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.
DebtDebt 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
-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) = (-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.
On the Road
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
This makes sense, since if they go for an hour
Now suppose a car passes you going east at
How about a car going west at
3 x -3 = -9 2 x -3 = -6 1 x -3 = -3 0 x -3 = 0 -1 x -3 = 3
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
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:
Les ennemis de nos ennemis sont nos amis.
Negative times negative is positive:
- French and English texts from Julio González Cabillón
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