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  Negative x Negative = Positive  

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Minus times minus results in a plus,
The reason for this, we needn't discuss.
      - Ogden Nash

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Why does a negative times a negative equal a positive? Does this ever happen in the real world?

Here are a few ways of thinking about how a negative times a negative could equal a positive in the real world. For other interesting explanations, see a discussion from amte, the mailing list of the American Association of Mathematics Teachers.

 

  Paying bills

• Let's say you get five bills in the mail for seven dollars each. You'd have 5 x -7 dollars, or -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'd have gotten negative five bills, so you'd have -5 x -7 = 35 more dollars than you started with.

• Imagine that you buy five gift certificates worth $5 each and pay for them using your credit card. 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 worth of gift certificates without having paid anything.

  Taking away a debt is analogous to negating a negative. Taking away five debts of $5 (-5*-5) equals a gain of $25.

 

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  Number Line

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.

 

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  On the Road

Suppose you're standing on a road, and you measure mileage to the east as positive, and to the west as negative. So you are at zero, and a town one mile east is at +1 mile, while a town two miles to the west is at -2 miles.

A car travelling east will have a positive velocity, and a car travelling west will have a negative one. So a car going east at 60 mph goes at +60 mph, and a car going west at the same speed goes at -60 mph.

This makes sense, since if they go for an hour (+1 hour), the east-going car will be at (+1)(+60) = 60 miles, and the car going west will be at (+1)(-60) = -60 miles (= 60 miles west).

Now suppose a car passes you going east at 60 mph. Where was it one hour ago? Or at -1 hour? Just multiply: (-1)(60) = -60 = 60 miles west.

How about a car going west at 60 mph? Where was it an hour ago? Its velocity is -60, the time is -1, so it was at (-1)(-60), and the answer should be 60 miles east, or +60.

So (-1)(-60) = +60.

 

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  Patterns

Here's a plausibility argument drawn from multiplication patterns:

          3 x -3 = -9
          2 x -3 = -6
          1 x -3 = -3
          0 x -3 =  0
         -1 x -3 =  3

 

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  Two Proofs

1. When we say -1 we mean the additive inverse of 1 so we have:

     1 + -1 = 0       (since they are inverses)
   
     1 + (1)*-1 = 0   (1 multiplied by anything is that thing)
   
     1 = -(1)*-1      (subtract (1)*-1 from both sides)
   
     1 = -1*-1        (the parentheses don't mean anything)
   
   

   So a negative times a negative means a positive.

    

   - Doctor Steven, The Math Forum

2. Proof: Let a and b be any two real numbers. Consider the number x defined by

    

   x = ab + (-a)(b) + (-a)(-b).

   We can write

   x = ab + (-a)[ (b) + (-b) ]          (factor out -a)
     = ab + (-a)(0)
     = ab + 0
     = ab.
   

   Also,

   x = [ a + (-a) ]b + (-a)(-b)         (factor out b)
     = 0 * b + (-a)(-b)
     = 0 + (-a)(-b)
     = (-a)(-b).
   

   So we have

    

   x = ab

   and

    

   x = (-a)(-b)

   Hence, by the transitivity of equality, we have

    

   ab = (-a)(-b).

    

   - Doctor Matt, The Math Forum

 

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  From the Dr. Math archives:

 

Integer Operations

A number line explanation from Suzanne Alejandre.

Middle School Negative Numbers

Answers to many questions about integer operations.

Negative Number Rules

When do two negative numbers equal a positive?

A Negative Times a Negative

Why do you take out the negative sign when you multiply two negative numbers?

Why Does a Negative Times a Negative Equal a Positive?

A chain of reasoning or mathematical argument that shows why the rule has to be.

 

  On the Web:

 

Re: -1 x -1 ? by Dave Seaman, from sci.math

Why is a Negative times a Negative a Positive?

A compilation of a long running discussion on the NCTM-L mailing list on how best to teach integer multiplication. Rex Boggs says, "If you can't find a method here that you like, I think you may never find one."

Mnemonics - Algebraical Mnemonic Poem by Jean Hervé-Bazin (1911-1996). Excerpt:

 

Moins par moins donne plus:
Les ennemis de nos ennemis sont nos amis.

Negative times negative is positive:
The enemies of our enemies are our friends.

  - French and English texts from Julio González Cabillón

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