Algebraic Proof That Positive Times Negative Equals Negative

Date: 03/28/2004 at 15:30:58
From: Jamie
Subject: Proof?

Is there an algebraic proof that a positive times a negative equals a
negative?

Date: 03/29/2004 at 22:37:26
From: Doctor Justin
Subject: Re: Proof?

Hi Jamie,

Thanks for writing to Dr. Math.  Here's an algebraic proof that should
convince you that a positive times a negative equals a negative:

For a, b > 0:

Consider:

               0*b = 0

Because [a + (-a)] = 0, use the Transitive Property:

      [a + (-a)]*b = 0*b

Distributive Property:

      ab + (-a)(b) = 0*b

Again, use the Transitive Property:

      ab + (-a)(b) = 0

It is assumed that a positive multiplied by a positive equals a 
positive, so the quantity ab must be positive.  If ab is positive and 
(-a)(b) is added to it to equal zero, then (-a)(b) must be the 
additive inverse of ab.  In other words, (-a)(b) must be less than 
zero, or negative.

If you have any more questions, please write back.

- Doctor Justin, The Math Forum
  http://mathforum.org/dr.math/