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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

Why is a negative times a negative a positive?

People have suggested many ways of picturing what is going on when a negative number is multiplied by a negative number. It's not easy to do, however, and there doesn't seem to be a visualization that works for everyone.



Debt

Debt is a good example of a negative number. One common form of debt is a mortgage in which you owe the bank money because the bank paid for your house. It is also common for an employer to deduct a mortgage payment from an employee's paycheck to help the employee keep on schedule with the payments.

Suppose $700 is being deducted each month to pay the mortgage. After six months, how much money has been taken out of the pay for the mortgage? We can figure out the answer by doing multiplication.

6 * -$700 = -$4,200

This is an illustration of a positive times a negative resulting in a negative.

Now suppose that, as a bonus, the employer decides to pay the mortgage for one year. The employer removes the mortgage deduction from the monthly paychecks. How much money is gained by the employee in our example? We can represent "removes" by a negative number and figure out the answer by multiplying.

-12 * -$700 = $8,400

This is an illustration of a negative times a negative resulting in a positive.

If one thinks of multiplication as grouping, then we have made a positive group by taking away a negative number twelve times.

This example may not work for you, and you might want to read others by following the related links below.



Visualizing isn't the same as understanding. Let's see how a mathematician might understand what's going on when a negative number is multiplied by a negative number.

A Mathematical Explanation

If we can agree that a negative number is just a positive number multiplied by -1, then we can always write the product of two negative numbers this way:

   (-a)(-b) = (-1)(a)(-1)(b) = (-1)(-1)ab 

For example,

    -2 * -3 = (-1)(2)(-1)(3)
 
            = (-1)(-1)(2)(3)

            = (-1)(-1) * 6

So the real question is,

   (-1)(-1) = ?

and the answer is that the following convention has been adopted:

   (-1)(-1) = +1

This convention has been adopted for the simple reason that any other convention would cause something to break.

For example, if we adopted the convention that (-1)(-1) = -1, the distributive property of multiplication wouldn't work for negative numbers:

   (-1)(1 + -1) = (-1)(1) + (-1)(-1)
	
        (-1)(0) = -1 + -1

              0 = -2

As Sherlock Holmes observed, "When you have excluded the impossible, whatever remains, however improbable, must be the truth."

Since everything except +1 can be excluded as impossible, it follows that, however improbable it seems, (-1)(-1) = +1.



Paying bills

  1. 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.

  2. 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.



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.


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.



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
    

A 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).



For other interesting explanations, see a discussion from amte, the mailing list of the American Association of Mathematics Teachers.

  From the Dr. Math archives:

  On the Web:

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