Connaître vs Savoir

I am lucky in that one of the languages I learnt as a child has the exact equivalents of connaître and savoir. The easiest way I can think of explaining this is to say that "savoir" means to know a fact or how to do something. If you mean to say "know" in any other sense, the correct word is  "connaître". To clarify what these other meanings of "know" might be, "connaître" refers to knowing a person, knowing a person's character or being familiar with something.

Below are a few examples:

1. I know he has left the company. -  I know a fact - savoir
2. I know how to cook an omelette. - I know how to do something - savoir
3. I know Mathematics. - Mathematics is not a fact; it is a discipline. - connaître
4. He is good at Accounting. - If you paraphrase this as "He (really) knows Accounting." - connaître
5. I know Paris./I am familiar or acquainted with Paris. - connaître