Expert: Jack Cheng - 12/3/2006

Question
I am having problems with a couple of math proofs, and I was hoping you could help?  Proofs are not my favorite.

1.  (-a) * b = -(a*b)

2.  (-1) * a = -a


Answer
Hi Nathan,

Proofs aren't my favorites either (except mathematical induction), but we go:

1) (-a) * b = -(a*b)
  (-1) * a * b = -(a*b)        
  (-1)(a*b) = -(a*b)          by Associative Property
  -(a*b) = -(a*b)             see 2) below

2) (-1) * a = -a

Let's do a proof by case. There are three special cases for a: when a is negative, when a is positive, and when a=0.

a is negative:
-1 * -n = n = -(-n)

a is positive:
-1 * n = -n = -(n)

a is 0:
-1 * 0 = 0 = -(0)

Because all the cases are valid, 2) is proved.

I hope this helps,
Jack