Why Does a Negative Times a Negative Equal a Positive?

Date: 03/01/98 at 21:12:30
From: Jessica
Subject: Multiplying Negative numbers

How does a negative number times another negative number equal 
a positive number?

Date: 03/02/98 at 09:15:10
From: Doctor Bruce
Subject: Re: Multiplying Negative numbers

Hello Jessica,

I detect in your question a measure of annoyance at having to learn 
the rule for multiplying negative numbers. I'll bet it seems like 
someone just made the rule up out of thin air, with no particular 
reason why the answer should be positive. I want to reassure you that 
this rule is not just "made up." There is a chain of reasoning -- a 
mathematical "argument" -- that shows why the rule *has* to be that 
negative times negative equals positive.

Mathematical argument takes a little getting used to. This might look 
rather strange at first. Here's how the reasoning goes:

  (1) Zero times anything equals zero.

  (2) Every number has exactly one additive inverse. This means if N
      is a positive number, then -N is its additive inverse, so that
      N + (-N) = 0. Likewise, the additive inverse of -N is N.

  (3) We want negative numbers to obey the distributive law.  This  
      says that 

           a*(b+c) = a*b + a*c.

  (4) Now, we are forced to accept a new law, that negative times  
      positive equals negative. This is because we can use the 
      distributive law on an expression like

           2*(3 + (-3)).

      This equals 2*(0), which is zero. But by the distributive law, 
      it also equals

           2*3 + 2*(-3).
      
      So 2*(-3) does the job of the additive inverse of 2*3, and 
      therefore 2*(-3) is the additive inverse of 2*3. But the 
      additive inverse of 6 is just -6. So 2 times -3 equals -6.

  (5) Next, we are forced to accept another new law, that negative 
      times negative equals positive. It's a lot like the example 
      in (4). We use the distributive law on, say,

           -3*(5 + (-5)).

      This is again equal to zero. But by the distributive law, it
      also equals 

           -3*5 + (-3)*(-5).  

      We know the first thing, (-3*5) equals -15 because of the law 
      in (4). So (-3)*(-5) is doing the job of the additive inverse  
      of -15. We know -15 has exactly one additive inverse, namely 15.
      Therefore,
  
           (-3)*(-5) = 15.

I hope this doesn't frighten you! The main thing is, keep right on 
questioning the things that don't make sense. In mathematics, you are 
always entitled to an explanation of WHY things are the way your 
teacher (or I) say they are.

Good luck,

-Doctor Bruce, The Math Forum
Check out our web site http://mathforum.org/dr.math/