Ask A Scientist Mathematics Archive

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.
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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
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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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