[HN Gopher] Product of Additive Inverses
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       Product of Additive Inverses
        
       Author : blenderob
       Score  : 6 points
       Date   : 2025-07-05 08:50 UTC (3 days ago)
        
 (HTM) web link (susam.net)
 (TXT) w3m dump (susam.net)
        
       | JadeNB wrote:
       | This is a formal justification, from the ring axioms, of the
       | formula (-a)(-b) = ab. As the article mentions, this is often
       | phrased as "the product of two negatives is positive," but, of
       | course, the presence of a minus sign in front of a _variable_
       | does not indicate a negative number (for example, if a = -3, then
       | -a is positive); and the formula makes sense even in a ring with
       | no notion of positive and negative numbers.
        
       | empath75 wrote:
       | A simple example of how this is true _even if you don't have
       | negative numbers_:
       | 
       | Let's use mod 5 arithmetic. You have 5 elements in the ring --
       | 0,1,2,3,4
       | 
       | The additive inverses are as follows:                 1 + 4 = 0
       | 2 + 3 = 0
       | 
       | Which is to say that 1 is the additive inverse of 4 and 2 is the
       | additive inverse of 3, and vice versa. 0 is the identity, of
       | course.
       | 
       | So what happens if you multiply 2 * -3 (2 times the additive
       | inverse of 3).
       | 
       | The additive inverse of 3 is just 2, so the answer is 2 * -3 = 2
       | * 2 = 4.
       | 
       | The other way to calculate it is to find the additive inverse of
       | the product:
       | 
       | 2 * -3 = -(2 * 3) = -(1) which is the additive inverse of 1: 4
       | again.
        
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       (page generated 2025-07-08 23:01 UTC)