Unravelling the Mystery of Negative Multiplication

[jacobrhcp]

I wondered why it is that when you multiply a negative with a negative, you get a positive?

in example; why is it that -3 x -3 = 9?

when you do -3 x 3, the answer is intuitively -9, because you just go three times as negative. But when I multiply two minus signs, I don't have this intuition it should be positive, except for the fact that it's the only thing left.

 

[dextercioby]

Do you agree that $(-1)(-1)=-(-1)\cdot 1$ ? Then what is the opose of the opose of 1 ?

 

[jacobrhcp]

but then, how do you know that a minus sign means taking the opposite side 0? is it defined that way?

 

[dextercioby]

Yes, -1 is/can be defined as the single solution to the equation x+1=0.

 

[jacobrhcp]

thanks =)

 

[JonF]

but then, how do you know that a minus sign means taking the opposite side 0? is it defined that way?

Negative means “additive inverse”.

For any given a we define –a to be the number such that a + –a = 0

In addition by convention we normally drop the “+” symbol and call it subtraction. But all subtraction is, is adding by the “additive inverse” of what ever number you are subtracting. But that doesn’t tell us anything about two negatives multiplying, consider the following:

1 + (-1) = 0 additive inverse
-1(1 + (-1)) = -1*0 multiplying both sides by 0
-1(1 + (-1)) = 0 anything times 0, is 0
(-1)1 + (-1)(-1) = 0 distribution
(-1)(-1) = 1 uniqueness of additive inverse

now consider

-a*-b = (-1)a*(-1)b = (-1)(-1)a*b=(1)a*b=a*b