

Negative multiplied by negative = positive; Why?
Name: Name
Status: N/A
Age: N/A
Location: N/A
Country: N/A
Date: Around 1993
Question:
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.
Replies:
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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Update: June 2012

