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

Something such as -4 x (-3) = 12 and 4 x 3 = 12, why do the negatives cancel out?Why do two negative numbers, multiplied or divided, equal a positive number?
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.Why do two negative numbers, multiplied or divided, equal a positive number?
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.
why does this matter?


as long as you know thats the rule, why do you care?


thats how i always look at math...
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.
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

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