Logical Fallacies

So, I'm a computer engineer and therefore logic is pretty much the backbone of my career path. I'm fascinated by all sorts of logic puzzles, and the nature of logic in general. Logical Fallacies are also of interest. Wikipedia defines Fallacies as: "A fallacy is a component of an argument which, being demonstrably flawed in its logic or form, renders the argument invalid in whole. " There are many kinds of logical fallacies. For instance because someone is true for one element in a set, does not imply it is true for everything in a set. There are many such fallacies, many of which appear in arguments so subtly we never see them. Below is one of many math proofs which proves 2 = 1. Obviously this is not true, so where does the proof go wrong?

Assume a = b and neither a nor b are zero.

1. ab = ab Therefore
2. a^2 = ab Subtract b^2 from both side
3. a^2 - b^2 = ab - b^2 Factor out both sides
4. (a-b)(a+b)=b(a-b) Now divide (a-b) out of both sides
   
5. a+b = b Since a = b
   
6. b+b = b
   
7. 2b = b
8. 2 = 1

Can see you where the proof went wrong?

Also, I'm still looking for a job! o.O