I have come again. I am just a lover of maths (maths is fun)!

Please take a look at this other approach for explaining why the product of two negative numbers is always POSITIVE.

Let x = a^2 ……..1

And y = (-a)^2 ……… 2

Now eqn 1 minus eqn 2 will give

x – y = a^2 – (-a)^2

= (a – -a) (a + -a) [difference between two squares]

= (a – -a) (0)

x – y = 0

x = y

i.e. a^2 = (-a)^2

a x a = (–a) x (–a)

Note that “a” is any real number.

Example

-2 x -5 = (-1)(2) x (-1)(9)

= (-1)(-1) x (2)(9)

-2 x -5 = (-1) ^2 x 18

-2 x -5 = (1) ^2 x 18 [since a^2 = (-a)^2 or (1)^2 = (-1)^2]

-2 x -5 = 1 x 18

-2 x -5 = 18

This is a proof of product of two negative numbers is positive.

Once again, I will be grateful for your feedback on this other approach.

Thanks.

• AUTHOR

Larry Fields

5 years ago from Northern California

Hi Gershon,

Sorry to be the bearer of bad tidings. This is still not a proof. The first step:

(-a)^2 – (-b)^2 = a^2 – b^2

is a leap of faith. It makes intuitive sense when we already know that the product of two negative numbers is positive, but not when we don't.

5 years ago

Thanks Larry. I perfectly agree with your point but just want to approach the problem from another perspective and will be grateful for your feedback. Actually, I am teaching my house help mathematics and she asked me why the product of two negative numbers is positive. Something I have not thought of before, this is how come I began to search for the answer on the internet and finally got to your website.

Now my professor told me that when we cannot readily find an answer to a problem, we have to make reasonable assumptions (LOL). Based on this, I want us to assume that [ (-a)^2 – (-b)^2 = a^2 – b^2 ]:

Then we can say,

(-a)^2 = a^2 – b^2 + (-b)^2

= a^2 + (-b)^2 – b^2 ; Difference between two squares again

= a^2 + (-b –b)(-b + b)

= a^2 + (-b –b)(0)

= a^2 + 0

(-a)^2 = a^2

that is (–a) x (–a) = a x a

similarly, we can show that

(-b)^2 = b^2

that is (–b) x (–b) = b x b

• AUTHOR

Larry Fields

5 years ago from Northern California

Hi Gershon,

Thanks for stopping by. Nice try.

If Equations 1 and 2 are both true, then the line beginning with "Therefore" is also true. But does the logic flow in the reverse direction? It is not obvious that Equations 1 and 2 necessarily follow from the line beginning with "Therefore."

5 years ago

To prove that the product of two negative numbers is positive can also be done by using the principle of the difference between two squares. That is,

(-a)^2 – (-b)^2 = (-a – -b)(-a + -b)

= (-a + b)(-1)(a + b)

= (a – b) (a + b)

Therefore (-a)^2-(-b)^2 = a^2 – b^2

Then it follows that:

1. (-a)^2 = a^2, that is (–a) x (–a) = a x a

2. (-b)^2 = b^2, that is (–b) x (–b) = b x b

Note That (a^2) is the square of "a"

• AUTHOR

Larry Fields

5 years ago from Northern California

Hi Glenn. Thanks for stopping by.

• Glenn Stok

5 years ago from Long Island, NY

This goes to show how much we take for granted. When I read this hub, memories returned about how I had difficulty accepting this concept in high school. But along the way I must've simply given into it without any further consideration. Something like groupthink that you discussed in another hub of yours. Nevertheless, your explanation makes it clear why the product of two negative numbers is positive. Now I can move on rather than just accepting what was taught.

• AUTHOR

Larry Fields

6 years ago from Northern California

Hi Teresa. I'm glad that you liked it. Thanks for your comment.

• Teresa Coppens

6 years ago from Ontario, Canada

Wow, Larry, this is the first time I've been able to understand the NNP concept. And truly, I have not found any teacher Able to explain it to me but in my supply teaching career I have had many students ask for an explanation. Thanks so much for giving me the tools to go out and enlighten! It's one of those, "so simple, why didn't I think of it myself" explanations! Voted up and useful!

• AUTHOR

Larry Fields

6 years ago from Northern California

Hi Tina,

Thanks for stopping by. I especially liked this part of your comment:

"This hub is very enlightening although I really have to admit I spent almost ten minutes unraveling the essence..."

It reveals an important truth about math education. Math is something that you learn mostly on your own. A skilled math teacher will organize the material into bite-size pieces, each of which you can comprehend on your own, given a reasonable amount of 'cave' time, to use your metaphor.

If there's an aha moment at the end, a reasonable amount struggling makes your mind a bit stronger, in the same way that practicing a challenging piece of music enhances overall brain function.

• Tina Siuagan

6 years ago from Rizal, Philippines

When I read my Math textbooks and notes, English terms, letters, and numbers suddenly turns to some "alibata" or archaic symbols of some sort. You know those things you find in caves, written by early cavemen, wherein you never have the single clue what they mean. But now, I had second thoughts thanks to that Purple Elephant. Maybe I could paint my white elephant hot pink whenever I feel like it. Hahaha!

This hub is very enlightening although I really have to admit I spent almost ten minutes unraveling the essence of your:

25 = 5*5 = (10 - 5)*(10 - 5) = (10 + (-5))*(10 + (-5))

= 10*10 + 10*(-5) + (-5)*10 + (-5)*(-5) = 100 - 50 - 50 + (-5)*(-5)

= (-5)*(-5) "

I am so not good with numbers. As much as how I tried in vain to love them, they never loved me back. I don't even know how I maintained my scholarship back in college with Math getting in the way. Lol.

• AUTHOR

Larry Fields

6 years ago from Northern California

Thanks for catching my error. It's now corrected.

• al

6 years ago

isn't

a(b + c) = ab + bc (Equation 1)

be

a(b + c) = ab + ac (Equation 1)

• AUTHOR

Larry Fields

6 years ago from Northern California

Hi, Taleb80. Thanks for the compliment.

• Taleb AlDris

6 years ago

Mr. Larry

Welcome to HP

Really you are a good writer.

Thank you for this useful Hub.

It is a simple direct proof.

I voted "Awesome"

• AUTHOR

Larry Fields

6 years ago from Northern California

Thanks Tina. I'm glad that you liked it.

• Christina Lornemark

6 years ago from Sweden

Thanks for this very educational hub and I appreciated the repetition. Somehow I liked negative numbers and the Distributed law when I studied math. I had a marvelous professor teaching math and she could make any math problem look beautiful! Beautiful was the word she used to describe math and after a few years with her I had to agree. You did a great job explaining this problem that can be a bit hard to accept. I enjoyed this hub very much.

Tina

• AUTHOR

Larry Fields

7 years ago from Northern California

Hi BizGenGirl. Thanks for stopping by.

• Bema Self

7 years ago from Seattle

I always thought that a negative number times a negative number equaled a larger negative number.... so I guess I don't know why it equals a positive number. Just to confuse people maybe? lol

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