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Date: Around 1993

Question: I was wondering if someone out there might be able to explain why the result of a negative number multiplied by a negative number is positive. Also, in this explanation, please feel free to get as technical as necessary. Thank you for your time.

It is kind of like two wrongs make a right. Let us say that x and y are both positive. Then the meaning of -(x) + (y) is just y - x and (y) - - (x) = y + x, subtracting a negative number is the same as adding the positive number. Now, let us say n is a positive integer. Then n * x can be thought of as adding x to itself n times which explains why a positive number times a negative number is always negative. So, what does -(n) * x? Well, multiplication is commutative, i.e., x * y = y * x. So -(n) * x = x * -(n) or -(n) * x = (-1 * n ) * x = ( n * -1 ) * x = n * ( -1 * x ) because multiplication is also associative = n * -(x). All of which is to say that -n * x is the same as adding -x to itself n times. Therefore, -n * -x is the same as adding -(-x) to itself n times. And, I think we all agree that -(-x) better be +x. Hope this is not too confusing!

John Hawley

Negative numbers are additive inverses of positive numbers. Symbolically, x + (-x)=(-x)+x=0. Note that this also says that positive numbers are also additive inverses of corresponding negative numbers since the equations display symmetry (commutativity of addition). As a consequence, one can identify a number as negative if one knows that its additive inverse is positive, and one can identify a number as positive if one knows that its additive inverse is negative. Now, suppose that x represents a positive number and -y a negative number. We can then see that (-y)x is negative by adding it to the positive number yx as follows: (-y)x+yx=(-y+y)x=0x=0. Note that this uses both the distributive law and the fact that -y and y are additive inverses. It follows that (-y)x is negative since it is the additive inverse of the positive number xy. It is equally easy to show that x(-y) is negative. Now we get to your question. Let -x and -y denote two negative numbers. Since (-y)x is negative, we can show that (-x)(-y) must be positive by showing that it is the additive inverse of (-y)x as follows: (-y)(-x)+(- y)x=(-y)(-x+x)=(-y)0=0.

Robert Allan Chaffer

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