Why is the Product of Two Negative Numbers Positive?

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.

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.

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. If you're not entirely comfortable with parentheses in algebra, here's a link to a useful hub on the subject.
http://hubpages.com/hub/Math-Help-How-Do-Parentheses-Work-in-Algebra

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

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.

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 by Larry Fields