Expert: Jack Cheng - 12/5/2006

Question
I am having some problem with a couple of math proofs, and I hope you can help.  On each proof, I need a REASON for each step.  Thanks

1.  If (a) not equal to 0<-(zero) , then (1/a) not equal to 0 <-(zero)

2.  If (a) not equal to 0 <-(zero), then (-1/a = 1/-a)

3.  If a>0, then (1/a)> 0

4.  a-b<0, <-> a<b

5.   c<0 and a,b, then bc<ac

Answer
Hi Nathan,

1) The only way a division can be zero is if the numerator is zero, but it's one. So the division is always non-zero.

2) -1/a = (-1/-1) * (-1/a) = 1/-a

3) a>0 -> a is positive        definition of positive
  1>0 -> 1 is positive        definition of positive
  1/a is positive -> (1/a)>0  quotient of two positive number is positive; definition of positive

4) a-b < 0        
  a-b+b < 0+b    adding b to both sides
  a < b          

  a < b
  a-b < b-b      subtracting b from both sides
  a-b < 0

5) I assume you mean c<0 and a<b, then bc<ac.

c<0 -> c=-k for a positive number k
a < b
ac = a(-k)      substitution
bc = b(-k)      substitution
-ak > -bk       algebra rule that mean multiple with negative number, inequality symbol changes direction
ac > bc

I am not very good with proves, but I hope I helped.

~ Jack