Why do two negative numbers, multiplied or divided, equal a positive number?

Question

Something such as -4 x (-3) = 12 and 4 x 3 = 12, why do the negatives cancel out?

Answer 1

Because... when you divide it is seeing how many times it goes in.. ex: how many times does -3 go into -9.. -9 divided by negative 3 . It goes into it 3 times. not -3 times.

Answer 2

My math teacher always told me this trick for remembering what happens with two negative numbers: Whenever you multiply or divide two negative numbers, the two negative signs - and - combine to form a plus + In other words, the two negative signs cancel each other out and form a positive. Thus, when you divide or multiply a negative by a negative, you will get a positive answer.

Answer 3

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.

Answer 4

Formally, it's a consequence of the distributive property. Consider the following:

xy + x(-y) = x(y+(-y)) = x*0 = 0
(-x)(-y) + x(-y) = ((-x)+x)(-y) = 0*(-y) = 0
Thus by transitivity:
(-x)(-y) + x(-y) = xy + x(-y)
And subtracting x(-y) from both sides yields:
(-x)(-y) = xy

Informally, consider that you probably have no trouble believing that 4*(-3) = -12, because you know that multiplying by a negative number is just the opposite of multiplying by a positive number. So multiplying a positive number by a negative number gives you the opposite of a positive number, which is a negative number. Thus multiplying a negative number by a negative number must give you the opposite of a positive number times a negative number. And since a positive number times a negative number is negative, the product of two negatives must be the opposite of negative, which is positive.

Answer 5

It's just the way it works.

Visit Wikipedia or Google it to find out more, or if you're having trouble and your at School/College - Just ask the Tutor/teacher, if you get a detention, my bad!!!

Hope it helped you! Good luck!
:D

Answer 6

why does this matter?
as long as you know thats the rule, why do you care?
thats how i always look at math...